%------------------------------------------------------------------------------
% File     : ITP291_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Intf_Imperative 00092_002426
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0096_VEBT_Intf_Imperative_00092_002426 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11958 (4335 unt;1541 typ;   0 def)
%            Number of atoms       : 18988 (8013 equ)
%            Maximal formula atoms :   47 (   1 avg)
%            Number of connectives : 19098 (1820   ~; 303   |;1950   &)
%                                         (2017 <=>;13008  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   36 (   2 avg)
%            Number of FOOLs       : 1067 ( 578 fml; 489 var)
%            Number of X terms     : 1001 (   0  []; 740 ite; 261 let)
%            Number of types       :   18 (  17 usr)
%            Number of type conns  : 1263 (1096   >; 167   *;   0   +;   0  <<)
%            Number of predicates  :  217 ( 214 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1332 (1332 usr;  83 con; 0-6 aty)
%            Number of variables   : 32115 (29084   !; 682   ?;32115   :)
%                                         (2349  !>;   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 23:24:47.814
%------------------------------------------------------------------------------
% Could-be-implicit typings (31)
tff(ty_t_VEBT__BuildupMemImp_OVEBTi,type,
    vEBT_VEBTi: $tType ).

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

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

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

tff(ty_t_Heap_Oheap_Oheap__ext,type,
    heap_ext: $tType > $tType ).

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

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

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

tff(ty_t_Numeral__Type_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_Uint32_Ouint32,type,
    uint32: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

% Explicit typings (1510)
tff(sy_cl_Finite__Set_Ofinite,type,
    finite_finite: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_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_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_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_Fields_Ofield__abs__sgn,type,
    field_abs_sgn: 
      !>[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_Fields_Olinordered__field,type,
    linordered_field: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Least__significant__bit_Olsb,type,
    least_6119777620449941438nt_lsb: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

tff(sy_cl_Topological__Spaces_Ot1__space,type,
    topological_t1_space: 
      !>[A: $tType] : $o ).

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

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

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

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

tff(sy_cl_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_Euclidean__Division_Oeuclidean__ring,type,
    euclid5891614535332579305n_ring: 
      !>[A: $tType] : $o ).

tff(sy_cl_Limits_Otopological__semigroup__mult,type,
    topolo4211221413907600880p_mult: 
      !>[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_Groups_Olinordered__ab__semigroup__add,type,
    linord4140545234300271783up_add: 
      !>[A: $tType] : $o ).

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

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

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

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

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

tff(sy_cl_Limits_Otopological__comm__monoid__mult,type,
    topolo4987421752381908075d_mult: 
      !>[A: $tType] : $o ).

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

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

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

tff(sy_cl_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_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] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__um____,type,
    aTP_Lamp_um: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__un____,type,
    aTP_Lamp_un: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__yx____,type,
    aTP_Lamp_yx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Array__Time_Ofreeze,type,
    array_freeze: 
      !>[A: $tType] : ( array(A) > heap_Time_Heap(list(A)) ) ).

tff(sy_c_Array__Time_Olen,type,
    array_len: 
      !>[A: $tType] : ( array(A) > heap_Time_Heap(nat) ) ).

tff(sy_c_Array__Time_Omake,type,
    array_make: 
      !>[A: $tType] : ( ( nat * fun(nat,A) ) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Array__Time_Onew,type,
    array_new: 
      !>[A: $tType] : ( ( nat * A ) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Array__Time_Onth,type,
    array_nth: 
      !>[A: $tType] : ( ( array(A) * nat ) > heap_Time_Heap(A) ) ).

tff(sy_c_Array__Time_Oof__list,type,
    array_of_list: 
      !>[A: $tType] : ( list(A) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Array__Time_Oupd,type,
    array_upd: 
      !>[A: $tType] : ( ( nat * A * array(A) ) > heap_Time_Heap(array(A)) ) ).

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

tff(sy_c_Assertions_Oex__assn,type,
    ex_assn: 
      !>[A: $tType] : ( fun(A,assn) > assn ) ).

tff(sy_c_Assertions_Opure__assn,type,
    pure_assn: $o > assn ).

tff(sy_c_Assertions_Osnga__assn,type,
    snga_assn: 
      !>[A: $tType] : ( array(A) > fun(list(A),assn) ) ).

tff(sy_c_BNF__Def_Orel__fun,type,
    bNF_rel_fun: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,$o)) * fun(B,fun(D,$o)) ) > fun(fun(A,B),fun(fun(C,D),$o)) ) ).

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

tff(sy_c_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_Oand__not__num,type,
    bit_and_not_num: ( num * num ) > option(num) ).

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] : fun(nat,fun(A,A)) ).

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: ( nat * num ) > option(num) ).

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

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

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

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

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

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

tff(sy_c_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_Odup,type,
    code_dup: code_integer > code_integer ).

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

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

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

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

tff(sy_c_Complex_OArg,type,
    arg: complex > real ).

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

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

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

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

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

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

tff(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod(int,int) > int ).

tff(sy_c_Divides_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_Finite__Set_Ofinite,type,
    finite_finite2: 
      !>[A: $tType] : ( 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_Othe__inv__into,type,
    the_inv_into: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * B ) > A ) ).

tff(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
    semiri4206861660011772517g_char: 
      !>[A: $tType] : ( itself(A) > nat ) ).

tff(sy_c_Groups_Oabs__class_Oabs,type,
    abs_abs: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Ominus__class_Ominus,type,
    minus_minus: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Oone__class_Oone,type,
    one_one: 
      !>[A: $tType] : A ).

tff(sy_c_Groups_Oplus__class_Oplus,type,
    plus_plus: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Lattices_Oinf__class_Oinf,type,
    inf_inf: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

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

tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
    lattic643756798349783984er_Max: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
    lattic643756798350308766er_Min: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
    lattic7623131987881927897min_on: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).

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

tff(sy_c_List_Oconcat,type,
    concat: 
      !>[A: $tType] : ( list(list(A)) > list(A) ) ).

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

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

tff(sy_c_List_Olist_Ohd,type,
    hd: 
      !>[A: $tType] : ( list(A) > A ) ).

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

tff(sy_c_List_Olist_Oset,type,
    set2: 
      !>[A: $tType] : ( 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_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_Oreplicate,type,
    replicate: 
      !>[A: $tType] : ( ( nat * A ) > list(A) ) ).

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

tff(sy_c_List_Osubseqs,type,
    subseqs: 
      !>[A: $tType] : ( 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__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_Map_Oran,type,
    ran: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).

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_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_Nat_OSuc,type,
    suc: fun(nat,nat) ).

tff(sy_c_Nat_Onat_Ocase__nat,type,
    case_nat: 
      !>[A: $tType] : ( ( A * fun(nat,A) * nat ) > A ) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
    semiring_1_of_nat: 
      !>[A: $tType] : fun(nat,A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
    semiri8178284476397505188at_aux: 
      !>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).

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

tff(sy_c_Nat__Bijection_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: fun(num,num) ).

tff(sy_c_Num_Onum_OOne,type,
    one2: num ).

tff(sy_c_Num_Onum_Ocase__num,type,
    case_num: 
      !>[A: $tType] : fun(A,fun(fun(num,A),fun(fun(num,A),fun(num,A)))) ).

tff(sy_c_Num_Onum_Orec__num,type,
    rec_num: 
      !>[A: $tType] : fun(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)))) ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: fun(nat,num) ).

tff(sy_c_Num_Onumeral__class_Onumeral,type,
    numeral_numeral: 
      !>[A: $tType] : fun(num,A) ).

tff(sy_c_Num_Opow,type,
    pow: ( num * num ) > num ).

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

tff(sy_c_Option_Ooption_ONone,type,
    none: 
      !>[A: $tType] : option(A) ).

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

tff(sy_c_Option_Ooption_Ocase__option,type,
    case_option: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,B) * option(A) ) > B ) ).

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

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

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

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

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

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

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

tff(sy_c_Orderings_Otop__class_Otop,type,
    top_top: 
      !>[A: $tType] : A ).

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

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

tff(sy_c_Product__Type_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_Oproduct,type,
    product_product: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Pure_Otype,type,
    type2: 
      !>[A: $tType] : itself(A) ).

tff(sy_c_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_Oquotient__of,type,
    quotient_of: rat > product_prod(int,int) ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear,type,
    real_V3181309239436604168linear: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_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_Refine__Imp__Hol_Orefines,type,
    refine_Imp_refines: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_Time_Heap(A) ) > $o ) ).

tff(sy_c_Rings_Odivide__class_Odivide,type,
    divide_divide: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Rings_Odvd__class_Odvd,type,
    dvd_dvd: 
      !>[A: $tType] : ( A > fun(A,$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_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( ( fun(A,$o) * set(A) ) > set(A) ) ).

tff(sy_c_Set_Oimage,type,
    image: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(set(A),set(B)) ) ).

tff(sy_c_Set_Oinsert,type,
    insert: 
      !>[A: $tType] : fun(A,fun(set(A),set(A))) ).

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

tff(sy_c_Set_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_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_OgreaterThanLessThan,type,
    set_or5935395276787703475ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
    set_ord_lessThan: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Signed__Division_Osigned__division__class_Osigned__divide,type,
    signed7115095781618012415divide: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo,type,
    signed6721504322012087516modulo: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_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_Otime,type,
    time_time: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > nat ) ).

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] : fun(A,A) ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun(nat,real) ).

tff(sy_c_Transcendental_Ocosh,type,
    cosh: 
      !>[A: $tType] : fun(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] : fun(A,A) ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun(nat,real) ).

tff(sy_c_Transcendental_Osinh,type,
    sinh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Otan,type,
    tan: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Otanh,type,
    tanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Type__Length_Olen0__class_Olen__of,type,
    type_len0_len_of: 
      !>[A: $tType] : ( itself(A) > nat ) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
    vEBT_T_i_n_s_e_r_t: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
    vEBT_T_i_n_s_e_r_t2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
    vEBT_T5076183648494686801_t_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
    vEBT_T9217963907923527482_t_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
    vEBT_T_m_a_x_t: vEBT_VEBT > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
    vEBT_T_m_a_x_t_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
    vEBT_T_m_e_m_b_e_r: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
    vEBT_T_m_e_m_b_e_r2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
    vEBT_T8099345112685741742_r_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
    vEBT_T5837161174952499735_r_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
    vEBT_T_m_i_n_N_u_l_l: vEBT_VEBT > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
    vEBT_T_m_i_n_t: vEBT_VEBT > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
    vEBT_T_m_i_n_t_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
    vEBT_T_p_r_e_d: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
    vEBT_T_p_r_e_d2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
    vEBT_T_p_r_e_d_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
    vEBT_T_p_r_e_d_rel2: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
    vEBT_T_s_u_c_c: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
    vEBT_T_s_u_c_c2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
    vEBT_T_s_u_c_c_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
    vEBT_T_s_u_c_c_rel2: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: nat > nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
    vEBT_V9176841429113362141ildupi: nat > int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
    vEBT_VEBT_Tb: nat > int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
    vEBT_VEBT_Tb2: nat > nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
    vEBT_VEBT_highi: ( nat * nat ) > heap_Time_Heap(nat) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
    vEBT_VEBT_lowi: ( nat * nat ) > heap_Time_Heap(nat) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli,type,
    vEBT_VEBT_minNulli: vEBT_VEBTi > heap_Time_Heap($o) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei,type,
    vEBT_VEBT_replicatei: 
      !>[A: $tType] : ( ( nat * heap_Time_Heap(A) ) > heap_Time_Heap(list(A)) ) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H,type,
    vEBT_V739175172307565963ildupi: nat > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__inserti_H,type,
    vEBT_V3964819847710782039nserti: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__memberi_H,type,
    vEBT_V854960066525838166emberi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap($o) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi,type,
    vEBT_case_VEBTi: 
      !>[A: $tType] : ( ( fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))) * fun($o,fun($o,A)) * vEBT_VEBTi ) > A ) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: fun(vEBT_VEBT,fun(vEBT_VEBTi,assn)) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
    vEBT_vebt_buildupi: nat > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__inserti,type,
    vEBT_vebt_inserti: ( vEBT_VEBTi * nat ) > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti,type,
    vEBT_vebt_maxti: vEBT_VEBTi > heap_Time_Heap(option(nat)) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__memberi,type,
    vEBT_vebt_memberi: ( vEBT_VEBTi * nat ) > heap_Time_Heap($o) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
    vEBT_vebt_minti: vEBT_VEBTi > heap_Time_Heap(option(nat)) ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( $o * $o ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option(product_prod(nat,nat)) * nat * list(vEBT_VEBT) * vEBT_VEBT ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > fun(nat,$o) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list(vEBT_VEBT) * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__DelImperative_OVEBT__internal_Ovebt__deletei_H,type,
    vEBT_V1365221501068881998eletei: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__DelImperative_Ovebt__deletei,type,
    vEBT_vebt_deletei: ( vEBT_VEBTi * nat ) > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
    vEBT_T_d_e_l_e_t_e: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
    vEBT_T8441311223069195367_e_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
    vEBT_V1232361888498592333_e_t_e: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H,type,
    vEBT_VEBT_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H__rel,type,
    vEBT_VEBT_insert_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Intf__Imperative_Ovebt__assn,type,
    vEBT_Intf_vebt_assn: ( nat * set(nat) ) > fun(vEBT_VEBTi,assn) ).

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

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

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

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: vEBT_VEBT > $o ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > fun(nat,$o) ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift,type,
    vEBT_V2048590022279873568_shift: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > fun(option(A),fun(option(A),option(A))) ) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > option(nat) ).

tff(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option(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__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

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

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

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

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

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: vEBT_VEBT > real ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__predi_H,type,
    vEBT_VEBT_vebt_predi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap(option(nat)) ).

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__succi_H,type,
    vEBT_VEBT_vebt_succi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap(option(nat)) ).

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

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

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

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Uniqueness_OVEBT__internal_OperInsTrans,type,
    vEBT_V6289311342943941716sTrans: ( vEBT_VEBT * vEBT_VEBT ) > $o ).

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

tff(sy_c_Word_OWord,type,
    word2: 
      !>[A: $tType] : fun(int,word(A)) ).

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

tff(sy_c_Word_Oof__nat,type,
    of_nat: 
      !>[A: $tType] : fun(nat,word(A)) ).

tff(sy_c_Word_Opcr__word,type,
    pcr_word: 
      !>[B: $tType] : fun(int,fun(word(B),$o)) ).

tff(sy_c_Word_Orevcast,type,
    revcast: 
      !>[A: $tType,B: $tType] : fun(word(A),word(B)) ).

tff(sy_c_Word_Oring__1__class_Osigned,type,
    ring_1_signed: 
      !>[B: $tType,A: $tType] : ( 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__drop__bit,type,
    signed_drop_bit: 
      !>[A: $tType] : fun(nat,fun(word(A),word(A))) ).

tff(sy_c_Word_Oslice,type,
    slice2: 
      !>[A: $tType,B: $tType] : ( ( nat * word(A) ) > word(B) ) ).

tff(sy_c_Word_Oslice1,type,
    slice1: 
      !>[A: $tType,B: $tType] : ( ( nat * word(A) ) > word(B) ) ).

tff(sy_c_Word_Osource__size,type,
    source_size: 
      !>[A: $tType,B: $tType] : fun(fun(word(A),B),nat) ).

tff(sy_c_Word_Otarget__size,type,
    target_size: 
      !>[A: $tType,B: $tType] : fun(fun(A,word(B)),nat) ).

tff(sy_c_Word_Othe__nat,type,
    the_nat: 
      !>[A: $tType] : fun(word(A),nat) ).

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

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

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

tff(sy_c_Word_Oword__pred,type,
    word_pred: 
      !>[A: $tType] : fun(word(A),word(A)) ).

tff(sy_c_Word_Oword__rcat,type,
    word_rcat: 
      !>[A: $tType,B: $tType] : fun(list(word(A)),word(B)) ).

tff(sy_c_Word_Oword__reverse,type,
    word_reverse: 
      !>[A: $tType] : ( word(A) > word(A) ) ).

tff(sy_c_Word_Oword__roti,type,
    word_roti: 
      !>[A: $tType] : fun(int,fun(word(A),word(A))) ).

tff(sy_c_Word_Oword__rotl,type,
    word_rotl: 
      !>[A: $tType] : fun(nat,fun(word(A),word(A))) ).

tff(sy_c_Word_Oword__rotr,type,
    word_rotr: 
      !>[A: $tType] : fun(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] : fun(word(A),word(A)) ).

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

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

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

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

tff(sy_v_n,type,
    n: nat ).

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

tff(sy_v_ti,type,
    ti: vEBT_VEBTi ).

tff(sy_v_x,type,
    x: nat ).

% Relevant facts (9133)
tff(fact_0_bit__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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),D2))),L) ).

% bit_concat_def
tff(fact_1_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Na: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Na)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),Na)),Xa) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_2_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,I: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),Na)) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3_perIT__concat,axiom,
    ! [Sa: vEBT_VEBT,T: vEBT_VEBT,U: vEBT_VEBT] :
      ( vEBT_V6289311342943941716sTrans(Sa,T)
     => ( vEBT_V6289311342943941716sTrans(T,U)
       => vEBT_V6289311342943941716sTrans(Sa,U) ) ) ).

% perIT_concat
tff(fact_4_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),Xa) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_5_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,B2: nat,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Xa: num,Na: nat,Y: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Na) = aa(nat,A,semiring_1_of_nat(A),Y) )
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xa)),Na) = Y ) ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Y: nat,Xa: num,Na: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Y) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Na) )
        <=> ( Y = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xa)),Na) ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_8_ceiling__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: num,Na: nat] : archimedean_ceiling(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Na)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na) ) ).

% ceiling_numeral_power
tff(fact_9_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xa)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_add_numeral
tff(fact_10_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),V)),Xa) ) ) ).

% numeral_less_ceiling
tff(fact_11_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,aa(nat,fun(nat,nat),power_power(nat),M),Na)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),M)),Na) ) ).

% semiring_1_class.of_nat_power
tff(fact_12_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [B2: nat,W: nat,Xa: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W) = aa(nat,A,semiring_1_of_nat(A),Xa) )
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W) = Xa ) ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_13_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_14_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,Na: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),M))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),Na))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)))),B2) ) ).

% power_add_numeral2
tff(fact_15_power__add__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,Na: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),Na))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na))) ) ).

% power_add_numeral
tff(fact_16_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Xa: nat,B2: nat,W: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Xa) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W) )
        <=> ( Xa = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W) ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_17_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_18_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_19_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_20_power__commuting__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xa: A,Y: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)) ) ) ) ).

% power_commuting_commutes
tff(fact_21_power__mult__distrib,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na)) ) ).

% power_mult_distrib
tff(fact_22_power__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ).

% power_commutes
tff(fact_23_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% reals_Archimedean2
tff(fact_24_power__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: nat,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),Na) ) ).

% power_mult
tff(fact_25_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),archimedean_ceiling(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% ceiling_less_cancel
tff(fact_26_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: nat,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),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,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ).

% power_add
tff(fact_27_power__numeral__even,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z: A,W: num] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),bit0(W))) = $let(
            w: A,
            w:= aa(nat,A,aa(A,fun(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_28_power__numeral__odd,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z: A,W: num] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,W))) = $let(
            w: A,
            w:= aa(nat,A,aa(A,fun(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_29_power2__eq__square,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) ) ).

% power2_eq_square
tff(fact_30_power4__eq__xxxx,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xa: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),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),Xa),Xa)),Xa)),Xa) ) ).

% power4_eq_xxxx
tff(fact_31_power3__eq__cube,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2)),A2) ) ).

% power3_eq_cube
tff(fact_32_power__even__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power_even_eq
tff(fact_33_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% of_nat_less_two_power
tff(fact_34_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xa: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),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,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)),Y)) ) ).

% power2_sum
tff(fact_35_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xa)),Na)),aa(int,nat,nat2,A2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na)),A2) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_36_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xa: num,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xa)),Na))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_37_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,Y: int] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xa)),Na) = aa(int,nat,nat2,Y) )
    <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_38_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xa: num,Na: nat] :
      ( ( aa(int,nat,nat2,Y) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xa)),Na) )
    <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_39_less__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% less_log2_of_power
tff(fact_40_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_41_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_42_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_43_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_44_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_45_mem__Collect__eq,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      ( member(A,A2,collect(A,P))
    <=> aa(A,$o,P,A2) ) ).

% mem_Collect_eq
tff(fact_46_Collect__mem__eq,axiom,
    ! [A: $tType,A3: set(A)] : collect(A,aTP_Lamp_a(set(A),fun(A,$o),A3)) = A3 ).

% Collect_mem_eq
tff(fact_47_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(A,$o,P,X)
        <=> aa(A,$o,Q,X) )
     => ( collect(A,P) = collect(A,Q) ) ) ).

% Collect_cong
tff(fact_48_HOL_Oext,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] :
      ( ! [X: A] : aa(A,B,F2,X) = aa(A,B,G,X)
     => ( F2 = G ) ) ).

% HOL.ext
tff(fact_49_semiring__norm_I3_J,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bit0(Na)) = aa(num,num,bit1,Na) ).

% semiring_norm(3)
tff(fact_50_semiring__norm_I4_J,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),aa(num,num,bit1,Na)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),Na),one2)) ).

% semiring_norm(4)
tff(fact_51_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_52_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_53_semiring__norm_I87_J,axiom,
    ! [M: num,Na: num] :
      ( ( bit0(M) = bit0(Na) )
    <=> ( M = Na ) ) ).

% semiring_norm(87)
tff(fact_54_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_55_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_56_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_57_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_58_semiring__norm_I90_J,axiom,
    ! [M: num,Na: num] :
      ( ( aa(num,num,bit1,M) = aa(num,num,bit1,Na) )
    <=> ( M = Na ) ) ).

% semiring_norm(90)
tff(fact_59_semiring__norm_I80_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),aa(num,num,bit1,M)),aa(num,num,bit1,Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(80)
tff(fact_60_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_61_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_62_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_63_semiring__norm_I85_J,axiom,
    ! [M: num] : bit0(M) != one2 ).

% semiring_norm(85)
tff(fact_64_semiring__norm_I83_J,axiom,
    ! [Na: num] : one2 != bit0(Na) ).

% semiring_norm(83)
tff(fact_65_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_66_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_67_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_68_semiring__norm_I15_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),bit0(Na)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),Na)) ).

% semiring_norm(15)
tff(fact_69_semiring__norm_I14_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M)),aa(num,num,bit1,Na)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),aa(num,num,bit1,Na))) ).

% semiring_norm(14)
tff(fact_70_semiring__norm_I89_J,axiom,
    ! [M: num,Na: num] : aa(num,num,bit1,M) != bit0(Na) ).

% semiring_norm(89)
tff(fact_71_semiring__norm_I88_J,axiom,
    ! [M: num,Na: num] : bit0(M) != aa(num,num,bit1,Na) ).

% semiring_norm(88)
tff(fact_72_semiring__norm_I81_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),aa(num,num,bit1,M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(81)
tff(fact_73_semiring__norm_I86_J,axiom,
    ! [M: num] : aa(num,num,bit1,M) != one2 ).

% semiring_norm(86)
tff(fact_74_semiring__norm_I84_J,axiom,
    ! [Na: num] : one2 != aa(num,num,bit1,Na) ).

% semiring_norm(84)
tff(fact_75_semiring__norm_I77_J,axiom,
    ! [Na: num] : aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),aa(num,num,bit1,Na)) ).

% semiring_norm(77)
tff(fact_76_power__mult__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,Na: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),M))),aa(num,nat,numeral_numeral(nat),Na)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),times_times(num),M),Na))) ) ).

% power_mult_numeral
tff(fact_77_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_78_nat__int,axiom,
    ! [Na: nat] : aa(int,nat,nat2,aa(nat,int,semiring_1_of_nat(int),Na)) = Na ).

% nat_int
tff(fact_79_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_80_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% distrib_left_numeral
tff(fact_81_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A2: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).

% distrib_right_numeral
tff(fact_82_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_83_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_84_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_85_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_86_semiring__norm_I16_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),aa(num,num,bit1,Na)) = aa(num,num,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_87_semiring__norm_I9_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),bit0(Na)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)) ).

% semiring_norm(9)
tff(fact_88_semiring__norm_I7_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),aa(num,num,bit1,Na)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)) ).

% semiring_norm(7)
tff(fact_89_semiring__norm_I10_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),aa(num,num,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_90_semiring__norm_I8_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),one2) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2)) ).

% semiring_norm(8)
tff(fact_91_semiring__norm_I5_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),one2) = aa(num,num,bit1,M) ).

% semiring_norm(5)
tff(fact_92_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_93_int__distrib_I1_J,axiom,
    ! [Z1: int,Z2: 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),Z2)),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),Z2),W)) ).

% int_distrib(1)
tff(fact_94_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z2: 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),Z2)) = 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),Z2)) ).

% int_distrib(2)
tff(fact_95_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X: A] :
              ( ! [Y2: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y2)),aa(A,B,F2,X))
                 => aa(A,$o,P,Y2) )
             => aa(A,$o,P,X) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct
tff(fact_96_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X: A] :
              ( ! [Y2: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y2)),aa(A,B,F2,X))
                 => aa(A,$o,P,Y2) )
             => aa(A,$o,P,X) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct_rule
tff(fact_97_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_98_is__num__normalize_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% is_num_normalize(1)
tff(fact_99_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_100_less__not__refl,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),Na) ).

% less_not_refl
tff(fact_101_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_102_less__not__refl3,axiom,
    ! [Sa: nat,T: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Sa),T)
     => ( Sa != T ) ) ).

% less_not_refl3
tff(fact_103_less__irrefl__nat,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),Na) ).

% less_irrefl_nat
tff(fact_104_nat__less__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
             => aa(nat,$o,P,M2) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Na) ) ).

% nat_less_induct
tff(fact_105_infinite__descent,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( ! [N: nat] :
          ( ~ aa(nat,$o,P,N)
         => ? [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
              & ~ aa(nat,$o,P,M2) ) )
     => aa(nat,$o,P,Na) ) ).

% infinite_descent
tff(fact_106_linorder__neqE__nat,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa != Y )
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa) ) ) ).

% linorder_neqE_nat
tff(fact_107_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,$o),V2: fun(A,nat),Xa: A] :
      ( ! [X: A] :
          ( ~ aa(A,$o,P,X)
         => ? [Y2: A] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V2,Y2)),aa(A,nat,V2,X))
              & ~ aa(A,$o,P,Y2) ) )
     => aa(A,$o,P,Xa) ) ).

% infinite_descent_measure
tff(fact_108_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xa: nat,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Xa)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),Xa)) ) ).

% mult_of_nat_commute
tff(fact_109_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_110_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_111_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_112_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_113_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_114_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_115_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_116_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_117_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_118_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_119_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_120_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_121_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_122_mult__numeral__1,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),one2)),A2) = A2 ) ).

% mult_numeral_1
tff(fact_123_mult__numeral__1__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),one2)) = A2 ) ).

% mult_numeral_1_right
tff(fact_124_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one2 )
     => ( ! [X2: num] : Y != bit0(X2)
       => ~ ! [X3: num] : Y != aa(num,num,bit1,X3) ) ) ).

% num.exhaust
tff(fact_125_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_126_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_127_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_128_nat__int__add,axiom,
    ! [A2: nat,B2: nat] : aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2) ).

% nat_int_add
tff(fact_129_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_130_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_131_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_132_left__add__twice,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),B2) ) ).

% left_add_twice
tff(fact_133_log2__of__power__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(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_134_nat__add__offset__less,axiom,
    ! [Y: nat,Na: nat,Xa: nat,M: nat,Sz: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
       => ( ( Sz = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),Y)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Sz)) ) ) ) ).

% nat_add_offset_less
tff(fact_135_singleton__conv,axiom,
    ! [A: $tType,A2: A] : collect(A,aTP_Lamp_aa(A,fun(A,$o),A2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ).

% singleton_conv
tff(fact_136_singleton__conv2,axiom,
    ! [A: $tType,A2: A] : collect(A,aa(A,fun(A,$o),fequal(A),A2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_137_Un__insert__left,axiom,
    ! [A: $tType,A2: A,B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)),C3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) ).

% Un_insert_left
tff(fact_138_Un__insert__right,axiom,
    ! [A: $tType,A3: set(A),A2: A,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ).

% Un_insert_right
tff(fact_139_Un__empty,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> ( ( A3 = bot_bot(set(A)) )
        & ( B3 = bot_bot(set(A)) ) ) ) ).

% Un_empty
tff(fact_140_sup__bot__left,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),Xa) = Xa ) ).

% sup_bot_left
tff(fact_141_sup__bot__right,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),bot_bot(A)) = Xa ) ).

% sup_bot_right
tff(fact_142_bot__eq__sup__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A,Y: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) )
        <=> ( ( Xa = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% bot_eq_sup_iff
tff(fact_143_sup__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = bot_bot(A) )
        <=> ( ( Xa = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% sup_eq_bot_iff
tff(fact_144_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = bot_bot(A) )
        <=> ( ( A2 = bot_bot(A) )
            & ( B2 = bot_bot(A) ) ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_145_sup__bot_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),A2) = A2 ) ).

% sup_bot.left_neutral
tff(fact_146_empty__Collect__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( bot_bot(set(A)) = collect(A,P) )
    <=> ! [X4: A] : ~ aa(A,$o,P,X4) ) ).

% empty_Collect_eq
tff(fact_147_Collect__empty__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( collect(A,P) = bot_bot(set(A)) )
    <=> ! [X4: A] : ~ aa(A,$o,P,X4) ) ).

% Collect_empty_eq
tff(fact_148_all__not__in__conv,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X4: A] : ~ member(A,X4,A3)
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% all_not_in_conv
tff(fact_149_empty__iff,axiom,
    ! [A: $tType,C2: A] : ~ member(A,C2,bot_bot(set(A))) ).

% empty_iff
tff(fact_150_insert__absorb2,axiom,
    ! [A: $tType,Xa: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3) ).

% insert_absorb2
tff(fact_151_insert__iff,axiom,
    ! [A: $tType,A2: A,B2: A,A3: set(A)] :
      ( member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),A3))
    <=> ( ( A2 = B2 )
        | member(A,A2,A3) ) ) ).

% insert_iff
tff(fact_152_insertCI,axiom,
    ! [A: $tType,A2: A,B3: set(A),B2: A] :
      ( ( ~ member(A,A2,B3)
       => ( A2 = B2 ) )
     => member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B3)) ) ).

% insertCI
tff(fact_153_sup__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(B,A),G: fun(B,A),Xa: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),sup_sup(fun(B,A)),F2),G),Xa) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F2,Xa)),aa(B,A,G,Xa)) ) ).

% sup_apply
tff(fact_154_sup_Oright__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ).

% sup.right_idem
tff(fact_155_sup__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) ) ).

% sup_left_idem
tff(fact_156_sup_Oleft__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ).

% sup.left_idem
tff(fact_157_sup__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Xa) = Xa ) ).

% sup_idem
tff(fact_158_sup_Oidem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),A2) = A2 ) ).

% sup.idem
tff(fact_159_Un__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
    <=> ( member(A,C2,A3)
        | member(A,C2,B3) ) ) ).

% Un_iff
tff(fact_160_UnCI,axiom,
    ! [A: $tType,C2: A,B3: set(A),A3: set(A)] :
      ( ( ~ member(A,C2,B3)
       => member(A,C2,A3) )
     => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ) ).

% UnCI
tff(fact_161_singletonI,axiom,
    ! [A: $tType,A2: A] : member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) ).

% singletonI
tff(fact_162_sup__bot_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),bot_bot(A)) = A2 ) ).

% sup_bot.right_neutral
tff(fact_163_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) )
        <=> ( ( A2 = bot_bot(A) )
            & ( B2 = bot_bot(A) ) ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_164_bot__set__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = collect(A,bot_bot(fun(A,$o))) ).

% bot_set_def
tff(fact_165_sup__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = collect(A,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A3)),aTP_Lamp_a(set(A),fun(A,$o),B3))) ).

% sup_set_def
tff(fact_166_not__psubset__empty,axiom,
    ! [A: $tType,A3: set(A)] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),bot_bot(set(A))) ).

% not_psubset_empty
tff(fact_167_ex__in__conv,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ? [X4: A] : member(A,X4,A3)
    <=> ( A3 != bot_bot(set(A)) ) ) ).

% ex_in_conv
tff(fact_168_equals0I,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [Y3: A] : ~ member(A,Y3,A3)
     => ( A3 = bot_bot(set(A)) ) ) ).

% equals0I
tff(fact_169_equals0D,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ( A3 = bot_bot(set(A)) )
     => ~ member(A,A2,A3) ) ).

% equals0D
tff(fact_170_emptyE,axiom,
    ! [A: $tType,A2: A] : ~ member(A,A2,bot_bot(set(A))) ).

% emptyE
tff(fact_171_mk__disjoint__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( member(A,A2,A3)
     => ? [B4: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B4) )
          & ~ member(A,A2,B4) ) ) ).

% mk_disjoint_insert
tff(fact_172_insert__commute,axiom,
    ! [A: $tType,Xa: A,Y: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) ).

% insert_commute
tff(fact_173_insert__eq__iff,axiom,
    ! [A: $tType,A2: A,A3: set(A),B2: A,B3: set(A)] :
      ( ~ member(A,A2,A3)
     => ( ~ member(A,B2,B3)
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B3) )
        <=> $ite(
              A2 = B2,
              A3 = B3,
              ? [C4: set(A)] :
                ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),C4) )
                & ~ member(A,B2,C4)
                & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),C4) )
                & ~ member(A,A2,C4) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_174_insert__absorb,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( member(A,A2,A3)
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3) = A3 ) ) ).

% insert_absorb
tff(fact_175_insert__ident,axiom,
    ! [A: $tType,Xa: A,A3: set(A),B3: set(A)] :
      ( ~ member(A,Xa,A3)
     => ( ~ member(A,Xa,B3)
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),B3) )
        <=> ( A3 = B3 ) ) ) ) ).

% insert_ident
tff(fact_176_Set_Oset__insert,axiom,
    ! [A: $tType,Xa: A,A3: set(A)] :
      ( member(A,Xa,A3)
     => ~ ! [B4: set(A)] :
            ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),B4) )
           => member(A,Xa,B4) ) ) ).

% Set.set_insert
tff(fact_177_insertI2,axiom,
    ! [A: $tType,A2: A,B3: set(A),B2: A] :
      ( member(A,A2,B3)
     => member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B3)) ) ).

% insertI2
tff(fact_178_insertI1,axiom,
    ! [A: $tType,A2: A,B3: set(A)] : member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)) ).

% insertI1
tff(fact_179_insertE,axiom,
    ! [A: $tType,A2: A,B2: A,A3: set(A)] :
      ( member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),A3))
     => ( ( A2 != B2 )
       => member(A,A2,A3) ) ) ).

% insertE
tff(fact_180_sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_sup(B)
     => ! [F2: fun(A,B),G: fun(A,B),X5: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),sup_sup(fun(A,B)),F2),G),X5) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,X5)),aa(A,B,G,X5)) ) ).

% sup_fun_def
tff(fact_181_sup__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z)) ) ).

% sup_left_commute
tff(fact_182_sup_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)) ) ).

% sup.left_commute
tff(fact_183_sup__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Xa) ) ).

% sup_commute
tff(fact_184_sup_Ocommute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),A2) ) ).

% sup.commute
tff(fact_185_sup__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) ) ).

% sup_assoc
tff(fact_186_sup_Oassoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)) ) ).

% sup.assoc
tff(fact_187_inf__sup__aci_I5_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Xa) ) ).

% inf_sup_aci(5)
tff(fact_188_inf__sup__aci_I6_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) ) ).

% inf_sup_aci(6)
tff(fact_189_inf__sup__aci_I7_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z)) ) ).

% inf_sup_aci(7)
tff(fact_190_inf__sup__aci_I8_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) ) ).

% inf_sup_aci(8)
tff(fact_191_Un__left__commute,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C3)) ).

% Un_left_commute
tff(fact_192_Un__left__absorb,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) ).

% Un_left_absorb
tff(fact_193_Un__commute,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),A3) ).

% Un_commute
tff(fact_194_Un__absorb,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),A3) = A3 ).

% Un_absorb
tff(fact_195_Un__assoc,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) ).

% Un_assoc
tff(fact_196_ball__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),P: fun(A,$o)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
         => aa(A,$o,P,X4) )
    <=> ( ! [X4: A] :
            ( member(A,X4,A3)
           => aa(A,$o,P,X4) )
        & ! [X4: A] :
            ( member(A,X4,B3)
           => aa(A,$o,P,X4) ) ) ) ).

% ball_Un
tff(fact_197_bex__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
          & aa(A,$o,P,X4) )
    <=> ( ? [X4: A] :
            ( member(A,X4,A3)
            & aa(A,$o,P,X4) )
        | ? [X4: A] :
            ( member(A,X4,B3)
            & aa(A,$o,P,X4) ) ) ) ).

% bex_Un
tff(fact_198_UnI2,axiom,
    ! [A: $tType,C2: A,B3: set(A),A3: set(A)] :
      ( member(A,C2,B3)
     => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ) ).

% UnI2
tff(fact_199_UnI1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,A3)
     => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ) ).

% UnI1
tff(fact_200_UnE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
     => ( ~ member(A,C2,A3)
       => member(A,C2,B3) ) ) ).

% UnE
tff(fact_201_Set_Oempty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = collect(A,aTP_Lamp_ab(A,$o)) ).

% Set.empty_def
tff(fact_202_insert__Collect,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),collect(A,P)) = collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ac(A,fun(fun(A,$o),fun(A,$o)),A2),P)) ).

% insert_Collect
tff(fact_203_insert__compr,axiom,
    ! [A: $tType,A2: A,B3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_ad(A,fun(set(A),fun(A,$o)),A2),B3)) ).

% insert_compr
tff(fact_204_Collect__disj__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ae(fun(A,$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_205_Un__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_af(set(A),fun(set(A),fun(A,$o)),A3),B3)) ).

% Un_def
tff(fact_206_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.strict_coboundedI2
tff(fact_207_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.strict_coboundedI1
tff(fact_208_sup_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
tff(fact_209_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% sup.strict_boundedE
tff(fact_210_sup_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb4
tff(fact_211_sup_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb3
tff(fact_212_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% less_supI2
tff(fact_213_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% less_supI1
tff(fact_214_singleton__inject,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) )
     => ( A2 = B2 ) ) ).

% singleton_inject
tff(fact_215_insert__not__empty,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3) != bot_bot(set(A)) ).

% insert_not_empty
tff(fact_216_doubleton__eq__iff,axiom,
    ! [A: $tType,A2: A,B2: A,C2: A,D2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),C2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),D2),bot_bot(set(A)))) )
    <=> ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_217_singleton__iff,axiom,
    ! [A: $tType,B2: A,A2: A] :
      ( member(A,B2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_218_singletonD,axiom,
    ! [A: $tType,B2: A,A2: A] :
      ( member(A,B2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_219_Un__empty__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),bot_bot(set(A))) = A3 ).

% Un_empty_right
tff(fact_220_Un__empty__left,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),bot_bot(set(A))),B3) = B3 ).

% Un_empty_left
tff(fact_221_Collect__conv__if2,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ag(A,fun(fun(A,$o),fun(A,$o)),A2),P)) = $ite(aa(A,$o,P,A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if2
tff(fact_222_Collect__conv__if,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ah(A,fun(fun(A,$o),fun(A,$o)),A2),P)) = $ite(aa(A,$o,P,A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if
tff(fact_223_insert__def,axiom,
    ! [A: $tType,A2: A,B3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),collect(A,aTP_Lamp_aa(A,fun(A,$o),A2))),B3) ).

% insert_def
tff(fact_224_singleton__Un__iff,axiom,
    ! [A: $tType,Xa: A,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) )
    <=> ( ( ( A3 = bot_bot(set(A)) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ) )
        | ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) )
          & ( B3 = bot_bot(set(A)) ) )
        | ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ) ) ) ) ).

% singleton_Un_iff
tff(fact_225_Un__singleton__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),Xa: A] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) )
    <=> ( ( ( A3 = bot_bot(set(A)) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ) )
        | ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) )
          & ( B3 = bot_bot(set(A)) ) )
        | ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ) ) ) ) ).

% Un_singleton_iff
tff(fact_226_insert__is__Un,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))),A3) ).

% insert_is_Un
tff(fact_227_n__less__equal__power__2,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% n_less_equal_power_2
tff(fact_228_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_229_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_cancel_right
tff(fact_230_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_cancel_left
tff(fact_231_nat__times__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X5),Xa2) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_times_as_int
tff(fact_232_nat__plus__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X5),Xa2) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_plus_as_int
tff(fact_233_high__inv,axiom,
    ! [Xa: nat,Na: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),Xa),Na) = Y ) ) ).

% high_inv
tff(fact_234_low__inv,axiom,
    ! [Xa: nat,Na: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),Xa),Na) = Xa ) ) ).

% low_inv
tff(fact_235_mulcomm,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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),I) ).

% mulcomm
tff(fact_236_pow__sum,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A2)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2) ).

% pow_sum
tff(fact_237_four__x__squared,axiom,
    ! [Xa: real] : aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% four_x_squared
tff(fact_238_word__unat__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ).

% word_unat_power
tff(fact_239_high__bound__aux,axiom,
    ! [Ma: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Ma,Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ).

% high_bound_aux
tff(fact_240_bit__split__inv,axiom,
    ! [Xa: nat,D2: nat] : vEBT_VEBT_bit_concat(vEBT_VEBT_high(Xa,D2),vEBT_VEBT_low(Xa,D2),D2) = Xa ).

% bit_split_inv
tff(fact_241_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( bit0(X22) = bit0(Y22) )
    <=> ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
tff(fact_242_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add_left_cancel
tff(fact_243_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
        <=> ( B2 = C2 ) ) ) ).

% add_right_cancel
tff(fact_244_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( aa(num,num,bit1,X32) = aa(num,num,bit1,Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_245_high__def,axiom,
    ! [Xa: nat,Na: nat] : vEBT_VEBT_high(Xa,Na) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% high_def
tff(fact_246_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))) ) ) ).

% divide_less_eq_numeral1(1)
tff(fact_247_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))),B2) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_248_psubsetD,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
     => ( member(A,C2,A3)
       => member(A,C2,B3) ) ) ).

% psubsetD
tff(fact_249_less__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
    <=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A3)),aTP_Lamp_a(set(A),fun(A,$o),B3)) ) ).

% less_set_def
tff(fact_250_psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B3),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),C3) ) ) ).

% psubset_trans
tff(fact_251_enat__less__induct,axiom,
    ! [P: fun(extended_enat,$o),Na: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M2: extended_enat] :
              ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),M2),N)
             => aa(extended_enat,$o,P,M2) )
         => aa(extended_enat,$o,P,N) )
     => aa(extended_enat,$o,P,Na) ) ).

% enat_less_induct
tff(fact_252_power__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),Na) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na)) ) ).

% power_divide
tff(fact_253_divide__numeral__1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),one2)) = A2 ) ).

% divide_numeral_1
tff(fact_254_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),A2) ) ).

% verit_comp_simplify1(1)
tff(fact_255_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_256_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A,K: A,A2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add1
tff(fact_257_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B3: A,K: A,B2: A,A2: A] :
          ( ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add2
tff(fact_258_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% add.assoc
tff(fact_259_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add.left_cancel
tff(fact_260_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
        <=> ( B2 = C2 ) ) ) ).

% add.right_cancel
tff(fact_261_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_262_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_263_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
         => ( B2 = C2 ) ) ) ).

% add_left_imp_eq
tff(fact_264_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
         => ( B2 = C2 ) ) ) ).

% add_right_imp_eq
tff(fact_265_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% mult.assoc
tff(fact_266_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_267_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_268_nat__int__comparison_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( aa(nat,int,semiring_1_of_nat(int),A2) = aa(nat,int,semiring_1_of_nat(int),B2) ) ) ).

% nat_int_comparison(1)
tff(fact_269_int__if,axiom,
    ! [P: $o,A2: nat,B2: nat] :
      aa(nat,int,semiring_1_of_nat(int),
        $ite((P),A2,B2)) = $ite((P),aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_if
tff(fact_270_field__sum__of__halves,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),aa(num,A,numeral_numeral(A),bit0(one2)))) = Xa ) ).

% field_sum_of_halves
tff(fact_271_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% field_less_half_sum
tff(fact_272_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_273_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_274_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_275_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_strict_mono
tff(fact_276_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% add_strict_left_mono
tff(fact_277_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ).

% add_strict_right_mono
tff(fact_278_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_imp_less_left
tff(fact_279_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_imp_less_right
tff(fact_280_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one2 != bit0(X22) ).

% verit_eq_simplify(10)
tff(fact_281_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] : bit0(X22) != aa(num,num,bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_282_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one2 != aa(num,num,bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_283_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_284_nat__numeral__as__int,axiom,
    ! [X5: num] : aa(num,nat,numeral_numeral(nat),X5) = aa(int,nat,nat2,aa(num,int,numeral_numeral(int),X5)) ).

% nat_numeral_as_int
tff(fact_285_nat__int__comparison_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ).

% nat_int_comparison(2)
tff(fact_286_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_287_int__ops_I5_J,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(5)
tff(fact_288_int__ops_I7_J,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(7)
tff(fact_289_nat__less__as__int,axiom,
    ! [X5: nat,Xa2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X5),Xa2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa2)) ) ).

% nat_less_as_int
tff(fact_290_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_291_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: 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),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) ) ).

% div_exp_eq
tff(fact_292_bot__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( bot(A)
     => ! [Xa: B] : aa(B,A,bot_bot(fun(B,A)),Xa) = bot_bot(A) ) ).

% bot_apply
tff(fact_293_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,C2: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),C2) ) ).

% times_divide_eq_left
tff(fact_294_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% divide_divide_eq_left
tff(fact_295_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ).

% divide_divide_eq_right
tff(fact_296_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ).

% times_divide_eq_right
tff(fact_297_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),aa(num,num,bit1,Na))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(num,A,numeral_numeral(A),Na) ) ).

% numeral_Bit1_div_2
tff(fact_298_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_299_sup1CI,axiom,
    ! [A: $tType,B3: fun(A,$o),Xa: A,A3: fun(A,$o)] :
      ( ( ~ aa(A,$o,B3,Xa)
       => aa(A,$o,A3,Xa) )
     => aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),A3),B3),Xa) ) ).

% sup1CI
tff(fact_300_low__def,axiom,
    ! [Xa: nat,Na: nat] : vEBT_VEBT_low(Xa,Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% low_def
tff(fact_301_mod__mod__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% mod_mod_trivial
tff(fact_302_real__divide__square__eq,axiom,
    ! [R: real,A2: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),R),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),R),R)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),R) ).

% real_divide_square_eq
tff(fact_303_mod__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% mod_add_self2
tff(fact_304_mod__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% mod_add_self1
tff(fact_305_mod__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) = M ) ) ).

% mod_less
tff(fact_306_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: 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),B2),C2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% mod_mult_self4
tff(fact_307_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: 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),B2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% mod_mult_self3
tff(fact_308_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: 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),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% mod_mult_self2
tff(fact_309_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% mod_mult_self1
tff(fact_310_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_311_mod__add__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: 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),A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ).

% mod_add_right_eq
tff(fact_312_mod__add__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: 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),A2),C2)),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ).

% mod_add_left_eq
tff(fact_313_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A4: A,B2: A,B5: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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),B2),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B5),C2) )
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B5)),C2) ) ) ) ) ).

% mod_add_cong
tff(fact_314_mod__add__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: 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),A2),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ).

% mod_add_eq
tff(fact_315_mod__mult__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: 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),A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ).

% mod_mult_right_eq
tff(fact_316_mod__mult__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),C2)),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ).

% mod_mult_left_eq
tff(fact_317_mult__mod__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% mult_mod_right
tff(fact_318_mod__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),C2) ) ).

% mod_mult_mult2
tff(fact_319_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A4: A,B2: A,B5: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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),B2),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B5),C2) )
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B5)),C2) ) ) ) ) ).

% mod_mult_cong
tff(fact_320_mod__mult__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ).

% mod_mult_eq
tff(fact_321_power__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,Na: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),Na)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),B2) ) ).

% power_mod
tff(fact_322_real__of__nat__div__aux,axiom,
    ! [Xa: nat,D2: nat] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Xa)),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),Xa),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),Xa),D2))),aa(nat,real,semiring_1_of_nat(real),D2))) ).

% real_of_nat_div_aux
tff(fact_323_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2) )
         => ~ ! [D3: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3)) ) ) ).

% mod_eqE
tff(fact_324_div__add1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2)) ) ).

% div_add1_eq
tff(fact_325_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2)) ) ).

% div_mult1_eq
tff(fact_326_mod__mult2__eq,axiom,
    ! [M: nat,Na: nat,Q2: 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),Q2)) = 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)),Q2))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)) ).

% mod_mult2_eq
tff(fact_327_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_328_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,M: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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),A2),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),A2),aa(nat,A,semiring_1_of_nat(A),M))) ) ).

% mod_mult2_eq'
tff(fact_329_div__mod__decomp,axiom,
    ! [A3: nat,Na: nat] : A3 = 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),A3),Na)),Na)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A3),Na)) ).

% div_mod_decomp
tff(fact_330_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% order_less_imp_not_less
tff(fact_331_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( Y != Xa ) ) ) ).

% order_less_imp_not_eq2
tff(fact_332_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( Xa != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_333_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
          | ( Xa = Y )
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% linorder_less_linear
tff(fact_334_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,P: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
           => (P) ) ) ) ).

% order_less_imp_triv
tff(fact_335_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% order_less_not_sym
tff(fact_336_order__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B2)),C2)
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_less_subst2
tff(fact_337_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_subst1
tff(fact_338_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Xa) ) ).

% order_less_irrefl
tff(fact_339_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_340_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F2,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_341_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z) ) ) ) ).

% order_less_trans
tff(fact_342_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% order_less_asym'
tff(fact_343_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ) ).

% linorder_neq_iff
tff(fact_344_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% order_less_asym
tff(fact_345_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ) ).

% linorder_neqE
tff(fact_346_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( A2 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_347_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X5: A] :
        ? [X_1: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),X_1) ) ).

% linordered_field_no_ub
tff(fact_348_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X5: A] :
        ? [Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X5) ) ).

% linordered_field_no_lb
tff(fact_349_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( A2 != B2 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_350_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% dual_order.strict_trans
tff(fact_351_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
            | ( Xa = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_352_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% order.strict_trans
tff(fact_353_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A2: A,B2: A] :
          ( ! [A5: A,B6: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A5),B6)
             => aa(A,$o,aa(A,fun(A,$o),P,A5),B6) )
         => ( ! [A5: A] : aa(A,$o,aa(A,fun(A,$o),P,A5),A5)
           => ( ! [A5: A,B6: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),P,B6),A5)
                 => aa(A,$o,aa(A,fun(A,$o),P,A5),B6) )
             => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) ) ) ) ) ).

% linorder_less_wlog
tff(fact_354_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_12: A] : aa(A,$o,P,X_12)
        <=> ? [N2: A] :
              ( aa(A,$o,P,N2)
              & ! [M3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M3),N2)
                 => ~ aa(A,$o,P,M3) ) ) ) ) ).

% exists_least_iff
tff(fact_355_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),A2) ) ).

% dual_order.irrefl
tff(fact_356_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% dual_order.asym
tff(fact_357_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( ( Xa != Y )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ) ).

% linorder_cases
tff(fact_358_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xa: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% antisym_conv3
tff(fact_359_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [X: A] :
              ( ! [Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y2),X)
                 => aa(A,$o,P,Y2) )
             => aa(A,$o,P,X) )
         => aa(A,$o,P,A2) ) ) ).

% less_induct
tff(fact_360_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ( B2 = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% ord_less_eq_trans
tff(fact_361_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% ord_eq_less_trans
tff(fact_362_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% order.asym
tff(fact_363_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( Xa != Y ) ) ) ).

% less_imp_neq
tff(fact_364_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ? [Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),Y) ) ) ) ).

% dense
tff(fact_365_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xa: A] :
        ? [X_1: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X_1) ) ).

% gt_ex
tff(fact_366_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xa: A] :
        ? [Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),Xa) ) ).

% lt_ex
tff(fact_367_boolean__algebra__cancel_Osup2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,K: A,B2: A,A2: A] :
          ( ( B3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_368_boolean__algebra__cancel_Osup1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,K: A,A2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),A2) )
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_369_nat__div__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X5),Xa2) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_div_as_int
tff(fact_370_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: 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),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% div_exp_mod_exp_eq
tff(fact_371_sup1I2,axiom,
    ! [A: $tType,B3: fun(A,$o),Xa: A,A3: fun(A,$o)] :
      ( aa(A,$o,B3,Xa)
     => 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)),A3),B3),Xa) ) ).

% sup1I2
tff(fact_372_sup1I1,axiom,
    ! [A: $tType,A3: fun(A,$o),Xa: A,B3: fun(A,$o)] :
      ( aa(A,$o,A3,Xa)
     => 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)),A3),B3),Xa) ) ).

% sup1I1
tff(fact_373_sup1E,axiom,
    ! [A: $tType,A3: fun(A,$o),B3: fun(A,$o),Xa: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),A3),B3),Xa)
     => ( ~ aa(A,$o,A3,Xa)
       => aa(A,$o,B3,Xa) ) ) ).

% sup1E
tff(fact_374_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( ( A2 != bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),A2) ) ) ).

% bot.not_eq_extremum
tff(fact_375_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),bot_bot(A)) ) ).

% bot.extremum_strict
tff(fact_376_add__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ).

% add_divide_distrib
tff(fact_377_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% divide_divide_eq_left'
tff(fact_378_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xa: 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),Xa),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),Xa),W)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ).

% divide_divide_times_eq
tff(fact_379_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xa: 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),Xa),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),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),W)) ) ).

% times_divide_times_eq
tff(fact_380_boolean__algebra_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),bot_bot(A)) = Xa ) ).

% boolean_algebra.disj_zero_right
tff(fact_381_div__mult2__eq,axiom,
    ! [M: nat,Na: nat,Q2: 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),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),Q2) ).

% div_mult2_eq
tff(fact_382_bot__fun__def,axiom,
    ! [A: $tType,B: $tType] :
      ( bot(B)
     => ! [X5: A] : aa(A,B,bot_bot(fun(A,B)),X5) = bot_bot(B) ) ).

% bot_fun_def
tff(fact_383_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,M: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),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),A2),aa(nat,A,semiring_1_of_nat(A),M))),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% div_mult2_eq'
tff(fact_384_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_385_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_386_div__mult2__numeral__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: 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),A2),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),A2),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_387_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),aa(num,num,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_388_nat__mod__eq_H,axiom,
    ! [A2: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),Na) = A2 ) ) ).

% nat_mod_eq'
tff(fact_389_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_390_mult__div__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) = A2 ) ).

% mult_div_mod_eq
tff(fact_391_mod__mult__div__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))) = A2 ) ).

% mod_mult_div_eq
tff(fact_392_mod__div__mult__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)) = A2 ) ).

% mod_div_mult_eq
tff(fact_393_div__mult__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) = A2 ) ).

% div_mult_mod_eq
tff(fact_394_mod__div__decomp,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) ) ).

% mod_div_decomp
tff(fact_395_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) ) ).

% cancel_div_mod_rules(1)
tff(fact_396_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) ) ).

% cancel_div_mod_rules(2)
tff(fact_397_cong__exp__iff__simps_I10_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: 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(Q2))) != aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Na))),aa(num,A,numeral_numeral(A),bit0(Q2))) ) ).

% cong_exp_iff_simps(10)
tff(fact_398_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_399_zmod__helper,axiom,
    ! [Na: int,M: int,K: int,A2: 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),A2)),M) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),A2)),M) ) ) ).

% zmod_helper
tff(fact_400_zmod__int,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zmod_int
tff(fact_401_div__mod__decomp__int,axiom,
    ! [A3: int,Na: int] : A3 = 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),A3),Na)),Na)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),Na)) ).

% div_mod_decomp_int
tff(fact_402_pos__mod__bound2,axiom,
    ! [A2: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% pos_mod_bound2
tff(fact_403_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_404_nat__mod__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X5),Xa2) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_mod_as_int
tff(fact_405_mod__exp__less__eq__exp,axiom,
    ! [A2: int,Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% mod_exp_less_eq_exp
tff(fact_406_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ring_class.ring_distribs(2)
tff(fact_407_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% ring_class.ring_distribs(1)
tff(fact_408_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( comm_semiring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% comm_semiring_class.distrib
tff(fact_409_distrib__left,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% distrib_left
tff(fact_410_distrib__right,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% distrib_right
tff(fact_411_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,E: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),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),A2),B2)),E)),C2) ) ).

% combine_common_factor
tff(fact_412_nat__mod__eq,axiom,
    ! [B2: nat,Na: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),Na)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B2),Na) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),Na) = B2 ) ) ) ).

% nat_mod_eq
tff(fact_413_zdiv__int,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zdiv_int
tff(fact_414_mod__plus__right,axiom,
    ! [A2: nat,Xa: nat,M: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),Xa)),M) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),Xa)),M) )
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),M) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B2),M) ) ) ).

% mod_plus_right
tff(fact_415_cong__exp__iff__simps_I9_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: 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(Q2))) = 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(Q2))) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Q2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),Na)),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_416_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_417_nat__mod__eq__iff,axiom,
    ! [Xa: nat,Na: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Y),Na) )
    <=> ? [Q1: nat,Q22: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_418_cong__exp__iff__simps_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: 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(Q2))) != 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(Q2))) ) ).

% cong_exp_iff_simps(8)
tff(fact_419_cong__exp__iff__simps_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: 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(Q2))) != 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(Q2))) ) ).

% cong_exp_iff_simps(6)
tff(fact_420_cong__exp__iff__simps_I13_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,Na: num] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),aa(num,A,numeral_numeral(A),bit0(Q2))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Na))),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Q2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),Na)),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_421_cong__exp__iff__simps_I12_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,Na: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),aa(num,A,numeral_numeral(A),bit0(Q2))) != 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(Q2))) ) ).

% cong_exp_iff_simps(12)
tff(fact_422_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_423_mlex__bound,axiom,
    ! [A2: nat,A3: nat,B2: nat,N3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),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),A2),N3)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N3)) ) ) ).

% mlex_bound
tff(fact_424_mlex__fst__decrI,axiom,
    ! [A2: nat,A4: nat,B2: nat,N3: nat,B5: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),N3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B5),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),A2),N3)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N3)),B5)) ) ) ) ).

% mlex_fst_decrI
tff(fact_425_mlex__snd__decrI,axiom,
    ! [A2: nat,A4: nat,B2: nat,B5: nat,N3: nat] :
      ( ( A2 = A4 )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),B5)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N3)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N3)),B5)) ) ) ).

% mlex_snd_decrI
tff(fact_426_Abs__fnat__hom__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A2: nat,B2: nat] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),A2)),aa(nat,A,semiring_1_of_nat(A),B2)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)) ) ).

% Abs_fnat_hom_add
tff(fact_427_sup__Un__eq,axiom,
    ! [A: $tType,R2: set(A),S: set(A),X5: A] :
      ( aa(A,$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_a(set(A),fun(A,$o),R2)),aTP_Lamp_a(set(A),fun(A,$o),S)),X5)
    <=> member(A,X5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),R2),S)) ) ).

% sup_Un_eq
tff(fact_428_log2__of__power__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ).

% log2_of_power_less
tff(fact_429_power__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num,L: num] : aa(nat,A,aa(A,fun(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_430_the__elem__eq,axiom,
    ! [A: $tType,Xa: A] : the_elem(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = Xa ).

% the_elem_eq
tff(fact_431_bot__empty__eq,axiom,
    ! [A: $tType,X5: A] :
      ( aa(A,$o,bot_bot(fun(A,$o)),X5)
    <=> member(A,X5,bot_bot(set(A))) ) ).

% bot_empty_eq
tff(fact_432_Collect__empty__eq__bot,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( collect(A,P) = bot_bot(set(A)) )
    <=> ( P = bot_bot(fun(A,$o)) ) ) ).

% Collect_empty_eq_bot
tff(fact_433_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_434_double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% double_eq_0_iff
tff(fact_435_add__0,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% add_0
tff(fact_436_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A,Y: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_437_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y) = zero_zero(A) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_438_add__cancel__right__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_right
tff(fact_439_add__cancel__right__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_left
tff(fact_440_add__cancel__left__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = A2 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_right
tff(fact_441_add__cancel__left__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = A2 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_left
tff(fact_442_double__zero__sym,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% double_zero_sym
tff(fact_443_add_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% add.right_neutral
tff(fact_444_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_right
tff(fact_445_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_left
tff(fact_446_mult__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% mult_eq_0_iff
tff(fact_447_mult__zero__right,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% mult_zero_right
tff(fact_448_mult__zero__left,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% mult_zero_left
tff(fact_449_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_450_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_451_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_452_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% bits_div_0
tff(fact_453_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_454_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% division_ring_divide_zero
tff(fact_455_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% div_0
tff(fact_456_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_457_bits__mod__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% bits_mod_0
tff(fact_458_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% mod_0
tff(fact_459_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),zero_zero(A)) = A2 ) ).

% mod_by_0
tff(fact_460_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),A2) = zero_zero(A) ) ).

% mod_self
tff(fact_461_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_462_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_463_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),A2) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_464_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_465_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_466_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_467_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_468_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_469_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_470_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_471_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_472_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_473_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ).

% less_add_same_cancel2
tff(fact_474_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ).

% less_add_same_cancel1
tff(fact_475_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% add_less_same_cancel2
tff(fact_476_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% add_less_same_cancel1
tff(fact_477_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_478_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_479_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_480_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_481_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_482_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_483_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% div_mult_mult1
tff(fact_484_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% div_mult_mult2
tff(fact_485_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% div_mult_mult1_if
tff(fact_486_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),A2) = B2 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_487_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),B2) = A2 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_488_power__zero__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(num,nat,numeral_numeral(nat),K)) = zero_zero(A) ) ).

% power_zero_numeral
tff(fact_489_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),B2) = zero_zero(A) ) ).

% mod_mult_self2_is_0
tff(fact_490_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),B2) = zero_zero(A) ) ).

% mod_mult_self1_is_0
tff(fact_491_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_492_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_493_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_494_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),B2) = zero_zero(A) ) ).

% mod_div_trivial
tff(fact_495_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),B2) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_496_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_497_div__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = zero_zero(nat) ) ) ).

% div_less
tff(fact_498_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_499_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_500_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_501_nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xa),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Xa)
        | ( Na = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_502_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_503_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W: num] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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),A2),aa(num,A,numeral_numeral(A),W)) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_504_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)) = A2 )
        <=> $ite(aa(num,A,numeral_numeral(A),W) != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W)),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_505_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self4
tff(fact_506_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self3
tff(fact_507_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self2
tff(fact_508_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self1
tff(fact_509_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,Na: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% power_eq_0_iff
tff(fact_510_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_511_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_512_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa) ) ) ).

% zero_less_ceiling
tff(fact_513_zero__eq__power2,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_power2
tff(fact_514_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_515_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_516_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = zero_zero(A) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_517_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),Xa)),Na))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Xa)
            | ( Na = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_518_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_519_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xa: A] :
          ( ( zero_zero(A) = Xa )
        <=> ( Xa = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_520_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_521_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_522_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Na) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_523_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D1: A,D22: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D1)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D22)
           => ? [E2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E2),D1)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E2),D22) ) ) ) ) ).

% field_lbound_gt_zero
tff(fact_524_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_525_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_526_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_527_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_528_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_529_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_530_verit__sum__simplify,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% verit_sum_simplify
tff(fact_531_add_Ogroup__left__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% add.group_left_neutral
tff(fact_532_add_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% add.comm_neutral
tff(fact_533_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% comm_monoid_add_class.add_0
tff(fact_534_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_right_cancel
tff(fact_535_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_left_cancel
tff(fact_536_no__zero__divisors,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) != zero_zero(A) ) ) ) ) ).

% no_zero_divisors
tff(fact_537_divisors__zero,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = zero_zero(A) )
         => ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divisors_zero
tff(fact_538_mult__not__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) != zero_zero(A) )
         => ( ( A2 != zero_zero(A) )
            & ( B2 != zero_zero(A) ) ) ) ) ).

% mult_not_zero
tff(fact_539_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,Na: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na) != zero_zero(A) ) ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_540_infinite__descent0__measure,axiom,
    ! [A: $tType,V2: fun(A,nat),P: fun(A,$o),Xa: A] :
      ( ! [X: A] :
          ( ( aa(A,nat,V2,X) = zero_zero(nat) )
         => aa(A,$o,P,X) )
     => ( ! [X: A] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,V2,X))
           => ( ~ aa(A,$o,P,X)
             => ? [Y2: A] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V2,Y2)),aa(A,nat,V2,X))
                  & ~ aa(A,$o,P,Y2) ) ) )
       => aa(A,$o,P,Xa) ) ) ).

% infinite_descent0_measure
tff(fact_541_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,$o,P,N)
             => ? [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
                  & ~ aa(nat,$o,P,M2) ) ) )
       => aa(nat,$o,P,Na) ) ) ).

% infinite_descent0
tff(fact_542_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_543_less__zeroE,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),zero_zero(nat)) ).

% less_zeroE
tff(fact_544_not__less0,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),zero_zero(nat)) ).

% not_less0
tff(fact_545_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_546_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_547_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),zero_zero(nat)) ).

% bot_nat_0.extremum_strict
tff(fact_548_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_549_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_550_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_551_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_552_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_ai(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_553_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_554_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_555_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% pos_add_strict
tff(fact_556_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ ! [C5: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C5) )
               => ( C5 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_557_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_pos_pos
tff(fact_558_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_neg_neg
tff(fact_559_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A)) ) ) ) ).

% add_less_zeroD
tff(fact_560_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_561_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_562_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_strict_right_mono
tff(fact_563_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_564_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_565_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_strict_left_mono
tff(fact_566_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_567_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_568_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_569_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ).

% zero_less_mult_pos2
tff(fact_570_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ).

% zero_less_mult_pos
tff(fact_571_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_572_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),zero_zero(A)) ) ) ) ).

% mult_pos_neg2
tff(fact_573_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_pos_pos
tff(fact_574_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_pos_neg
tff(fact_575_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_neg_pos
tff(fact_576_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ) ).

% mult_less_0_iff
tff(fact_577_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2)),zero_zero(A)) ) ).

% not_square_less_zero
tff(fact_578_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_neg_neg
tff(fact_579_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_580_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% divide_strict_right_mono
tff(fact_581_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_582_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            & ( C2 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_583_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ) ).

% divide_less_0_iff
tff(fact_584_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_pos_pos
tff(fact_585_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_pos_neg
tff(fact_586_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_neg_pos
tff(fact_587_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_neg_neg
tff(fact_588_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ).

% zero_less_power
tff(fact_589_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xa: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),W),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_590_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 )
        <=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq
tff(fact_591_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq
tff(fact_592_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 ) ) ) ) ).

% divide_eq_imp
tff(fact_593_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 )
           => ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_594_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 )
          <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_595_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_596_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_597_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),B2) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_598_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = A2 )
        <=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_599_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ? [K2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_600_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = zero_zero(nat) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( Na = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_601_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_602_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_603_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_604_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_605_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Na))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% nat_power_less_imp_less
tff(fact_606_mod__less__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)),Na) ) ).

% mod_less_divisor
tff(fact_607_nat__mod__lem,axiom,
    ! [Na: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),Na)
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B2),Na) = B2 ) ) ) ).

% nat_mod_lem
tff(fact_608_pow_Osimps_I1_J,axiom,
    ! [Xa: num] : pow(Xa,one2) = Xa ).

% pow.simps(1)
tff(fact_609_mod__eq__0D,axiom,
    ! [M: nat,D2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),D2) = zero_zero(nat) )
     => ? [Q3: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D2),Q3) ) ).

% mod_eq_0D
tff(fact_610_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))
        <=> ( ( Xa != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_611_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xa: A,Y: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)) ) ).

% not_sum_squares_lt_zero
tff(fact_612_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_613_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_614_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_615_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xa: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),Z) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_616_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% pos_less_divide_eq
tff(fact_617_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% pos_divide_less_eq
tff(fact_618_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% neg_less_divide_eq
tff(fact_619_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% neg_divide_less_eq
tff(fact_620_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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),A2),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% less_divide_eq
tff(fact_621_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% divide_less_eq
tff(fact_622_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B2: A,C2: A] :
          ( ( aa(num,A,numeral_numeral(A),W) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2) = B2,aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_623_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(num,A,numeral_numeral(A),W) )
        <=> $ite(C2 != zero_zero(A),B2 = 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_624_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_625_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),Y)),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_626_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xa: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_num_frac
tff(fact_627_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Xa: A,Z: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_frac_num
tff(fact_628_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xa: 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),Xa),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),Xa),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_629_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,Z: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = $ite(Z = zero_zero(A),A2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),Z)),B2)),Z)) ) ).

% add_divide_eq_if_simps(1)
tff(fact_630_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = $ite(Z = zero_zero(A),B2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(2)
tff(fact_631_memb__imp__not__empty,axiom,
    ! [A: $tType,Xa: A,S: set(A)] :
      ( member(A,Xa,S)
     => ( S != bot_bot(set(A)) ) ) ).

% memb_imp_not_empty
tff(fact_632_set__notEmptyE,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => ~ ! [X: A] : ~ member(A,X,S) ) ).

% set_notEmptyE
tff(fact_633_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),Xa)) ) ) ).

% ex_less_of_nat_mult
tff(fact_634_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num,Q2: 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(Q2))) = 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),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_635_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_636_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Q2)),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),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_637_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_638_td__gal__lt,axiom,
    ! [C2: nat,A2: nat,B2: 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),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),C2))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),C2)),B2) ) ) ).

% td_gal_lt
tff(fact_639_mod__plus__cong,axiom,
    ! [B2: int,B5: int,Xa: int,X6: int,Y: int,Y4: int,Z4: int] :
      ( ( B2 = B5 )
     => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xa),B5) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X6),B5) )
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Y),B5) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Y4),B5) )
         => ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X6),Y4) = Z4 )
           => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Y)),B2) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Z4),B5) ) ) ) ) ) ).

% mod_plus_cong
tff(fact_640_div__less__mono,axiom,
    ! [A3: nat,B3: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
     => ( 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),A3),Na) = zero_zero(nat) )
         => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B3),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),A3),Na)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B3),Na)) ) ) ) ) ).

% div_less_mono
tff(fact_641_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$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),B2),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)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$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_642_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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),B2),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,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),aa(A,$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_643_zero__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) ) ) ).

% zero_power2
tff(fact_644_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num,Q2: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Na))),aa(num,A,numeral_numeral(A),bit0(Q2))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_645_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_646_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_647_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_648_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,$o,P,zero_zero(nat)) )
        & ( ( Na != zero_zero(nat) )
         => ! [I2: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Na)
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),I2)),J2) )
               => aa(nat,$o,P,I2) ) ) ) ) ) ).

% split_div
tff(fact_649_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_650_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,$o,P,M) )
        & ( ( Na != zero_zero(nat) )
         => ! [I2: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Na)
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),I2)),J2) )
               => aa(nat,$o,P,J2) ) ) ) ) ) ).

% split_mod
tff(fact_651_mod__lemma,axiom,
    ! [C2: nat,R: nat,B2: nat,Q2: 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),B2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Q2),C2))),R)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),C2)) ) ) ).

% mod_lemma
tff(fact_652_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_653_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% half_gt_zero_iff
tff(fact_654_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% half_gt_zero
tff(fact_655_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A)) ) ).

% power2_less_0
tff(fact_656_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_657_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_left
tff(fact_658_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: 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(Q2))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Na))),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),Na)),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_659_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),aa(num,A,numeral_numeral(A),bit0(Q2))) = 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(Q2))) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_660_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
        <=> ( ( Xa != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_661_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A)) ) ).

% not_sum_power2_lt_zero
tff(fact_662_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_663_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_664_Abs__fnat__hom__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: nat,B2: 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)),A2)),aa(nat,word(A),semiring_1_of_nat(word(A)),B2)) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),B2)) ) ).

% Abs_fnat_hom_mult
tff(fact_665_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_666_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_low(Xa,Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_667_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_668_artanh__0,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ( aa(A,A,artanh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% artanh_0
tff(fact_669_arsinh__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arsinh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% arsinh_0
tff(fact_670_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),zero_zero(nat)),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% unset_bit_0
tff(fact_671_T__vebt__buildupi__gq__0,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),vEBT_V441764108873111860ildupi(Na)) ).

% T_vebt_buildupi_gq_0
tff(fact_672_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),M: nat,Na: nat] :
      ( ! [M4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,M4),zero_zero(nat))
     => ( ! [M4: nat,N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),P,N),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M4),N))
             => aa(nat,$o,aa(nat,fun(nat,$o),P,M4),N) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),P,M),Na) ) ) ).

% gcd_nat_induct
tff(fact_673_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R: A,A2: A,B2: A,C2: A,D2: A] :
          ( ( R != zero_zero(A) )
         => ( ( ( A2 = B2 )
              & ( C2 != D2 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),R),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R),D2)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_674_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ) ).

% mult_less_iff1
tff(fact_675_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),aa(num,num,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_676_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,Xa: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),U),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% arith_geo_mean
tff(fact_677_min__in__set__def,axiom,
    ! [Xs: set(nat),Xa: nat] :
      ( vEBT_VEBT_min_in_set(Xs,Xa)
    <=> ( member(nat,Xa,Xs)
        & ! [X4: nat] :
            ( member(nat,X4,Xs)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),X4) ) ) ) ).

% min_in_set_def
tff(fact_678_max__in__set__def,axiom,
    ! [Xs: set(nat),Xa: nat] :
      ( vEBT_VEBT_max_in_set(Xs,Xa)
    <=> ( member(nat,Xa,Xs)
        & ! [X4: nat] :
            ( member(nat,X4,Xs)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Xa) ) ) ) ).

% max_in_set_def
tff(fact_679_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Xa) ) ).

% order_refl
tff(fact_680_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),A2) ) ).

% dual_order.refl
tff(fact_681_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_682_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_683_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),A2) ).

% bot_nat_0.extremum
tff(fact_684_le0,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Na) ).

% le0
tff(fact_685_subset__empty,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),bot_bot(set(A)))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% subset_empty
tff(fact_686_empty__subsetI,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),bot_bot(set(A))),A3) ).

% empty_subsetI
tff(fact_687_semiring__norm_I73_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit1,M)),aa(num,num,bit1,Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(73)
tff(fact_688_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_689_insert__subset,axiom,
    ! [A: $tType,Xa: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)),B3)
    <=> ( member(A,Xa,B3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ) ).

% insert_subset
tff(fact_690_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_691_Un__subset__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C3)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3) ) ) ).

% Un_subset_iff
tff(fact_692_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_693_psubsetI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( ( A3 != B3 )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3) ) ) ).

% psubsetI
tff(fact_694_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_695_T__vebt__buildupi__univ,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V441764108873111860ildupi(Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),U)) ) ).

% T_vebt_buildupi_univ
tff(fact_696_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_697_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_698_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_cancel_left
tff(fact_699_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_cancel_right
tff(fact_700_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_701_mult_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),one_one(A)) = A2 ) ).

% mult.right_neutral
tff(fact_702_mult__1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A2) = A2 ) ).

% mult_1
tff(fact_703_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),one_one(A)) = A2 ) ).

% div_by_1
tff(fact_704_bits__div__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),one_one(A)) = A2 ) ).

% bits_div_by_1
tff(fact_705_power__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),Na) = one_one(A) ) ).

% power_one
tff(fact_706_semiring__norm_I72_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),aa(num,num,bit1,Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(72)
tff(fact_707_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_708_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit1,M)),one2) ).

% semiring_norm(70)
tff(fact_709_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% le_sup_iff
tff(fact_710_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% sup.bounded_iff
tff(fact_711_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_712_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_713_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_714_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B2: A,A2: A,A3: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3) )
    <=> ( ( A2 = B2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq
tff(fact_715_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A2: A,A3: set(A),B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) )
    <=> ( ( A2 = B2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_716_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_717_not__real__square__gt__zero,axiom,
    ! [Xa: real] :
      ( ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Xa))
    <=> ( Xa = zero_zero(real) ) ) ).

% not_real_square_gt_zero
tff(fact_718_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_719_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_720_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_721_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).

% le_add_same_cancel2
tff(fact_722_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).

% le_add_same_cancel1
tff(fact_723_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% add_le_same_cancel2
tff(fact_724_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% add_le_same_cancel1
tff(fact_725_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B2: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_726_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_left2
tff(fact_727_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B2: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_728_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_right2
tff(fact_729_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_730_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_731_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = one_one(A) ) ) ) ).

% div_self
tff(fact_732_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = one_one(A) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_733_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_734_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = one_one(A) ) ) ) ).

% divide_self
tff(fact_735_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% divide_self_if
tff(fact_736_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = one_one(A) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_737_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_738_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_739_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_740_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na) )
          <=> ( M = Na ) ) ) ) ).

% power_inject_exp
tff(fact_741_mod__by__1,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),one_one(A)) = zero_zero(A) ) ).

% mod_by_1
tff(fact_742_bits__mod__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),one_one(A)) = zero_zero(A) ) ).

% bits_mod_by_1
tff(fact_743_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_744_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_745_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_746_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_747_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_748_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_749_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_750_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_751_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_752_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_753_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_754_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_755_semiring__norm_I79_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit0(M)),aa(num,num,bit1,Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(79)
tff(fact_756_semiring__norm_I74_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit1,M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(74)
tff(fact_757_nat__ceiling__le__eq,axiom,
    ! [Xa: real,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,archimedean_ceiling(real,Xa))),A2)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,semiring_1_of_nat(real),A2)) ) ).

% nat_ceiling_le_eq
tff(fact_758_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_759_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% divide_le_0_1_iff
tff(fact_760_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% zero_le_divide_1_iff
tff(fact_761_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% divide_less_0_1_iff
tff(fact_762_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_763_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_764_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_765_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_766_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% zero_less_divide_1_iff
tff(fact_767_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_768_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))),B2) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_769_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xa: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Y) ) ) ) ).

% power_increasing_iff
tff(fact_770_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_771_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_772_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_773_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xa: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y) ) ) ) ).

% power_strict_increasing_iff
tff(fact_774_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),Xa) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_775_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,B2: nat,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_776_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A)) ) ) ).

% ceiling_le_zero
tff(fact_777_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(num,A,numeral_numeral(A),V)) ) ) ).

% ceiling_le_numeral
tff(fact_778_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa) ) ) ).

% one_le_ceiling
tff(fact_779_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),one_one(A)) ) ) ).

% ceiling_le_one
tff(fact_780_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_781_int__div__same__is__1,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A2)
     => ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = A2 )
      <=> ( B2 = one_one(int) ) ) ) ).

% int_div_same_is_1
tff(fact_782_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_783_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_784_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_785_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_786_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ) ) ) ) ).

% power_decreasing_iff
tff(fact_787_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_788_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_789_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ) ).

% power_mono_iff
tff(fact_790_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_791_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_792_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_793_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_794_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_795_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_796_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A)) ) ) ).

% ceiling_less_one
tff(fact_797_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa) ) ) ).

% one_less_ceiling
tff(fact_798_ceiling__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xa)),one_one(int)) ) ).

% ceiling_add_one
tff(fact_799_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_800_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xa: num,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xa)),Na))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_801_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xa)),Na)),aa(int,nat,nat2,A2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na)),A2) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_802_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_803_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_804_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_805_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
            <=> ( Xa = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_806_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Na: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Na)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),Na)),Xa) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_807_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,I: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),Na)) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_808_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_809_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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))) = one_one(int) ).

% one_mod_exp_eq_one
tff(fact_810_div__le__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Xa)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),Xa)),K) ) ) ) ).

% div_le_mult
tff(fact_811_div__lt__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Xa)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),Xa)),K) ) ) ) ).

% div_lt_mult
tff(fact_812_More__Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C2: word(A),A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),C2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),B2),C2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),C2)),B2) ) ) ) ).

% More_Word.word_div_mult
tff(fact_813_word__mod__div__equality,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),B2: 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),B2)),B2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Na),B2)) = Na ) ).

% word_mod_div_equality
tff(fact_814_word__div__mult__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),B2)),B2)),A2) ) ).

% word_div_mult_le
tff(fact_815_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_816_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_817_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ( B2 != A2 ) ) ) ) ).

% nle_le
tff(fact_818_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) )
         => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z) )
           => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
               => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y) )
             => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) )
               => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
                   => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xa) )
                 => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xa)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_819_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa = Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ) ).

% order_class.order_eq_iff
tff(fact_820_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% ord_eq_le_trans
tff(fact_821_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( B2 = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% ord_le_eq_trans
tff(fact_822_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
           => ( Xa = Y ) ) ) ) ).

% order_antisym
tff(fact_823_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% order.trans
tff(fact_824_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z) ) ) ) ).

% order_trans
tff(fact_825_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A2: A,B2: A] :
          ( ! [A5: A,B6: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),B6)
             => aa(A,$o,aa(A,fun(A,$o),P,A5),B6) )
         => ( ! [A5: A,B6: A] :
                ( aa(A,$o,aa(A,fun(A,$o),P,B6),A5)
               => aa(A,$o,aa(A,fun(A,$o),P,A5),B6) )
           => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) ) ) ) ).

% linorder_wlog
tff(fact_826_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% dual_order.eq_iff
tff(fact_827_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( A2 = B2 ) ) ) ) ).

% dual_order.antisym
tff(fact_828_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% dual_order.trans
tff(fact_829_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => ( A2 = B2 ) ) ) ) ).

% antisym
tff(fact_830_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),Xa: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,G,Xa)) ) ) ).

% le_funD
tff(fact_831_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),Xa: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,G,Xa)) ) ) ).

% le_funE
tff(fact_832_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( ! [X: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X))
         => aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G) ) ) ).

% le_funI
tff(fact_833_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),$o),ord_less_eq(fun(A,B)),F2),G)
        <=> ! [X4: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) ) ) ).

% le_fun_def
tff(fact_834_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_835_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F2,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_subst1
tff(fact_836_order__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,B2)),C2)
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_subst2
tff(fact_837_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa = Y )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% order_eq_refl
tff(fact_838_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% linorder_linear
tff(fact_839_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F2,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_840_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_841_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% linorder_le_cases
tff(fact_842_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_843_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_844_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),A2) ) ).

% verit_comp_simplify1(2)
tff(fact_845_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [Xa: A] :
          ( ( one_one(A) = Xa )
        <=> ( Xa = one_one(A) ) ) ) ).

% one_reorient
tff(fact_846_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% verit_la_disequality
tff(fact_847_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N3: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N3)) ) ) ) ).

% power_increasing
tff(fact_848_div__to__mult__word__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Y),Z))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),Z)),Y) ) ) ).

% div_to_mult_word_lt
tff(fact_849_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_850_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ).

% one_le_power
tff(fact_851_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_852_nat__int__comparison_I3_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ).

% nat_int_comparison(3)
tff(fact_853_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_854_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)
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I3)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_ge_induct
tff(fact_855_nat__mono,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),Y)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,Xa)),aa(int,nat,nat2,Y)) ) ).

% nat_mono
tff(fact_856_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_857_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_858_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_859_all__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ! [X_12: nat] : aa(nat,$o,P,X_12)
    <=> ! [X4: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X4)
         => aa(nat,$o,P,aa(int,nat,nat2,X4)) ) ) ).

% all_nat
tff(fact_860_ex__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ? [X_12: nat] : aa(nat,$o,P,X_12)
    <=> ? [X4: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X4)
          & aa(nat,$o,P,aa(int,nat,nat2,X4)) ) ) ).

% ex_nat
tff(fact_861_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_862_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_863_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Y)),archimedean_ceiling(A,Xa)) ) ) ).

% ceiling_mono
tff(fact_864_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N3: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ) ) ).

% power_decreasing
tff(fact_865_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% power_le_imp_le_exp
tff(fact_866_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_867_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_868_pos__mult__pos__ge,axiom,
    ! [Xa: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),Na),one_one(int))),aa(int,int,aa(int,fun(int,int),times_times(int),Na),Xa)) ) ) ).

% pos_mult_pos_ge
tff(fact_869_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_870_nat__leq__as__int,axiom,
    ! [X5: nat,Xa2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X5),Xa2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa2)) ) ).

% nat_leq_as_int
tff(fact_871_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_872_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),A2) ) ) ) ).

% mult_left_le
tff(fact_873_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A)) ) ) ) ) ).

% mult_le_one
tff(fact_874_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y)),Xa) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_875_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa)),Xa) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_876_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),B2) ) ) ).

% discrete
tff(fact_877_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),one_one(A)) ) ) ) ).

% power_le_one
tff(fact_878_mod__power__lem,axiom,
    ! [A2: int,Na: nat,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),A2)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),Na)),aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),M)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na),zero_zero(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),Na)) ) ) ).

% mod_power_lem
tff(fact_879_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_880_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_881_nat__le__iff,axiom,
    ! [Xa: int,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,Xa)),Na)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),aa(nat,int,semiring_1_of_nat(int),Na)) ) ).

% nat_le_iff
tff(fact_882_neg__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)) ) ) ).

% neg_mod_conj
tff(fact_883_pos__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)),B2) ) ) ).

% pos_mod_conj
tff(fact_884_int__mod__eq,axiom,
    ! [B2: int,Na: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),Na)
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),Na) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),Na) )
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),Na) = B2 ) ) ) ) ).

% int_mod_eq
tff(fact_885_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_886_int__mod__lem,axiom,
    ! [Na: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),Na) )
      <=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),Na) = B2 ) ) ) ).

% int_mod_lem
tff(fact_887_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_888_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_889_int__mod__ge,axiom,
    ! [A2: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),Na)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),Na)) ) ) ).

% int_mod_ge
tff(fact_890_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_891_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_892_zdiv__mono1,axiom,
    ! [A2: int,A4: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),A4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),B2)) ) ) ).

% zdiv_mono1
tff(fact_893_zdiv__mono2,axiom,
    ! [A2: int,B5: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B5)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B5),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B5)) ) ) ) ).

% zdiv_mono2
tff(fact_894_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_895_zdiv__mono1__neg,axiom,
    ! [A2: int,A4: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),A4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) ) ).

% zdiv_mono1_neg
tff(fact_896_zdiv__mono2__neg,axiom,
    ! [A2: int,B5: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B5)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B5),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B5)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_897_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_898_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_899_div__nonneg__neg__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_900_div__nonpos__pos__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_901_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_902_zdiv__le__dividend,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),A2) ) ) ).

% zdiv_le_dividend
tff(fact_903_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_904_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_905_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))
      <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),A2)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_906_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_907_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_908_zdiv__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: 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),A2),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_909_nonneg__mod__div,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2))
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) ) ) ).

% nonneg_mod_div
tff(fact_910_int__div__less__self,axiom,
    ! [Xa: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),K)),Xa) ) ) ).

% int_div_less_self
tff(fact_911_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_912_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa) ) ).

% zero_le
tff(fact_913_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B5: A,A4: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B5),A4)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B5) ) ) ).

% verit_comp_simplify1(3)
tff(fact_914_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% leD
tff(fact_915_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% leI
tff(fact_916_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% nless_le
tff(fact_917_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% antisym_conv1
tff(fact_918_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% antisym_conv2
tff(fact_919_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Y: A] :
          ( ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ).

% dense_ge
tff(fact_920_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z: A] :
          ( ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ).

% dense_le
tff(fact_921_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ) ).

% less_le_not_le
tff(fact_922_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xa: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% not_le_imp_less
tff(fact_923_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_924_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_925_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% order.strict_trans1
tff(fact_926_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% order.strict_trans2
tff(fact_927_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% order.strict_iff_not
tff(fact_928_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xa)
         => ( ! [W2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),W2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W2),Xa)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),W2) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% dense_ge_bounded
tff(fact_929_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( ! [W2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),W2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W2),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% dense_le_bounded
tff(fact_930_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
            | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_931_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_932_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% dual_order.strict_trans1
tff(fact_933_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% dual_order.strict_trans2
tff(fact_934_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_935_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% order.strict_implies_order
tff(fact_936_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% dual_order.strict_implies_order
tff(fact_937_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
            | ( Xa = Y ) ) ) ) ).

% order_le_less
tff(fact_938_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & ( Xa != Y ) ) ) ) ).

% order_less_le
tff(fact_939_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% linorder_not_le
tff(fact_940_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% linorder_not_less
tff(fact_941_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% order_less_imp_le
tff(fact_942_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( A2 != B2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% order_le_neq_trans
tff(fact_943_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% order_neq_le_trans
tff(fact_944_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z) ) ) ) ).

% order_le_less_trans
tff(fact_945_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z) ) ) ) ).

% order_less_le_trans
tff(fact_946_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F2,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_947_order__le__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B2)),C2)
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_le_less_subst2
tff(fact_948_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_949_order__less__le__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,B2)),C2)
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_less_le_subst2
tff(fact_950_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% linorder_le_less_linear
tff(fact_951_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
            | ( Xa = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_952_le__num__One__iff,axiom,
    ! [Xa: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Xa),one2)
    <=> ( Xa = one2 ) ) ).

% le_num_One_iff
tff(fact_953_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_954_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_955_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_956_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_mono
tff(fact_957_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% add_left_mono
tff(fact_958_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ~ ! [C5: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C5) ) ) ).

% less_eqE
tff(fact_959_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ).

% add_right_mono
tff(fact_960_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ? [C6: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C6) ) ) ).

% le_iff_add
tff(fact_961_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_imp_le_left
tff(fact_962_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_imp_le_right
tff(fact_963_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),bot_bot(A)),A2) ) ).

% bot.extremum
tff(fact_964_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),bot_bot(A))
        <=> ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_965_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),bot_bot(A))
         => ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_966_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_967_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),zero_zero(nat))
    <=> ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_968_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),zero_zero(nat))
     => ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_969_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_970_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_971_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_972_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_973_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_974_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_975_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_976_less__mono__imp__le__mono,axiom,
    ! [F2: fun(nat,nat),I: nat,J: nat] :
      ( ! [I3: nat,J3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,I3)),aa(nat,nat,F2,J3)) )
     => ( 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_977_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_978_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% real_arch_simple
tff(fact_979_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y: A,Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% inf_sup_ord(4)
tff(fact_980_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% inf_sup_ord(3)
tff(fact_981_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),Xa)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xa)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xa) ) ) ) ).

% le_supE
tff(fact_982_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,Xa: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),Xa) ) ) ) ).

% le_supI
tff(fact_983_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% sup_ge1
tff(fact_984_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% sup_ge2
tff(fact_985_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% le_supI1
tff(fact_986_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% le_supI2
tff(fact_987_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,D2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ) ).

% sup.mono
tff(fact_988_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2)) ) ) ) ).

% sup_mono
tff(fact_989_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xa: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)),Xa) ) ) ) ).

% sup_least
tff(fact_990_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = Y ) ) ) ).

% le_iff_sup
tff(fact_991_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ) ) ).

% sup.orderE
tff(fact_992_sup_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% sup.orderI
tff(fact_993_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(A,fun(A,A)),Xa: A,Y: A] :
          ( ! [X: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),F2,X),Y3))
         => ( ! [X: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),aa(A,A,aa(A,fun(A,A),F2,X),Y3))
           => ( ! [X: A,Y3: A,Z3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z3),X)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,Y3),Z3)),X) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),F2,Xa),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_994_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb1
tff(fact_995_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb2
tff(fact_996_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = Xa ) ) ) ).

% sup_absorb1
tff(fact_997_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = Y ) ) ) ).

% sup_absorb2
tff(fact_998_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% sup.boundedE
tff(fact_999_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2) ) ) ) ).

% sup.boundedI
tff(fact_1000_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ) ) ).

% sup.order_iff
tff(fact_1001_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ).

% sup.cobounded1
tff(fact_1002_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ).

% sup.cobounded2
tff(fact_1003_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb_iff1
tff(fact_1004_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb_iff2
tff(fact_1005_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.coboundedI1
tff(fact_1006_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.coboundedI2
tff(fact_1007_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A2) = A2 ) ).

% comm_monoid_mult_class.mult_1
tff(fact_1008_mult_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),one_one(A)) = A2 ) ).

% mult.comm_neutral
tff(fact_1009_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_1010_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_1011_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_1012_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_1013_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_1014_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_1015_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_1016_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_1017_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_1018_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_1019_nat__le__iff__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
    <=> ? [K3: nat] : Na = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3) ) ).

% nat_le_iff_add
tff(fact_1020_insert__mono,axiom,
    ! [A: $tType,C3: set(A),D4: set(A),A2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),D4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),C3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),D4)) ) ).

% insert_mono
tff(fact_1021_subset__insert,axiom,
    ! [A: $tType,Xa: A,A3: set(A),B3: set(A)] :
      ( ~ member(A,Xa,A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),B3))
      <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ) ).

% subset_insert
tff(fact_1022_subset__insertI,axiom,
    ! [A: $tType,B3: set(A),A2: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)) ).

% subset_insertI
tff(fact_1023_subset__insertI2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),B2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B3)) ) ).

% subset_insertI2
tff(fact_1024_less__eq__real__def,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
        | ( Xa = Y ) ) ) ).

% less_eq_real_def
tff(fact_1025_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_1026_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_1027_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_1028_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_1029_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_1030_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_1031_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_1032_Un__mono,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B3: set(A),D4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),D4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),D4)) ) ) ).

% Un_mono
tff(fact_1033_Un__least,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C3) ) ) ).

% Un_least
tff(fact_1034_Un__upper1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ).

% Un_upper1
tff(fact_1035_Un__upper2,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ).

% Un_upper2
tff(fact_1036_Un__absorb1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = B3 ) ) ).

% Un_absorb1
tff(fact_1037_Un__absorb2,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = A3 ) ) ).

% Un_absorb2
tff(fact_1038_subset__UnE,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
     => ~ ! [A6: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A6),A3)
           => ! [B7: set(A)] :
                ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B7),B3)
               => ( C3 != aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B7) ) ) ) ) ).

% subset_UnE
tff(fact_1039_subset__Un__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = B3 ) ) ).

% subset_Un_eq
tff(fact_1040_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_1041_psubsetE,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
     => ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3) ) ) ).

% psubsetE
tff(fact_1042_psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
        & ( A3 != B3 ) ) ) ).

% psubset_eq
tff(fact_1043_psubset__imp__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ).

% psubset_imp_subset
tff(fact_1044_psubset__subset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),C3) ) ) ).

% psubset_subset_trans
tff(fact_1045_subset__not__subset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
        & ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3) ) ) ).

% subset_not_subset_eq
tff(fact_1046_subset__psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B3),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),C3) ) ) ).

% subset_psubset_trans
tff(fact_1047_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
        | ( A3 = B3 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1048_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_1049_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_1050_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_1051_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_1052_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_1053_word__le__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
         => ? [N: nat] : Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),aa(nat,word(A),semiring_1_of_nat(word(A)),N)) ) ) ).

% word_le_add
tff(fact_1054_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_1055_verit__le__mono__div__int,axiom,
    ! [A3: int,B3: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),B3)
     => ( 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),A3),Na)),
                $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B3),Na) = zero_zero(int),one_one(int),zero_zero(int)))),
            aa(int,int,aa(int,fun(int,int),divide_divide(int),B3),Na)) ) ) ).

% verit_le_mono_div_int
tff(fact_1056_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$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),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1057_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A)) )
            & ( aa(A,$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)),A2) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1058_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$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),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1059_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A)) )
            & ( aa(A,$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)),A2) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1060_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$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),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1061_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) )
            & ( aa(A,$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)),A2) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1062_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$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),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1063_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) )
            & ( aa(A,$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)),A2) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1064_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( ! [Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),one_one(A))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z3),Xa)),Y) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% field_le_mult_one_interval
tff(fact_1065_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [Xa: A,A2: A,Y: A,U: A,V: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => aa(A,$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),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1066_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% le_divide_eq_1
tff(fact_1067_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_1068_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ) ).

% self_le_power
tff(fact_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_aj(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_1076_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q4: int,R3: int,Q2: 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),B2),Q4)),R3)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),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),B2),R)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q2),Q4) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1077_unique__quotient__lemma,axiom,
    ! [B2: int,Q4: int,R3: int,Q2: 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),B2),Q4)),R3)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R3)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R3),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q4),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1078_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q2: int,R: int,B5: int,Q4: int,R3: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B5),Q4)),R3) )
     => ( 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),B5),Q4)),R3)),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R3)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B5)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B5),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q4),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1079_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q2: int,R: int,B5: int,Q4: int,R3: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B5),Q4)),R3) )
     => ( 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),B5),Q4)),R3))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R3),B5)
         => ( 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)),B5)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B5),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q2),Q4) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1080_q__pos__lemma,axiom,
    ! [B5: int,Q4: int,R3: 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),B5),Q4)),R3))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R3),B5)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B5)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Q4) ) ) ) ).

% q_pos_lemma
tff(fact_1081_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_1082_int__mod__ge_H,axiom,
    ! [B2: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),Na)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),Na)) ) ) ).

% int_mod_ge'
tff(fact_1083_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_1084_nat__div__distrib_H,axiom,
    ! [Y: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,Xa)),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_1085_nat__div__distrib,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,Xa)),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_1086_nat__power__eq,axiom,
    ! [Z: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,nat,nat2,aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(int,nat,nat2,Z)),Na) ) ) ).

% nat_power_eq
tff(fact_1087_nat__mod__distrib,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(int,nat,nat2,Xa)),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_1088_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [Xa: A,A2: A,Y: A,U: A,V: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => aa(A,$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),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1089_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_1090_pos__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),A2))) ) ) ).

% pos_zmod_mult_2
tff(fact_1091_pos__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_1092_neg__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int))),A2) ) ) ).

% neg_zdiv_mult_2
tff(fact_1093_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A2)),archimedean_ceiling(A,B2))) ) ) ) ).

% mult_ceiling_le
tff(fact_1094_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1095_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1096_pos__mod__sign2,axiom,
    ! [A2: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% pos_mod_sign2
tff(fact_1097_not__exp__less__eq__0__int,axiom,
    ! [Na: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),zero_zero(int)) ).

% not_exp_less_eq_0_int
tff(fact_1098_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_1099_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_1100_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_1101_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_1102_real__archimedian__rdiv__eq__0,axiom,
    ! [Xa: real,C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C2)
       => ( ! [M4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M4)
             => 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),M4)),Xa)),C2) )
         => ( Xa = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_1103_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y) = zero_zero(A) )
            <=> ( ( Xa = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1104_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y) = zero_zero(A) )
            <=> ( ( Xa = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1105_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_nonpos_nonpos
tff(fact_1106_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_nonneg_nonneg
tff(fact_1107_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_increasing2
tff(fact_1108_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ) ) ).

% add_decreasing2
tff(fact_1109_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_increasing
tff(fact_1110_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ) ) ).

% add_decreasing
tff(fact_1111_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_mono
tff(fact_1112_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_mono'
tff(fact_1113_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2)) ) ).

% zero_le_square
tff(fact_1114_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ).

% split_mult_pos_le
tff(fact_1115_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_left_mono_neg
tff(fact_1116_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_1117_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_left_mono
tff(fact_1118_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_right_mono_neg
tff(fact_1119_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_right_mono
tff(fact_1120_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ) ) ).

% mult_le_0_iff
tff(fact_1121_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ).

% split_mult_neg_le
tff(fact_1122_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_1123_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_1124_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_1125_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1126_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_1127_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1128_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_1129_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_1130_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_le_less_mono
tff(fact_1131_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_less_le_mono
tff(fact_1132_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ) ) ).

% divide_le_0_iff
tff(fact_1133_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% divide_right_mono
tff(fact_1134_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_1135_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_1136_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_1137_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_1138_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_1139_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)) ) ) ) ).

% divide_right_mono_neg
tff(fact_1140_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_1141_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ).

% zero_le_power
tff(fact_1142_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na)) ) ) ) ).

% power_mono
tff(fact_1143_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_1144_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,$o),ord_less(int),zero_zero(int)),K)
         => ! [I2: int,J2: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J2)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J2),K)
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),J2) ) )
             => aa(int,$o,P,J2) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I2: int,J2: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J2)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J2),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),I2)),J2) ) )
             => aa(int,$o,P,J2) ) ) ) ) ).

% split_zmod
tff(fact_1145_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),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),B2),R)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = R ) ) ) ) ).

% int_mod_neg_eq
tff(fact_1146_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),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),B2)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = R ) ) ) ) ).

% int_mod_pos_eq
tff(fact_1147_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),A2) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1148_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,$o),ord_less(int),zero_zero(int)),K)
         => ! [I2: int,J2: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J2)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J2),K)
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),J2) ) )
             => aa(int,$o,P,I2) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I2: int,J2: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J2)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J2),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),I2)),J2) ) )
             => aa(int,$o,P,I2) ) ) ) ) ).

% split_zdiv
tff(fact_1149_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),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),B2),R)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1150_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),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),B2)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1151_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_1152_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_1153_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_1154_zmod__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: 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),A2),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),C2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_1155_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_1156_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_1157_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))) ) ).

% less_add_one
tff(fact_1158_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),one_one(A))) ) ) ).

% add_mono1
tff(fact_1159_one__plus__numeral__commute,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),Xa)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Xa)),one_one(A)) ) ).

% one_plus_numeral_commute
tff(fact_1160_ex__least__nat__le,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,Na)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Na)
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K2)
               => ~ aa(nat,$o,P,I4) )
            & aa(nat,$o,P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_1161_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_1162_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = one_one(A) )
          <=> ( A2 = B2 ) ) ) ) ).

% right_inverse_eq
tff(fact_1163_mono__nat__linear__lb,axiom,
    ! [F2: fun(nat,nat),M: nat,K: nat] :
      ( ! [M4: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,M4)),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_1164_subset__singletonD,axiom,
    ! [A: $tType,A3: set(A),Xa: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))
     => ( ( A3 = bot_bot(set(A)) )
        | ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_1165_subset__singleton__iff,axiom,
    ! [A: $tType,X7: set(A),A2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))
    <=> ( ( X7 = bot_bot(set(A)) )
        | ( X7 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_1166_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xa: A,Y: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y) = one_one(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Na)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_1167_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),Na) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ).

% power_one_over
tff(fact_1168_power__0,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),zero_zero(nat)) = one_one(A) ) ).

% power_0
tff(fact_1169_mlex__leI,axiom,
    ! [A2: nat,A4: nat,B2: nat,B5: nat,N3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),B5)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N3)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N3)),B5)) ) ) ).

% mlex_leI
tff(fact_1170_div__mult__le,axiom,
    ! [A2: nat,B2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),B2)),B2)),A2) ).

% div_mult_le
tff(fact_1171_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_1172_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_1173_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_1174_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_1175_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_1176_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)))
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I3)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_gr_induct
tff(fact_1177_nat__zero__as__int,axiom,
    zero_zero(nat) = aa(int,nat,nat2,zero_zero(int)) ).

% nat_zero_as_int
tff(fact_1178_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_1179_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_1180_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_1181_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)) ) ).

% neg_mod_bound
tff(fact_1182_zmod__eq__0__iff,axiom,
    ! [M: int,D2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),D2) = zero_zero(int) )
    <=> ? [Q5: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q5) ) ).

% zmod_eq_0_iff
tff(fact_1183_zmod__eq__0D,axiom,
    ! [M: int,D2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),D2) = zero_zero(int) )
     => ? [Q3: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q3) ) ).

% zmod_eq_0D
tff(fact_1184_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_1185_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A2) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_1186_div__neg__pos__less0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_neg_pos_less0
tff(fact_1187_axxmod2,axiom,
    ! [Xa: 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)),Xa)),Xa)),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)),Xa)),Xa)),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int) ) ) ).

% axxmod2
tff(fact_1188_axxdiv2,axiom,
    ! [Xa: 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)),Xa)),Xa)),aa(num,int,numeral_numeral(int),bit0(one2))) = Xa )
      & ( 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)),Xa)),Xa)),aa(num,int,numeral_numeral(int),bit0(one2))) = Xa ) ) ).

% axxdiv2
tff(fact_1189_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,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),K)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),K))
      <=> ! [I2: int,J2: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J2)
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J2),K)
              & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),J2) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I2),J2) ) ) ) ).

% split_pos_lemma
tff(fact_1190_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,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),K)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),K))
      <=> ! [I2: int,J2: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J2)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J2),zero_zero(int))
              & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),J2) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I2),J2) ) ) ) ).

% split_neg_lemma
tff(fact_1191_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_neg_nonpos
tff(fact_1192_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_nonneg_pos
tff(fact_1193_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_nonpos_neg
tff(fact_1194_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_pos_nonneg
tff(fact_1195_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_strict_increasing
tff(fact_1196_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_strict_increasing2
tff(fact_1197_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( ! [E2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E2)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% field_le_epsilon
tff(fact_1198_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$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),B2),A2) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_1199_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$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),B2),A2) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_1200_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
         => ( aa(A,$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),A2),B2) ) ) ) ).

% mult_left_less_imp_less
tff(fact_1201_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1202_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$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),B2),A2) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1203_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ( aa(A,$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),A2),B2) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1204_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1205_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$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),B2),A2) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1206_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1207_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1208_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
         => ( aa(A,$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),A2),B2) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1209_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ( aa(A,$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),A2),B2) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1210_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1211_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1212_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))) ) ).

% sum_squares_ge_zero
tff(fact_1213_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1214_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1215_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_positive
tff(fact_1216_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xa: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => ( aa(A,$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),Xa),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_le
tff(fact_1217_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W)
             => ( aa(A,$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),Xa),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_less
tff(fact_1218_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W)
             => ( aa(A,$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),Xa),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_less2
tff(fact_1219_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$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),B2),A2) ) ) ) ) ).

% divide_le_cancel
tff(fact_1220_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_neg
tff(fact_1221_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_nonneg_pos
tff(fact_1222_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_nonpos_neg
tff(fact_1223_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_pos
tff(fact_1224_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% power_less_imp_less_base
tff(fact_1225_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$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),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1226_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1227_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = A2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1228_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_1229_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_1230_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% less_divide_eq_1
tff(fact_1231_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_1232_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_1233_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B2) ) ) ).

% gt_half_sum
tff(fact_1234_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ) ).

% less_half_sum
tff(fact_1235_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))) ) ) ).

% power_gt1_lemma
tff(fact_1236_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))) ) ) ).

% power_less_power_Suc
tff(fact_1237_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(num,A,numeral_numeral(A),aa(num,num,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_1238_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Na) = $ite(Na = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% power_0_left
tff(fact_1239_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_1240_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_1241_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ) ).

% power_less_imp_less_exp
tff(fact_1242_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N3: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N3)) ) ) ) ).

% power_strict_increasing
tff(fact_1243_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_1244_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_1245_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),Q2) )
     => ( 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),Q2),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_1246_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),Q2) )
     => ( 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),Q2),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_1247_nat__mod__eq__lemma,axiom,
    ! [Xa: nat,Na: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Y),Na) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xa)
       => ? [Q3: nat] : Xa = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_1248_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xa)),archimedean_ceiling(A,Y))) ) ).

% ceiling_add_le
tff(fact_1249_real__of__nat__div4,axiom,
    ! [Na: nat,Xa: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),Xa))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,semiring_1_of_nat(real),Xa))) ).

% real_of_nat_div4
tff(fact_1250_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_1251_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_1252_reals__Archimedean3,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ! [Y2: real] :
        ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),Xa)) ) ).

% reals_Archimedean3
tff(fact_1253_zdiv__mult__self,axiom,
    ! [M: int,A2: 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),A2),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),A2),M)),Na) ) ) ).

% zdiv_mult_self
tff(fact_1254_zdiv__mono__strict,axiom,
    ! [A3: int,B3: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),B3)
     => ( 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),A3),Na) = zero_zero(int) )
         => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B3),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),A3),Na)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B3),Na)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_1255_le__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% le_log2_of_power
tff(fact_1256_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] :
          aa(num,A,numeral_numeral(A),aa(num,num,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_1257_real__nat__ceiling__ge,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,semiring_1_of_nat(real),aa(int,nat,nat2,archimedean_ceiling(real,Xa)))) ).

% real_nat_ceiling_ge
tff(fact_1258_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
        <=> $ite(
              aa(A,$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),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% divide_le_eq
tff(fact_1259_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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),A2),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ).

% le_divide_eq
tff(fact_1260_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_left_mono
tff(fact_1261_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% neg_divide_le_eq
tff(fact_1262_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% neg_le_divide_eq
tff(fact_1263_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% pos_divide_le_eq
tff(fact_1264_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% pos_le_divide_eq
tff(fact_1265_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xa: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),Z) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1266_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1267_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1268_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na) )
              <=> ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1269_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
               => ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1270_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)) ) ).

% self_le_ge2_pow
tff(fact_1271_power2__nat__le__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% power2_nat_le_eq_le
tff(fact_1272_power2__nat__le__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% power2_nat_le_imp_le
tff(fact_1273_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ) ).

% power_Suc_less
tff(fact_1274_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N3: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1275_one__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) ) ) ).

% one_power2
tff(fact_1276_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ) ).

% one_less_power
tff(fact_1277_td__gal,axiom,
    ! [C2: nat,B2: nat,A2: 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),B2),C2)),A2)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),C2)) ) ) ).

% td_gal
tff(fact_1278_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q2)
     => ( 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),Q2))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)),Na) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1279_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)))
             => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)))),one_one(A)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_1280_realpow__pos__nth,axiom,
    ! [Na: nat,A2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ? [R4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R4),Na) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_1281_realpow__pos__nth__unique,axiom,
    ! [Na: nat,A2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ? [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),X),Na) = A2 )
            & ! [Y2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y2)
                  & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y2),Na) = A2 ) )
               => ( Y2 = X ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1282_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_1283_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_1284_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_1285_log2__of__power__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ).

% log2_of_power_le
tff(fact_1286_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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),B2),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,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),aa(A,$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_1287_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$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),B2),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)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$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_1288_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% zero_le_power2
tff(fact_1289_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
             => ( Xa = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_1290_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ).

% power2_le_imp_le
tff(fact_1291_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na)) ) ) ) ) ).

% power_strict_mono
tff(fact_1292_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$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),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_1293_log__base__pow,axiom,
    ! [A2: real,Na: nat,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( aa(real,real,log(aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),Na)),Xa) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A2),Xa)),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ).

% log_base_pow
tff(fact_1294_log__nat__power,axiom,
    ! [Xa: real,B2: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,log(B2),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B2),Xa)) ) ) ).

% log_nat_power
tff(fact_1295_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ) ).

% power2_less_imp_less
tff(fact_1296_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% sum_power2_ge_zero
tff(fact_1297_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A))
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_1298_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ) ).

% zero_le_even_power'
tff(fact_1299_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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% two_pow_div_gt_le
tff(fact_1300_z1pmod2,axiom,
    ! [B2: 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))),B2)),one_one(int))),aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) ).

% z1pmod2
tff(fact_1301_z1pdiv2,axiom,
    ! [B2: 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))),B2)),one_one(int))),aa(num,int,numeral_numeral(int),bit0(one2))) = B2 ).

% z1pdiv2
tff(fact_1302_add__0__iff,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [B2: A,A2: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% add_0_iff
tff(fact_1303_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W: A,Y: A,Xa: 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),Xa),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),Xa),Y)) )
        <=> ( ( W = Xa )
            | ( Y = Z ) ) ) ) ).

% crossproduct_eq
tff(fact_1304_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ( A2 != B2 )
            & ( C2 != D2 ) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),D2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% crossproduct_noteq
tff(fact_1305_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),A2: nat,B2: nat] :
      ( ! [A5: nat,B6: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),P,A5),B6)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),P,B6),A5) )
     => ( ! [A5: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,A5),zero_zero(nat))
       => ( ! [A5: nat,B6: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,A5),B6)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,A5),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A5),B6)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,A2),B2) ) ) ) ).

% Euclid_induct
tff(fact_1306_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% sum_squares_bound
tff(fact_1307_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),times_times(real),A2),C2))),aa(real,real,aa(real,fun(real,real),times_times(real),B2),D2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),C2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_1308_p1mod22k_H,axiom,
    ! [B2: 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))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),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),B2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)))) ).

% p1mod22k'
tff(fact_1309_p1mod22k,axiom,
    ! [B2: 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))),B2)),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,aa(int,fun(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),B2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)))),one_one(int)) ).

% p1mod22k
tff(fact_1310_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Xa),M) )
              | ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Xa),M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_1311_not__mod__2__eq__0__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1312_not__mod__2__eq__1__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1313_vebt__buildup__bound,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V8346862874174094_d_u_p(Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))),U)) ) ).

% vebt_buildup_bound
tff(fact_1314_Tbuildupi__buildupi_H,axiom,
    ! [Na: nat] : aa(nat,int,semiring_1_of_nat(int),vEBT_V441764108873111860ildupi(Na)) = vEBT_V9176841429113362141ildupi(Na) ).

% Tbuildupi_buildupi'
tff(fact_1315_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_1316_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),zero_zero(nat)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% set_bit_0
tff(fact_1317_two__realpow__ge__two,axiom,
    ! [Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Na))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)) ) ).

% two_realpow_ge_two
tff(fact_1318_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Na)),one_one(int)) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Na)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_1319_incr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X: int] :
            ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X5: int] :
              ( aa(int,$o,P,X5)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1320_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Na)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
         => ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Na)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_1321_subsetI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ! [X: A] :
          ( member(A,X,A3)
         => member(A,X,B3) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ).

% subsetI
tff(fact_1322_subset__antisym,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
       => ( A3 = B3 ) ) ) ).

% subset_antisym
tff(fact_1323_power__one__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),one_one(nat)) = A2 ) ).

% power_one_right
tff(fact_1324_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_1325_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_1326_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_1327_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_1328_word__less__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),one_one(word(A)))
        <=> ( Xa = zero_zero(word(A)) ) ) ) ).

% word_less_1
tff(fact_1329_div__of__0__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),zero_zero(word(A))),Na) = zero_zero(word(A)) ) ).

% div_of_0_id
tff(fact_1330_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_1331_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_1332_log__one,axiom,
    ! [A2: real] : aa(real,real,log(A2),one_one(real)) = zero_zero(real) ).

% log_one
tff(fact_1333_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_1334_zero__less__log__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,log(A2),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_1335_log__less__zero__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A2),Xa)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_1336_one__less__log__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,log(A2),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xa) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_1337_log__less__one__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A2),Xa)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),A2) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_1338_log__less__cancel__iff,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A2),Xa)),aa(real,real,log(A2),Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_1339_log__eq__one,axiom,
    ! [A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),A2) = one_one(real) ) ) ) ).

% log_eq_one
tff(fact_1340_zero__le__log__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,log(A2),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_1341_log__le__zero__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A2),Xa)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_1342_one__le__log__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,log(A2),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Xa) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_1343_log__le__one__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A2),Xa)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),A2) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_1344_log__le__cancel__iff,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A2),Xa)),aa(real,real,log(A2),Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_1345_log__pow__cancel,axiom,
    ! [A2: real,B2: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),B2)) = aa(nat,real,semiring_1_of_nat(real),B2) ) ) ) ).

% log_pow_cancel
tff(fact_1346_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_1347_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_1348_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_1349_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_1350_word__div__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)),divide_divide(word(A)),W),V) = zero_zero(word(A)) ) ) ) ).

% word_div_less
tff(fact_1351_word__less__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y) = zero_zero(word(A)) )
         => ( ( Y = zero_zero(word(A)) )
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ).

% word_less_div
tff(fact_1352_word__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))))
          | ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% word_overflow
tff(fact_1353_plus__one__helper2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Na)
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Na),one_one(word(A))) != zero_zero(word(A)) )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Na),one_one(word(A)))) ) ) ) ).

% plus_one_helper2
tff(fact_1354_word__div__lt__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y) = zero_zero(word(A)) ) ) ) ).

% word_div_lt_eq_0
tff(fact_1355_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_1356_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_1357_less__is__non__zero__p1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),K)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),one_one(word(A))) != zero_zero(word(A)) ) ) ) ).

% less_is_non_zero_p1
tff(fact_1358_plus__le__left__cancel__nowrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y4: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y4))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y4)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y4),Y) ) ) ) ) ).

% plus_le_left_cancel_nowrap
tff(fact_1359_div__less__dividend__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: word(A)] :
          ( ( Xa != zero_zero(word(A)) )
         => ( ( Na != one_one(word(A)) )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Na)),Xa) ) ) ) ).

% div_less_dividend_word
tff(fact_1360_word__mod__by__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),K),zero_zero(word(A))) = K ) ).

% word_mod_by_0
tff(fact_1361_div__by__0__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),zero_zero(word(A))) = zero_zero(word(A)) ) ).

% div_by_0_word
tff(fact_1362_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_1363_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_1364_neq__0__no__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
         => ( ( Xa != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y) != zero_zero(word(A)) ) ) ) ) ).

% neq_0_no_wrap
tff(fact_1365_word__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
         => ( ( Y != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))) != zero_zero(word(A)) ) ) ) ) ).

% word_plus_one_nonzero
tff(fact_1366_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_1367_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_1368_lt1__neq0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),Xa)
        <=> ( Xa != zero_zero(word(A)) ) ) ) ).

% lt1_neq0
tff(fact_1369_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_1370_zero__one__enat__neq_I1_J,axiom,
    zero_zero(extended_enat) != one_one(extended_enat) ).

% zero_one_enat_neq(1)
tff(fact_1371_pred__subset__eq,axiom,
    ! [A: $tType,R2: set(A),S: set(A)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),R2)),aTP_Lamp_a(set(A),fun(A,$o),S))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),R2),S) ) ).

% pred_subset_eq
tff(fact_1372_bot__enat__def,axiom,
    bot_bot(extended_enat) = zero_zero(extended_enat) ).

% bot_enat_def
tff(fact_1373_word__greater__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),A2)
        <=> ( A2 != zero_zero(word(A)) ) ) ) ).

% word_greater_zero_iff
tff(fact_1374_word__gt__a__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Na: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),Na)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Na) ) ) ).

% word_gt_a_gt_0
tff(fact_1375_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_1376_word__le__less__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
        <=> ( ( Xa = Y )
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ).

% word_le_less_eq
tff(fact_1377_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_1378_word__coorder_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),zero_zero(word(A))) ) ).

% word_coorder.extremum_strict
tff(fact_1379_word__not__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),zero_zero(word(A))) ) ).

% word_not_simps(1)
tff(fact_1380_word__not__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Xa) ) ) ).

% word_not_le
tff(fact_1381_word__le__not__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),B2),A2)
        <=> ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2) ) ) ).

% word_le_not_less
tff(fact_1382_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_1383_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_1384_in__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),Xa: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( member(A,Xa,A3)
       => member(A,Xa,B3) ) ) ).

% in_mono
tff(fact_1385_subsetD,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( member(A,C2,A3)
       => member(A,C2,B3) ) ) ).

% subsetD
tff(fact_1386_equalityE,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 = B3 )
     => ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
         => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3) ) ) ).

% equalityE
tff(fact_1387_subset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
    <=> ! [X4: A] :
          ( member(A,X4,A3)
         => member(A,X4,B3) ) ) ).

% subset_eq
tff(fact_1388_equalityD1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 = B3 )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ).

% equalityD1
tff(fact_1389_Set_OequalityD2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 = B3 )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3) ) ).

% Set.equalityD2
tff(fact_1390_subset__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
    <=> ! [T2: A] :
          ( member(A,T2,A3)
         => member(A,T2,B3) ) ) ).

% subset_iff
tff(fact_1391_subset__refl,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),A3) ).

% subset_refl
tff(fact_1392_Collect__mono,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(A,$o,P,X)
         => aa(A,$o,Q,X) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,P)),collect(A,Q)) ) ).

% Collect_mono
tff(fact_1393_subset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3) ) ) ).

% subset_trans
tff(fact_1394_set__eq__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 = B3 )
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3) ) ) ).

% set_eq_subset
tff(fact_1395_less__eq__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
    <=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A3)),aTP_Lamp_a(set(A),fun(A,$o),B3)) ) ).

% less_eq_set_def
tff(fact_1396_Collect__mono__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,P)),collect(A,Q))
    <=> ! [X4: A] :
          ( aa(A,$o,P,X4)
         => aa(A,$o,Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_1397_complete__real,axiom,
    ! [S: set(real)] :
      ( ? [X5: real] : member(real,X5,S)
     => ( ? [Z5: real] :
          ! [X: real] :
            ( member(real,X,S)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Z5) )
       => ? [Y3: real] :
            ( ! [X5: real] :
                ( member(real,X5,S)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X5),Y3) )
            & ! [Z5: real] :
                ( ! [X: real] :
                    ( member(real,X,S)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Z5) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),Z5) ) ) ) ) ).

% complete_real
tff(fact_1398_le__refl,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),Na) ).

% le_refl
tff(fact_1399_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_1400_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_1401_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_1402_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_1403_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),B2) )
       => ? [X: nat] :
            ( aa(nat,$o,P,X)
            & ! [Y2: nat] :
                ( aa(nat,$o,P,Y2)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y2),X) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_1404_verit__la__generic,axiom,
    ! [A2: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),Xa)
      | ( A2 = Xa )
      | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),A2) ) ).

% verit_la_generic
tff(fact_1405_word__plus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Z),Xa)) ) ) ) ).

% word_plus_mono_left
tff(fact_1406_word__add__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),W: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),W)),Xa)
         => ( aa(word(A),$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),Xa) ) ) ) ).

% word_add_increasing
tff(fact_1407_word__plus__mcs__4_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),V: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),V)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),W))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),V))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),V),W) ) ) ) ).

% word_plus_mcs_4'
tff(fact_1408_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_1409_word__plus__mcs__4,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),Xa: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),V),W) ) ) ) ).

% word_plus_mcs_4
tff(fact_1410_word__plus__mcs__3,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A),Xa: word(A)] :
          ( aa(word(A),$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),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)) ) ) ) ).

% word_plus_mcs_3
tff(fact_1411_word__sub__mono2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A),D2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),C2),D2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),A2)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),B2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),D2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),C2),D2))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),B2),D2) ) ) ) ) ) ).

% word_sub_mono2
tff(fact_1412_word__random,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),X6: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),X6))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),X6)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),Xa)) ) ) ) ).

% word_random
tff(fact_1413_word__div__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Na),one_one(word(A))) = Na ) ).

% word_div_1
tff(fact_1414_word__plus__strict__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z)) ) ) ) ).

% word_plus_strict_mono_right
tff(fact_1415_plus__one__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Na),one_one(word(A))))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Na) ) ) ).

% plus_one_helper
tff(fact_1416_word__le__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Na: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Na))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),Na)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),A2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),A2)),one_one(word(A)))) ) ) ) ).

% word_le_plus_1
tff(fact_1417_word__le__plus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),C2),B2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2)) ) ) ) ).

% word_le_plus
tff(fact_1418_constraint__expand,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Lower: word(A),Upper: word(A)] :
          ( member(word(A),Xa,collect(word(A),aa(word(A),fun(word(A),$o),aTP_Lamp_ak(word(A),fun(word(A),fun(word(A),$o)),Lower),Upper)))
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Lower),Xa)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Upper) ) ) ) ).

% constraint_expand
tff(fact_1419_Collect__subset,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_al(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))),A3) ).

% Collect_subset
tff(fact_1420_plus__le__left__cancel__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y4: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y4)),Xa)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)),Xa)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y4)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y4),Y) ) ) ) ) ).

% plus_le_left_cancel_wrap
tff(fact_1421_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_1422_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_1423_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_1424_word__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),Xa: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),one_one(word(A)))),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2) ) ) ) ).

% word_1_0
tff(fact_1425_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_1426_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_1427_real__arch__pow,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),N)) ) ).

% real_arch_pow
tff(fact_1428_real__arch__pow__inv,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_1429_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_1430_nat__one__as__int,axiom,
    one_one(nat) = aa(int,nat,nat2,one_one(int)) ).

% nat_one_as_int
tff(fact_1431_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q6: fun(A,$o)] :
          ( ? [Z5: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z5),X)
             => ( aa(A,$o,P,X)
              <=> aa(A,$o,P3,X) ) )
         => ( ? [Z5: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z5),X)
               => ( aa(A,$o,Q,X)
                <=> aa(A,$o,Q6,X) ) )
           => ? [Z3: A] :
              ! [X5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
               => ( ( aa(A,$o,P,X5)
                    & aa(A,$o,Q,X5) )
                <=> ( aa(A,$o,P3,X5)
                    & aa(A,$o,Q6,X5) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_1432_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q6: fun(A,$o)] :
          ( ? [Z5: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z5),X)
             => ( aa(A,$o,P,X)
              <=> aa(A,$o,P3,X) ) )
         => ( ? [Z5: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z5),X)
               => ( aa(A,$o,Q,X)
                <=> aa(A,$o,Q6,X) ) )
           => ? [Z3: A] :
              ! [X5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
               => ( ( aa(A,$o,P,X5)
                    | aa(A,$o,Q,X5) )
                <=> ( aa(A,$o,P3,X5)
                    | aa(A,$o,Q6,X5) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_1433_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => ( X5 != T ) ) ) ).

% pinf(3)
tff(fact_1434_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => ( X5 != T ) ) ) ).

% pinf(4)
tff(fact_1435_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),T) ) ) ).

% pinf(5)
tff(fact_1436_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),T),X5) ) ) ).

% pinf(7)
tff(fact_1437_pinf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: B] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => ( F3 = F3 ) ) ) ).

% pinf(11)
tff(fact_1438_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q6: fun(A,$o)] :
          ( ? [Z5: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z5)
             => ( aa(A,$o,P,X)
              <=> aa(A,$o,P3,X) ) )
         => ( ? [Z5: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z5)
               => ( aa(A,$o,Q,X)
                <=> aa(A,$o,Q6,X) ) )
           => ? [Z3: A] :
              ! [X5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
               => ( ( aa(A,$o,P,X5)
                    & aa(A,$o,Q,X5) )
                <=> ( aa(A,$o,P3,X5)
                    & aa(A,$o,Q6,X5) ) ) ) ) ) ) ).

% minf(1)
tff(fact_1439_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q6: fun(A,$o)] :
          ( ? [Z5: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z5)
             => ( aa(A,$o,P,X)
              <=> aa(A,$o,P3,X) ) )
         => ( ? [Z5: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z5)
               => ( aa(A,$o,Q,X)
                <=> aa(A,$o,Q6,X) ) )
           => ? [Z3: A] :
              ! [X5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
               => ( ( aa(A,$o,P,X5)
                    | aa(A,$o,Q,X5) )
                <=> ( aa(A,$o,P3,X5)
                    | aa(A,$o,Q6,X5) ) ) ) ) ) ) ).

% minf(2)
tff(fact_1440_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => ( X5 != T ) ) ) ).

% minf(3)
tff(fact_1441_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => ( X5 != T ) ) ) ).

% minf(4)
tff(fact_1442_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),T) ) ) ).

% minf(5)
tff(fact_1443_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),T),X5) ) ) ).

% minf(7)
tff(fact_1444_minf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: B] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => ( F3 = F3 ) ) ) ).

% minf(11)
tff(fact_1445_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_1446_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_1447_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_1448_log__base__change,axiom,
    ! [A2: real,B2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(B2),Xa) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A2),Xa)),aa(real,real,log(A2),B2)) ) ) ) ).

% log_base_change
tff(fact_1449_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_1450_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_1451_two__realpow__ge__one,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)) ).

% two_realpow_ge_one
tff(fact_1452_log__mult,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
           => ( aa(real,real,log(A2),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(A2),Xa)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_1453_less__log__of__power,axiom,
    ! [B2: real,Na: nat,M: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Na)),M)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B2),M)) ) ) ).

% less_log_of_power
tff(fact_1454_log__of__power__eq,axiom,
    ! [M: nat,B2: real,Na: nat] :
      ( ( aa(nat,real,semiring_1_of_nat(real),M) = aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Na) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( aa(nat,real,semiring_1_of_nat(real),Na) = aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ) ).

% log_of_power_eq
tff(fact_1455_le__log__of__power,axiom,
    ! [B2: real,Na: nat,M: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Na)),M)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B2),M)) ) ) ).

% le_log_of_power
tff(fact_1456_p2__eq__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) = one_one(word(A)) )
        <=> ( Na = zero_zero(nat) ) ) ) ).

% p2_eq_1
tff(fact_1457_verit__le__mono__div,axiom,
    ! [A3: nat,B3: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
     => ( 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),A3),Na)),
                $ite(aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B3),Na) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B3),Na)) ) ) ).

% verit_le_mono_div
tff(fact_1458_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
       => ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_1459_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
       => ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_1460_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),T) ) ) ).

% pinf(6)
tff(fact_1461_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T),X5) ) ) ).

% pinf(8)
tff(fact_1462_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),T) ) ) ).

% minf(6)
tff(fact_1463_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T),X5) ) ) ).

% minf(8)
tff(fact_1464_log__of__power__less,axiom,
    ! [M: nat,B2: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Na))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_of_power_less
tff(fact_1465_log__of__power__le,axiom,
    ! [M: nat,B2: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Na))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_of_power_le
tff(fact_1466_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,aa(nat,fun(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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),K))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),K)) ) ) ).

% power_2_mult_step_le
tff(fact_1467_linear__plus__1__le__power,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),Xa)),one_one(real))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),one_one(real))),Na)) ) ).

% linear_plus_1_le_power
tff(fact_1468_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),aa(num,num,bit1,one2)) ) ) ).

% mod_exhaust_less_4
tff(fact_1469_buildup__build__time,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_V8346862874174094_d_u_p(Na)),vEBT_V8646137997579335489_i_l_d(Na)) ).

% buildup_build_time
tff(fact_1470_cnt__non__neg,axiom,
    ! [T: vEBT_VEBT] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),vEBT_VEBT_cnt(T)) ).

% cnt_non_neg
tff(fact_1471_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),aa(num,num,bit1,one2)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
tff(fact_1472_log__ceil__idem,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),Xa)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(int,real,ring_1_of_int(real),archimedean_ceiling(real,Xa)))) ) ) ).

% log_ceil_idem
tff(fact_1473_ceiling__log2__div2,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))))),one_one(int)) ) ) ).

% ceiling_log2_div2
tff(fact_1474_ceiling__log__eq__powr__iff,axiom,
    ! [Xa: real,B2: real,K: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( ( archimedean_ceiling(real,aa(real,real,log(B2),Xa)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,aa(nat,real,semiring_1_of_nat(real),K))),Xa)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),powr(real,B2,aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_1475_minus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( minus(A)
     => ! [A3: fun(B,A),B3: fun(B,A),Xa: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),minus_minus(fun(B,A)),A3),B3),Xa) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,A3,Xa)),aa(B,A,B3,Xa)) ) ).

% minus_apply
tff(fact_1476_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_1477_predicate1I,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(A,$o,P,X)
         => aa(A,$o,Q,X) )
     => aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q) ) ).

% predicate1I
tff(fact_1478_diff__self,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),A2) = zero_zero(A) ) ).

% diff_self
tff(fact_1479_diff__0__right,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),zero_zero(A)) = A2 ) ).

% diff_0_right
tff(fact_1480_zero__diff,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% zero_diff
tff(fact_1481_diff__zero,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),zero_zero(A)) = A2 ) ).

% diff_zero
tff(fact_1482_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),A2) = zero_zero(A) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1483_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = A2 ) ).

% add_diff_cancel
tff(fact_1484_diff__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),B2) = A2 ) ).

% diff_add_cancel
tff(fact_1485_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C2: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ).

% add_diff_cancel_left
tff(fact_1486_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),A2) = B2 ) ).

% add_diff_cancel_left'
tff(fact_1487_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ).

% add_diff_cancel_right
tff(fact_1488_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = A2 ) ).

% add_diff_cancel_right'
tff(fact_1489_minus__mod__self2,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% minus_mod_self2
tff(fact_1490_diff__0__eq__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),Na) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_1491_diff__self__eq__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),M) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_1492_of__int__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),minus_minus(int),W),Z)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_diff
tff(fact_1493_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

% diff_diff_left
tff(fact_1494_diff__diff__cancel,axiom,
    ! [I: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_1495_powr__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Z: A] : powr(A,zero_zero(A),Z) = zero_zero(A) ) ).

% powr_0
tff(fact_1496_powr__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [W: A,Z: A] :
          ( ( powr(A,W,Z) = zero_zero(A) )
        <=> ( W = zero_zero(A) ) ) ) ).

% powr_eq_0_iff
tff(fact_1497_powr__one__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [A2: A] : powr(A,one_one(A),A2) = one_one(A) ) ).

% powr_one_eq_one
tff(fact_1498_of__int__ceiling__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( ( aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xa)) = Xa )
        <=> ? [N2: int] : Xa = aa(int,A,ring_1_of_int(A),N2) ) ) ).

% of_int_ceiling_cancel
tff(fact_1499_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_1500_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% diff_ge_0_iff_ge
tff(fact_1501_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% diff_gt_0_iff_gt
tff(fact_1502_zero__comp__diff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% zero_comp_diff_simps(2)
tff(fact_1503_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = zero_zero(A) ) ).

% diff_add_zero
tff(fact_1504_diff__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),one_one(A)) = zero_zero(A) ) ) ).

% diff_numeral_special(9)
tff(fact_1505_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),B2) = A2 ) ) ) ).

% le_add_diff_inverse2
tff(fact_1506_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = A2 ) ) ) ).

% le_add_diff_inverse
tff(fact_1507_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A2: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).

% left_diff_distrib_numeral
tff(fact_1508_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),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_1509_zero__less__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% zero_less_diff
tff(fact_1510_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_1511_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_1512_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_1513_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_1514_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_1515_diff__is__0__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% diff_is_0_eq
tff(fact_1516_diff__is__0__eq_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_1517_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_1518_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_1519_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1520_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1521_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_1522_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_1523_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_1524_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xa: A] :
          powr(A,Xa,zero_zero(A)) = $ite(Xa = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% powr_zero_eq_one
tff(fact_1525_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_1526_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_1527_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_1528_of__int__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int,Na: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),Na)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),Z)),Na) ) ).

% of_int_power
tff(fact_1529_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [B2: int,W: nat,Xa: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W) = aa(int,A,ring_1_of_int(A),Xa) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) = Xa ) ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_1530_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xa: int,B2: int,W: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Xa) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W) )
        <=> ( Xa = aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_1531_ceiling__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,Xa)),Z) ) ).

% ceiling_diff_of_int
tff(fact_1532_powr__gt__zero,axiom,
    ! [Xa: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),powr(real,Xa,A2))
    <=> ( Xa != zero_zero(real) ) ) ).

% powr_gt_zero
tff(fact_1533_powr__nonneg__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,A2,Xa)),zero_zero(real))
    <=> ( A2 = zero_zero(real) ) ) ).

% powr_nonneg_iff
tff(fact_1534_powr__less__cancel__iff,axiom,
    ! [Xa: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xa,A2)),powr(real,Xa,B2))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2) ) ) ).

% powr_less_cancel_iff
tff(fact_1535_ceiling__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,Xa)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_diff_numeral
tff(fact_1536_ceiling__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,Xa)),one_one(int)) ) ).

% ceiling_diff_one
tff(fact_1537_ceiling__add__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xa)),Z) ) ).

% ceiling_add_of_int
tff(fact_1538_powr__eq__one__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( ( powr(real,A2,Xa) = one_one(real) )
      <=> ( Xa = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_1539_powr__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,one_one(real)) = Xa ) ) ).

% powr_one
tff(fact_1540_powr__one__gt__zero__iff,axiom,
    ! [Xa: real] :
      ( ( powr(real,Xa,one_one(real)) = Xa )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% powr_one_gt_zero_iff
tff(fact_1541_powr__le__cancel__iff,axiom,
    ! [Xa: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A2)),powr(real,Xa,B2))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2) ) ) ).

% powr_le_cancel_iff
tff(fact_1542_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,aa(real,fun(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_1543_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_1544_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_1545_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_1546_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_1547_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_1548_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_1549_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_1550_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_1551_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_1552_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_1553_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_1554_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_1555_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xa: num,Na: nat,Y: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Na) = aa(int,A,ring_1_of_int(A),Y) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na) = Y ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_1556_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,Xa: num,Na: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Na) )
        <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na) ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_1557_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: int,B2: int,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_1558_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,Xa: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)),aa(int,A,ring_1_of_int(A),Xa))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),Xa) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_1559_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,Xa: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)),aa(int,A,ring_1_of_int(A),Xa))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),Xa) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_1560_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: int,B2: int,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_1561_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_1562_log__powr__cancel,axiom,
    ! [A2: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),powr(real,A2,Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_1563_powr__log__cancel,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( powr(real,A2,aa(real,real,log(A2),Xa)) = Xa ) ) ) ) ).

% powr_log_cancel
tff(fact_1564_powr__numeral,axiom,
    ! [Xa: real,Na: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(num,real,numeral_numeral(real),Na)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),Na)) ) ) ).

% powr_numeral
tff(fact_1565_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))) ) ) ).

% ceiling_less_numeral
tff(fact_1566_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),Xa) ) ) ).

% numeral_le_ceiling
tff(fact_1567_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,Na: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Na)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na)),A2) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_1568_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xa: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na)) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_1569_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,Na: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Na)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na)),A2) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_1570_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xa: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na)) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_1571_predicate1D,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),Xa: A] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q)
     => ( aa(A,$o,P,Xa)
       => aa(A,$o,Q,Xa) ) ) ).

% predicate1D
tff(fact_1572_rev__predicate1D,axiom,
    ! [A: $tType,P: fun(A,$o),Xa: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,Xa)
     => ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q)
       => aa(A,$o,Q,Xa) ) ) ).

% rev_predicate1D
tff(fact_1573_powr__diff,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [W: A,Z1: A,Z2: A] : powr(A,W,aa(A,A,aa(A,fun(A,A),minus_minus(A),Z1),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),powr(A,W,Z1)),powr(A,W,Z2)) ) ).

% powr_diff
tff(fact_1574_fun__diff__def,axiom,
    ! [B: $tType,A: $tType] :
      ( minus(B)
     => ! [A3: fun(A,B),B3: fun(A,B),X5: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),minus_minus(fun(A,B)),A3),B3),X5) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A3,X5)),aa(A,B,B3,X5)) ) ).

% fun_diff_def
tff(fact_1575_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),K)),J) ).

% diff_commute
tff(fact_1576_powr__powr__swap,axiom,
    ! [Xa: real,A2: real,B2: real] : powr(real,powr(real,Xa,A2),B2) = powr(real,powr(real,Xa,B2),A2) ).

% powr_powr_swap
tff(fact_1577_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( ( A2 = B2 )
          <=> ( C2 = D2 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_1578_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_1579_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ) ) ).

% of_nat_diff
tff(fact_1580_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),Z3)) ) ).

% ex_le_of_int
tff(fact_1581_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),Z3)) ) ).

% ex_less_of_int
tff(fact_1582_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z3)),Xa) ) ).

% ex_of_int_less
tff(fact_1583_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: int,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Xa)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(int,A,ring_1_of_int(A),Xa)) ) ).

% mult_of_int_commute
tff(fact_1584_powr__powr,axiom,
    ! [Xa: real,A2: real,B2: real] : powr(real,powr(real,Xa,A2),B2) = powr(real,Xa,aa(real,real,aa(real,fun(real,real),times_times(real),A2),B2)) ).

% powr_powr
tff(fact_1585_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_1586_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)) ) ) ) ).

% diff_mono
tff(fact_1587_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)) ) ) ).

% diff_left_mono
tff(fact_1588_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) ) ) ).

% diff_right_mono
tff(fact_1589_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1590_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),D2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)) ) ) ) ).

% diff_strict_mono
tff(fact_1591_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2) ) ) ) ).

% diff_eq_diff_less
tff(fact_1592_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)) ) ) ).

% diff_strict_left_mono
tff(fact_1593_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) ) ) ).

% diff_strict_right_mono
tff(fact_1594_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,K: A,A2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_1595_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = C2 )
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) ) ) ) ).

% diff_eq_eq
tff(fact_1596_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = C2 ) ) ) ).

% eq_diff_eq
tff(fact_1597_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ).

% add_diff_eq
tff(fact_1598_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_diff_eq2
tff(fact_1599_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_add_eq
tff(fact_1600_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_1601_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) = A2 )
         => ( C2 = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ) ) ).

% add_implies_diff
tff(fact_1602_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% diff_diff_eq
tff(fact_1603_add__diff__add,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,C2: A,B2: A,D2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2)) ) ).

% add_diff_add
tff(fact_1604_left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% left_diff_distrib
tff(fact_1605_right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% right_diff_distrib
tff(fact_1606_left__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [B2: A,C2: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) ) ).

% left_diff_distrib'
tff(fact_1607_right__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% right_diff_distrib'
tff(fact_1608_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D4: A,Q: fun(A,$o)] :
          ( ! [X: A,K2: A] :
              ( aa(A,$o,P,X)
            <=> aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4))) )
         => ( ! [X: A,K2: A] :
                ( aa(A,$o,Q,X)
              <=> aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4))) )
           => ! [X5: A,K5: A] :
                ( ( aa(A,$o,P,X5)
                  | aa(A,$o,Q,X5) )
              <=> ( aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D4)))
                  | aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D4))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_1609_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D4: A,Q: fun(A,$o)] :
          ( ! [X: A,K2: A] :
              ( aa(A,$o,P,X)
            <=> aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4))) )
         => ( ! [X: A,K2: A] :
                ( aa(A,$o,Q,X)
              <=> aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4))) )
           => ! [X5: A,K5: A] :
                ( ( aa(A,$o,P,X5)
                  & aa(A,$o,Q,X5) )
              <=> ( aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D4)))
                  & aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D4))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_1610_diff__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ).

% diff_divide_distrib
tff(fact_1611_mod__diff__right__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) ) ).

% mod_diff_right_eq
tff(fact_1612_mod__diff__left__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),C2)),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) ) ).

% mod_diff_left_eq
tff(fact_1613_mod__diff__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,A4: A,B2: A,B5: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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),B2),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B5),C2) )
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A4),B5)),C2) ) ) ) ) ).

% mod_diff_cong
tff(fact_1614_mod__diff__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) ) ).

% mod_diff_eq
tff(fact_1615_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),zero_zero(nat)) = M ).

% minus_nat.diff_0
tff(fact_1616_diffs0__imp__equal,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na) = zero_zero(nat) )
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M) = zero_zero(nat) )
       => ( M = Na ) ) ) ).

% diffs0_imp_equal
tff(fact_1617_diff__less__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),Na)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M)) ) ) ).

% diff_less_mono2
tff(fact_1618_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),Na)),K) ) ).

% less_imp_diff_less
tff(fact_1619_diff__diff__less,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Na) ) ) ).

% diff_diff_less
tff(fact_1620_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na) ).

% Nat.diff_cancel
tff(fact_1621_diff__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na) ).

% diff_cancel2
tff(fact_1622_diff__add__inverse,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)),Na) = M ).

% diff_add_inverse
tff(fact_1623_diff__add__inverse2,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),Na) = M ).

% diff_add_inverse2
tff(fact_1624_diff__le__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),Na)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M)) ) ).

% diff_le_mono2
tff(fact_1625_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),C2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C2),A2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C2),B2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A2) ) ) ) ).

% le_diff_iff'
tff(fact_1626_diff__le__self,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),M) ).

% diff_le_self
tff(fact_1627_diff__le__mono,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),L)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),L)) ) ).

% diff_le_mono
tff(fact_1628_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1629_le__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% le_diff_iff
tff(fact_1630_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K) )
        <=> ( M = Na ) ) ) ) ).

% eq_diff_iff
tff(fact_1631_diff__mult__distrib,axiom,
    ! [M: nat,Na: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),K)) ).

% diff_mult_distrib
tff(fact_1632_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na)) ).

% diff_mult_distrib2
tff(fact_1633_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,aa(A,fun(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_1634_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xa))),one_one(A))),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xa))) ) ) ).

% ceiling_correct
tff(fact_1635_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),Z))
           => ( archimedean_ceiling(A,Xa) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_1636_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A2: int] :
          ( ( archimedean_ceiling(A,Xa) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),A2)),one_one(A))),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),A2)) ) ) ) ).

% ceiling_eq_iff
tff(fact_1637_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),T: A] :
          ( aa(int,$o,P,archimedean_ceiling(A,T))
        <=> ! [I2: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),I2)),one_one(A))),T)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T),aa(int,A,ring_1_of_int(A),I2)) )
             => aa(int,$o,P,I2) ) ) ) ).

% ceiling_split
tff(fact_1638_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))) ) ) ).

% ceiling_less_iff
tff(fact_1639_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xa) ) ) ).

% le_ceiling_iff
tff(fact_1640_powr__non__neg,axiom,
    ! [A2: real,Xa: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,A2,Xa)),zero_zero(real)) ).

% powr_non_neg
tff(fact_1641_powr__less__mono2__neg,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Y,A2)),powr(real,Xa,A2)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_1642_powr__mono2,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_mono2
tff(fact_1643_powr__ge__pzero,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),powr(real,Xa,Y)) ).

% powr_ge_pzero
tff(fact_1644_powr__less__cancel,axiom,
    ! [Xa: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xa,A2)),powr(real,Xa,B2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2) ) ) ).

% powr_less_cancel
tff(fact_1645_powr__less__mono,axiom,
    ! [A2: real,B2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xa,A2)),powr(real,Xa,B2)) ) ) ).

% powr_less_mono
tff(fact_1646_powr__mono,axiom,
    ! [A2: real,B2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A2)),powr(real,Xa,B2)) ) ) ).

% powr_mono
tff(fact_1647_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),zero_zero(A)) ) ) ).

% le_iff_diff_le_0
tff(fact_1648_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),zero_zero(A)) ) ) ).

% less_iff_diff_less_0
tff(fact_1649_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Na: A,J: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Na)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Na),K)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1650_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),aa(A,A,aa(A,fun(A,A),minus_minus(A),Na),K)) ) ) ).

% add_le_imp_le_diff
tff(fact_1651_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1652_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1653_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1654_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1655_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1656_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1657_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1658_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1659_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2)) ) ) ).

% le_add_diff
tff(fact_1660_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),A2) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_1661_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ) ).

% le_diff_eq
tff(fact_1662_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% diff_le_eq
tff(fact_1663_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = A2 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1664_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% diff_less_eq
tff(fact_1665_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ) ).

% less_diff_eq
tff(fact_1666_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E: A,C2: A,B2: 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),A2),E)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),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,aa(A,fun(A,A),minus_minus(A),A2),B2)),E)),C2) = D2 ) ) ) ).

% eq_add_iff1
tff(fact_1667_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E: A,C2: A,B2: 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),A2),E)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),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,aa(A,fun(A,A),minus_minus(A),B2),A2)),E)),D2) ) ) ) ).

% eq_add_iff2
tff(fact_1668_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),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),Xa),Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)) ) ).

% square_diff_square_factored
tff(fact_1669_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Xa: A,Y: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),A2)),B2)) ) ).

% mult_diff_mult
tff(fact_1670_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% diff_shunt_var
tff(fact_1671_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xa))) ) ).

% le_of_int_ceiling
tff(fact_1672_diff__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),M) ) ) ).

% diff_less
tff(fact_1673_diff__add__0,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)) = zero_zero(nat) ).

% diff_add_0
tff(fact_1674_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ).

% less_diff_conv
tff(fact_1675_add__diff__inverse__nat,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_1676_less__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ) ).

% less_diff_iff
tff(fact_1677_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),A2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),C2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),C2)) ) ) ).

% diff_less_mono
tff(fact_1678_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ).

% le_diff_conv
tff(fact_1679_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_1680_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1681_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1682_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I) = K )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1683_nat__minus__mod,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),M))),M) = zero_zero(nat) ).

% nat_minus_mod
tff(fact_1684_mod__nat__sub,axiom,
    ! [Xa: nat,Z: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Z)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),Y)),Z) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),Y) ) ) ).

% mod_nat_sub
tff(fact_1685_mod__geq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),Na) ) ) ).

% mod_geq
tff(fact_1686_mod__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na),M,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),Na)) ).

% mod_if
tff(fact_1687_nat__minus__mod__plus__right,axiom,
    ! [Na: nat,Xa: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),Xa)),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),Xa),M) ).

% nat_minus_mod_plus_right
tff(fact_1688_le__mod__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),Na) ) ) ).

% le_mod_geq
tff(fact_1689_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q2)))),one_one(A))),Q2)),P2) ) ) ).

% ceiling_divide_lower
tff(fact_1690_powr__less__mono2,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xa,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_less_mono2
tff(fact_1691_powr__mono2_H,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Y,A2)),powr(real,Xa,A2)) ) ) ) ).

% powr_mono2'
tff(fact_1692_powr__inj,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( ( powr(real,A2,Xa) = powr(real,A2,Y) )
        <=> ( Xa = Y ) ) ) ) ).

% powr_inj
tff(fact_1693_gr__one__powr,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),powr(real,Xa,Y)) ) ) ).

% gr_one_powr
tff(fact_1694_powr__le1,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A2)),one_one(real)) ) ) ) ).

% powr_le1
tff(fact_1695_powr__mono__both,axiom,
    ! [A2: real,B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A2)),powr(real,Y,B2)) ) ) ) ) ).

% powr_mono_both
tff(fact_1696_ge__one__powr__ge__zero,axiom,
    ! [Xa: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),powr(real,Xa,A2)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_1697_powr__divide,axiom,
    ! [Xa: real,Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),Y),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,Xa,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_divide
tff(fact_1698_powr__mult,axiom,
    ! [Xa: real,Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y),A2) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,Xa,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_mult
tff(fact_1699_log__base__powr,axiom,
    ! [A2: real,B2: real,Xa: real] :
      ( ( A2 != zero_zero(real) )
     => ( aa(real,real,log(powr(real,A2,B2)),Xa) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A2),Xa)),B2) ) ) ).

% log_base_powr
tff(fact_1700_log__powr,axiom,
    ! [Xa: real,B2: real,Y: real] :
      ( ( Xa != zero_zero(real) )
     => ( aa(real,real,log(B2),powr(real,Xa,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),aa(real,real,log(B2),Xa)) ) ) ).

% log_powr
tff(fact_1701_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),E)),C2)),D2) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1702_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2))
        <=> aa(A,$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,aa(A,fun(A,A),minus_minus(A),B2),A2)),E)),D2)) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1703_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),E)),C2)),D2) ) ) ).

% less_add_iff1
tff(fact_1704_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2))
        <=> aa(A,$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,aa(A,fun(A,A),minus_minus(A),B2),A2)),E)),D2)) ) ) ).

% less_add_iff2
tff(fact_1705_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1706_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),Y)),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1707_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xa: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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_1708_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,Z: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = $ite(Z = zero_zero(A),A2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),Z)),B2)),Z)) ) ).

% add_divide_eq_if_simps(4)
tff(fact_1709_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),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),Xa),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),one_one(A))) ) ).

% square_diff_one_factored
tff(fact_1710_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),A2))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),A2) ) ) ).

% ceiling_le
tff(fact_1711_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% ceiling_le_iff
tff(fact_1712_zmde,axiom,
    ! [A: $tType] :
      ( ( group_add(A)
        & semiring_modulo(A) )
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) ) ).

% zmde
tff(fact_1713_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% minus_mult_div_eq_mod
tff(fact_1714_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% minus_mod_eq_mult_div
tff(fact_1715_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2) ) ).

% minus_mod_eq_div_mult
tff(fact_1716_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% minus_div_mult_eq_mod
tff(fact_1717_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),Xa) ) ) ).

% less_ceiling_iff
tff(fact_1718_powr__add,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xa: A,A2: A,B2: A] : powr(A,Xa,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),powr(A,Xa,A2)),powr(A,Xa,B2)) ) ).

% powr_add
tff(fact_1719_real__of__int__div4,axiom,
    ! [Na: int,Xa: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),Xa))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Na)),aa(int,real,ring_1_of_int(real),Xa))) ).

% real_of_int_div4
tff(fact_1720_nat__diff__split,axiom,
    ! [P: fun(nat,$o),A2: nat,B2: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
         => aa(nat,$o,P,zero_zero(nat)) )
        & ! [D5: nat] :
            ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D5) )
           => aa(nat,$o,P,D5) ) ) ) ).

% nat_diff_split
tff(fact_1721_nat__diff__split__asm,axiom,
    ! [P: fun(nat,$o),A2: nat,B2: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2))
    <=> ~ ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
            & ~ aa(nat,$o,P,zero_zero(nat)) )
          | ? [D5: nat] :
              ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D5) )
              & ~ aa(nat,$o,P,D5) ) ) ) ).

% nat_diff_split_asm
tff(fact_1722_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ) ).

% less_diff_conv2
tff(fact_1723_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na) )
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M) = Na ) ) ) ).

% nat_eq_add_iff1
tff(fact_1724_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na) )
      <=> ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),Na) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1725_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M)),Na) ) ) ).

% nat_le_add_iff1
tff(fact_1726_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),Na)) ) ) ).

% nat_le_add_iff2
tff(fact_1727_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M)),Na) ) ) ).

% nat_diff_add_eq1
tff(fact_1728_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),Na)) ) ) ).

% nat_diff_add_eq2
tff(fact_1729_mod__nat__add,axiom,
    ! [Xa: nat,Z: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Z)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),Y)),Z) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),Y)),Z),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),Y),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),Y)),Z)) ) ) ) ).

% mod_nat_add
tff(fact_1730_modulo__nat__def,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) = aa(nat,nat,aa(nat,fun(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_1731_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_1732_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_1733_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_1734_powr__realpow,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(nat,real,semiring_1_of_nat(real),Na)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na) ) ) ).

% powr_realpow
tff(fact_1735_less__log__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,log(B2),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,Y)),Xa) ) ) ) ).

% less_log_iff
tff(fact_1736_log__less__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B2),Xa)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),powr(real,B2,Y)) ) ) ) ).

% log_less_iff
tff(fact_1737_less__powr__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),powr(real,B2,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B2),Xa)),Y) ) ) ) ).

% less_powr_iff
tff(fact_1738_powr__less__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,Y)),Xa)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,log(B2),Xa)) ) ) ) ).

% powr_less_iff
tff(fact_1739_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [X: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X)),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int))))
          & ! [Y2: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y2)),Xa)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y2),one_one(int)))) )
             => ( Y2 = X ) ) ) ) ).

% floor_exists1
tff(fact_1740_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z3)),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z3),one_one(int)))) ) ) ).

% floor_exists
tff(fact_1741_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_1742_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,Xa: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(int,A,ring_1_of_int(A),Xa))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Na)),Xa) ) ) ).

% of_nat_less_of_int_iff
tff(fact_1743_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xa: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A)) ) ) ) ) ).

% frac_le_eq
tff(fact_1744_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xa: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A)) ) ) ) ) ).

% frac_less_eq
tff(fact_1745_power2__commute,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xa: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_commute
tff(fact_1746_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_1747_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_1748_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,Na: nat,M: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ) ) ).

% power_diff
tff(fact_1749_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,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),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_1750_real__of__int__div__aux,axiom,
    ! [Xa: int,D2: int] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Xa)),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),Xa),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),Xa),D2))),aa(int,real,ring_1_of_int(real),D2))) ).

% real_of_int_div_aux
tff(fact_1751_mult__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) = $ite(M = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),Na))) ).

% mult_eq_if
tff(fact_1752_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M)),Na) ) ) ).

% nat_less_add_iff1
tff(fact_1753_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),Na)) ) ) ).

% nat_less_add_iff2
tff(fact_1754_nat__mult__power__less__eq,axiom,
    ! [B2: nat,A2: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Na))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),M))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na))) ) ) ).

% nat_mult_power_less_eq
tff(fact_1755_powr__mult__base,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),Xa),powr(real,Xa,Y)) = powr(real,Xa,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Y)) ) ) ).

% powr_mult_base
tff(fact_1756_le__log__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,log(B2),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Y)),Xa) ) ) ) ).

% le_log_iff
tff(fact_1757_log__le__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B2),Xa)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),powr(real,B2,Y)) ) ) ) ).

% log_le_iff
tff(fact_1758_le__powr__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),powr(real,B2,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B2),Xa)),Y) ) ) ) ).

% le_powr_iff
tff(fact_1759_powr__le__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Y)),Xa)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,log(B2),Xa)) ) ) ) ).

% powr_le_iff
tff(fact_1760_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V: A,R: A,Sa: A] :
          ( aa(A,$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,$o),ord_less_eq(A),R),Sa)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),R),aa(A,A,aa(A,fun(A,A),minus_minus(A),V),U))),Sa))),V) ) ) ) ) ).

% scaling_mono
tff(fact_1761_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat,M: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_1762_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)))) ) ) ) ).

% power_diff_power_eq
tff(fact_1763_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [P2: A,M: nat] :
          aa(nat,A,aa(A,fun(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,aa(A,fun(nat,A),power_power(A),P2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))) ) ).

% power_eq_if
tff(fact_1764_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Na: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))),A2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na) ) ) ) ).

% power_minus_mult
tff(fact_1765_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Na))) ) ).

% diff_le_diff_pow
tff(fact_1766_power__sub,axiom,
    ! [Na: nat,M: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),A2)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A2),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A2),Na)) ) ) ) ).

% power_sub
tff(fact_1767_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$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),Q2)))),Q2)) ) ) ).

% ceiling_divide_upper
tff(fact_1768_add__log__eq__powr,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Y),aa(real,real,log(B2),Xa)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B2,Y)),Xa)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_1769_log__add__eq__powr,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(B2),Xa)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),powr(real,B2,Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_1770_nat__power__less__diff,axiom,
    ! [Na: nat,Q2: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Q2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na))) ) ).

% nat_power_less_diff
tff(fact_1771_power__minus__is__div,axiom,
    ! [B2: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)) ) ) ).

% power_minus_is_div
tff(fact_1772_nat__le__power__trans,axiom,
    ! [Na: nat,M: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% nat_le_power_trans
tff(fact_1773_power__mod__div,axiom,
    ! [Xa: 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),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))) ).

% power_mod_div
tff(fact_1774_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Na)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Na)),one_one(A)))
           => ( archimedean_ceiling(A,Xa) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Na),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_1775_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xa: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)),Y)) ) ).

% power2_diff
tff(fact_1776_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_1777_less__two__pow__divD,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))) ) ) ).

% less_two_pow_divD
tff(fact_1778_less__two__pow__divI,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))) ) ) ).

% less_two_pow_divI
tff(fact_1779_nat__less__power__trans,axiom,
    ! [Na: nat,M: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% nat_less_power_trans
tff(fact_1780_word__less__two__pow__divD,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))) ) ) ) ).

% word_less_two_pow_divD
tff(fact_1781_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)))
             => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_1782_small__powers__of__2,axiom,
    ! [Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Xa)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),one_one(nat)))) ) ).

% small_powers_of_2
tff(fact_1783_cnt__cnt__eq,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_cnt(T) = aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,T)) ).

% cnt_cnt_eq
tff(fact_1784_space__cnt,axiom,
    ! [T: vEBT_VEBT] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2)))),vEBT_VEBT_cnt(T))) ).

% space_cnt
tff(fact_1785_Tb__T__vebt__buildupi_H_H,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V441764108873111860ildupi(Na)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),vEBT_VEBT_Tb2(Na)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% Tb_T_vebt_buildupi''
tff(fact_1786_t__build__cnt,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V8646137997579335489_i_l_d(Na))),aa(real,real,aa(real,fun(real,real),times_times(real),vEBT_VEBT_cnt(vEBT_vebt_buildup(Na))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))) ).

% t_build_cnt
tff(fact_1787_t__buildup__cnt,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V8346862874174094_d_u_p(Na))),aa(real,real,aa(real,fun(real,real),times_times(real),vEBT_VEBT_cnt(vEBT_vebt_buildup(Na))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))) ).

% t_buildup_cnt
tff(fact_1788_T__vebt__buildupi__cnt_H,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V441764108873111860ildupi(Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_cnt(vEBT_vebt_buildup(Na)))) ).

% T_vebt_buildupi_cnt'
tff(fact_1789_count__buildup_H,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),vEBT_VEBT_cnt(vEBT_vebt_buildup(Na))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))) ).

% count_buildup'
tff(fact_1790_count__buildup,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),vEBT_VEBT_cnt(vEBT_vebt_buildup(Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na))) ).

% count_buildup
tff(fact_1791_Diff__cancel,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),A3) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_1792_empty__Diff,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),bot_bot(set(A))),A3) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_1793_Diff__empty,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),bot_bot(set(A))) = A3 ).

% Diff_empty
tff(fact_1794_insert__Diff1,axiom,
    ! [A: $tType,Xa: A,B3: set(A),A3: set(A)] :
      ( member(A,Xa,B3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) ) ) ).

% insert_Diff1
tff(fact_1795_Diff__insert0,axiom,
    ! [A: $tType,Xa: A,A3: set(A),B3: set(A)] :
      ( ~ member(A,Xa,A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) ) ) ).

% Diff_insert0
tff(fact_1796_Un__Diff__cancel2,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),A3) ).

% Un_Diff_cancel2
tff(fact_1797_Un__Diff__cancel,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) ).

% Un_Diff_cancel
tff(fact_1798_idiff__0__right,axiom,
    ! [Na: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),Na),zero_zero(extended_enat)) = Na ).

% idiff_0_right
tff(fact_1799_idiff__0,axiom,
    ! [Na: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),zero_zero(extended_enat)),Na) = zero_zero(extended_enat) ).

% idiff_0
tff(fact_1800_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ).

% Diff_eq_empty_iff
tff(fact_1801_insert__Diff__single,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3) ).

% insert_Diff_single
tff(fact_1802_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_1803_Tb_H__cnt,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_VEBT_Tb2(Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_vebt_buildup(Na)))) ).

% Tb'_cnt
tff(fact_1804_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ).

% zle_diff1_eq
tff(fact_1805_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),aa(word(A),fun(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_1806_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),aa(word(A),fun(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_1807_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),aa(num,nat,numeral_numeral(nat),V3)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),V3))) ).

% diff_nat_numeral
tff(fact_1808_nat__numeral__diff__1,axiom,
    ! [V: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V)),one_one(int))) ).

% nat_numeral_diff_1
tff(fact_1809_Diff__mono,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),D4: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D4),B3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),D4)) ) ) ).

% Diff_mono
tff(fact_1810_Diff__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),A3) ).

% Diff_subset
tff(fact_1811_double__diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C3)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),A3)) = A3 ) ) ) ).

% double_diff
tff(fact_1812_insert__Diff__if,axiom,
    ! [A: $tType,Xa: A,A3: set(A),B3: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)),B3) = $ite(member(A,Xa,B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))) ).

% insert_Diff_if
tff(fact_1813_minus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),K),zero_zero(int)) = K ).

% minus_int_code(1)
tff(fact_1814_set__diff__diff__left,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) ).

% set_diff_diff_left
tff(fact_1815_Un__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),C3)) ).

% Un_Diff
tff(fact_1816_int__distrib_I3_J,axiom,
    ! [Z1: int,Z2: int,W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z1),Z2)),W) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z2),W)) ).

% int_distrib(3)
tff(fact_1817_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z2: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z1),Z2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z2)) ).

% int_distrib(4)
tff(fact_1818_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M4: nat,N: nat] : Z != aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),M4)),aa(nat,int,semiring_1_of_nat(int),N)) ).

% int_diff_cases
tff(fact_1819_word__le__imp__diff__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),Na: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),K),Na)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Na),M)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Na),K)),M) ) ) ) ).

% word_le_imp_diff_le
tff(fact_1820_word__le__minus__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Z: word(A),Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Z),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Z),Xa)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Z)) ) ) ) ) ).

% word_le_minus_mono_right
tff(fact_1821_sub__wrap__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Z))
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Z) ) ) ).

% sub_wrap_lt
tff(fact_1822_word__sub__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y))
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ).

% word_sub_less_iff
tff(fact_1823_word__less__minus__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),C2: word(A),D2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),C2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),D2),B2)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),A2),B2)),A2)
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),C2),D2)),C2)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),A2),B2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),C2),D2)) ) ) ) ) ) ).

% word_less_minus_mono
tff(fact_1824_Word_Omod__minus__cong,axiom,
    ! [B2: int,B5: int,Xa: int,X6: int,Y: int,Y4: int,Z4: int] :
      ( ( B2 = B5 )
     => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xa),B5) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X6),B5) )
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Y),B5) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Y4),B5) )
         => ( ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X6),Y4) = Z4 )
           => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),Y)),B2) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Z4),B5) ) ) ) ) ) ).

% Word.mod_minus_cong
tff(fact_1825_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
     => ? [B6: A] : member(A,B6,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)) ) ).

% psubset_imp_ex_mem
tff(fact_1826_word__numeral__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(num,word(A),numeral_numeral(word(A)),B2) = aa(int,word(A),ring_1_of_int(word(A)),aa(num,int,numeral_numeral(int),B2)) ) ).

% word_numeral_alt
tff(fact_1827_subset__minus__empty,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) = bot_bot(set(A)) ) ) ).

% subset_minus_empty
tff(fact_1828_set__minus__singleton__eq,axiom,
    ! [A: $tType,Xa: A,X7: set(A)] :
      ( ~ member(A,Xa,X7)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),X7),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = X7 ) ) ).

% set_minus_singleton_eq
tff(fact_1829_insert__minus__eq,axiom,
    ! [A: $tType,Xa: A,Y: A,A3: set(A)] :
      ( ( Xa != Y )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A))))) ) ) ).

% insert_minus_eq
tff(fact_1830_Diff__insert__absorb,axiom,
    ! [A: $tType,Xa: A,A3: set(A)] :
      ( ~ member(A,Xa,A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = A3 ) ) ).

% Diff_insert_absorb
tff(fact_1831_Diff__insert2,axiom,
    ! [A: $tType,A3: set(A),A2: A,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))),B3) ).

% Diff_insert2
tff(fact_1832_insert__Diff,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( member(A,A2,A3)
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = A3 ) ) ).

% insert_Diff
tff(fact_1833_Diff__insert,axiom,
    ! [A: $tType,A3: set(A),A2: A,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_1834_subset__Diff__insert,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),Xa: A,C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),C3)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),C3))
        & ~ member(A,Xa,A3) ) ) ).

% subset_Diff_insert
tff(fact_1835_wi__hom__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int,B2: 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)),A2)),aa(int,word(A),ring_1_of_int(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),B2)) ) ).

% wi_hom_add
tff(fact_1836_Diff__partition,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)) = B3 ) ) ).

% Diff_partition
tff(fact_1837_Diff__subset__conv,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),C3)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) ) ).

% Diff_subset_conv
tff(fact_1838_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)
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I3),K)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_le_induct
tff(fact_1839_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,aa(int,fun(int,int),minus_minus(int),K),one_one(int)))
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I3),K)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_less_induct
tff(fact_1840_le__step__down__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),Na: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),Na)
         => ( ( I != Na )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Na),one_one(word(A)))) ) ) ) ).

% le_step_down_word
tff(fact_1841_le__step__down__word__2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
         => ( ( Xa != Y )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),one_one(word(A)))) ) ) ) ).

% le_step_down_word_2
tff(fact_1842_word__of__int__power__hom,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int,Na: nat] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A2)),Na) = aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),Na)) ) ).

% word_of_int_power_hom
tff(fact_1843_word__diff__ls_H_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)) ) ) ) ).

% word_diff_ls''(3)
tff(fact_1844_word__diff__ls_H_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xa)) ) ) ) ).

% word_diff_ls''(4)
tff(fact_1845_word__l__diffs_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z) ) ) ) ).

% word_l_diffs'(3)
tff(fact_1846_word__l__diffs_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa)) ) ) ) ).

% word_l_diffs'(4)
tff(fact_1847_word__diff__ls_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),W)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)) ) ) ) ).

% word_diff_ls'(3)
tff(fact_1848_word__diff__ls_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),W) ) ) ) ).

% word_diff_ls'(4)
tff(fact_1849_More__Word_Oword__l__diffs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z) ) ) ) ).

% More_Word.word_l_diffs(3)
tff(fact_1850_More__Word_Oword__l__diffs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa)) ) ) ) ).

% More_Word.word_l_diffs(4)
tff(fact_1851_word__less__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y)
         => ( ( Xa = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),one_one(word(A))) )
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),one_one(word(A)))) ) ) ) ).

% word_less_cases
tff(fact_1852_word__less__minus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa)) ) ) ) ).

% word_less_minus_mono_left
tff(fact_1853_word__less__minus__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Z) ) ) ) ).

% word_less_minus_cancel
tff(fact_1854_word__less__imp__diff__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),Na: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),K),Na)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Na),M)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Na),K)),M) ) ) ) ).

% word_less_imp_diff_less
tff(fact_1855_wi__hom__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int,B2: 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)),A2)),aa(int,word(A),ring_1_of_int(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2)) ) ).

% wi_hom_mult
tff(fact_1856_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,Xa: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Z),Y)
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Xa),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),Y),Z)) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Xa),Y)),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_1857_Bolzano,axiom,
    ! [A2: real,B2: real,P: fun(real,fun(real,$o))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [A5: real,B6: real,C5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),P,A5),B6)
           => ( aa(real,$o,aa(real,fun(real,$o),P,B6),C5)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A5),B6)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B6),C5)
                 => aa(real,$o,aa(real,fun(real,$o),P,A5),C5) ) ) ) )
       => ( ! [X: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2)
               => ? [D6: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                    & ! [A5: real,B6: real] :
                        ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A5),X)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B6)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B6),A5)),D6) )
                       => aa(real,$o,aa(real,fun(real,$o),P,A5),B6) ) ) ) )
         => aa(real,$o,aa(real,fun(real,$o),P,A2),B2) ) ) ) ).

% Bolzano
tff(fact_1858_Diff__single__insert,axiom,
    ! [A: $tType,A3: set(A),Xa: A,B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))),B3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),B3)) ) ).

% Diff_single_insert
tff(fact_1859_subset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),Xa: A,B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),B3))
    <=> $ite(member(A,Xa,A3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))),B3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)) ) ).

% subset_insert_iff
tff(fact_1860_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)
     => ( ! [X: int,K2: int] :
            ( aa(int,$o,P3,X)
          <=> aa(int,$o,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2))) )
       => ( ? [Z5: int] :
            ! [X: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z5),X)
             => ( aa(int,$o,P,X)
              <=> aa(int,$o,P3,X) ) )
         => ( ? [X_13: int] : aa(int,$o,P3,X_13)
           => ? [X_1: int] : aa(int,$o,P,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_1861_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)
     => ( ! [X: int,K2: int] :
            ( aa(int,$o,P1,X)
          <=> aa(int,$o,P1,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2))) )
       => ( ? [Z5: int] :
            ! [X: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Z5)
             => ( aa(int,$o,P,X)
              <=> aa(int,$o,P1,X) ) )
         => ( ? [X_13: int] : aa(int,$o,P1,X_13)
           => ? [X_1: int] : aa(int,$o,P,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_1862_remove__subset,axiom,
    ! [A: $tType,Xa: A,S: set(A)] :
      ( member(A,Xa,S)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))),S) ) ).

% remove_subset
tff(fact_1863_less__1__helper,axiom,
    ! [Na: int,M: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Na),M)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Na),one_one(int))),M) ) ).

% less_1_helper
tff(fact_1864_int__induct,axiom,
    ! [P: fun(int,$o),K: int,I: int] :
      ( aa(int,$o,P,K)
     => ( ! [I3: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I3)
           => ( aa(int,$o,P,I3)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))) ) )
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I3),K)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_induct
tff(fact_1865_int__mod__le_H,axiom,
    ! [B2: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),Na))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),Na)),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),Na)) ) ).

% int_mod_le'
tff(fact_1866_int__minus,axiom,
    ! [Na: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) = aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),Na)),aa(nat,int,semiring_1_of_nat(int),M)))) ).

% int_minus
tff(fact_1867_word__must__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Na),one_one(word(A))))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Na),Xa)
           => ( Na = zero_zero(word(A)) ) ) ) ) ).

% word_must_wrap
tff(fact_1868_word__sub__1__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( Xa != zero_zero(word(A)) )
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A)))),Xa) ) ) ).

% word_sub_1_le
tff(fact_1869_word__le__minus__one__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),one_one(word(A)))) ) ) ).

% word_le_minus_one_leq
tff(fact_1870_word__minus__one__le__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A)))),Y)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y) ) ) ).

% word_minus_one_le_leq
tff(fact_1871_gt0__iff__gem1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Xa)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A)))),Xa) ) ) ).

% gt0_iff_gem1
tff(fact_1872_word__less__sub1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( Xa != zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),one_one(word(A))),Xa)
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A)))) ) ) ) ).

% word_less_sub1
tff(fact_1873_word__diff__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Na)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),M)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Na),M)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),M),Na)),M) ) ) ) ) ).

% word_diff_less
tff(fact_1874_sub__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Z))
        <=> ( ( Z = zero_zero(word(A)) )
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Z) ) ) ) ).

% sub_wrap
tff(fact_1875_word__diff__ls_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xa: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xb)) ) ) ) ).

% word_diff_ls(2)
tff(fact_1876_word__diff__ls_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W: word(A),Xaa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)) ) ) ) ).

% word_diff_ls(1)
tff(fact_1877_Word_Oword__l__diffs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Z: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xb)) ) ) ) ).

% Word.word_l_diffs(2)
tff(fact_1878_Word_Oword__l__diffs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xb))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z) ) ) ) ).

% Word.word_l_diffs(1)
tff(fact_1879_word__plus__mcs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),Xb: word(A),W: word(A),Xa: word(A),Xc: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xc),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),Xc)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xc)) ) ) ) ).

% word_plus_mcs(2)
tff(fact_1880_word__plus__mcs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),Xb: word(A),Xa: word(A),W: word(A),Xaa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)) ) ) ) ).

% word_plus_mcs(1)
tff(fact_1881_word__less__add__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Z))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Z),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z)),Y) ) ) ) ).

% word_less_add_right
tff(fact_1882_word__less__sub__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Z))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)),Z) ) ) ) ).

% word_less_sub_right
tff(fact_1883_plus__minus__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Ab),C2))
         => ( aa(word(A),$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)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),C2)) ) ) ) ).

% plus_minus_no_overflow
tff(fact_1884_word__diff__ls_H_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)) ) ) ) ).

% word_diff_ls''(1)
tff(fact_1885_word__diff__ls_H_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xa)) ) ) ) ).

% word_diff_ls''(2)
tff(fact_1886_word__l__diffs_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z) ) ) ) ).

% word_l_diffs'(1)
tff(fact_1887_word__l__diffs_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa)) ) ) ) ).

% word_l_diffs'(2)
tff(fact_1888_word__diff__ls_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),W)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)) ) ) ) ).

% word_diff_ls'(1)
tff(fact_1889_word__diff__ls_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),W) ) ) ) ).

% word_diff_ls'(2)
tff(fact_1890_More__Word_Oword__l__diffs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z) ) ) ) ).

% More_Word.word_l_diffs(1)
tff(fact_1891_More__Word_Oword__l__diffs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa)) ) ) ) ).

% More_Word.word_l_diffs(2)
tff(fact_1892_mod__div__equality__div__eq,axiom,
    ! [A2: int,B2: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),A2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)) ).

% mod_div_equality_div_eq
tff(fact_1893_nat__minus__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X5),Xa2) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_minus_as_int
tff(fact_1894_psubset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),Xa: A,B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),B3))
    <=> $ite(
          member(A,Xa,B3),
          aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3),
          $ite(member(A,Xa,A3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))),B3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)) ) ) ).

% psubset_insert_iff
tff(fact_1895_int__ops_I6_J,axiom,
    ! [A2: nat,B2: nat] :
      aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) ).

% int_ops(6)
tff(fact_1896_word__of__int__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
         => ( aa(int,word(A),ring_1_of_int(word(A)),Xa) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(int,nat,nat2,Xa)) ) ) ) ).

% word_of_int_nat
tff(fact_1897_decr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X: int] :
            ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X5: int] :
              ( aa(int,$o,P,X5)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1898_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,aa(int,fun(int,int),minus_minus(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) ) ) ) ).

% nat_diff_distrib
tff(fact_1899_nat__diff__distrib_H,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Xa)),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_1900_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_1901_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),aa(word(A),fun(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_1902_word__leq__minus__one__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),one_one(word(A))))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ).

% word_leq_minus_one_le
tff(fact_1903_word__leq__le__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
         => ( ( Xa != zero_zero(word(A)) )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A)))),Y) ) ) ) ).

% word_leq_le_minus_one
tff(fact_1904_le__m1__iff__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Xa)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A))))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Xa) ) ) ) ).

% le_m1_iff_lt
tff(fact_1905_less__1__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Na),one_one(word(A)))),M)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Na),M)
            & ( Na != zero_zero(word(A)) ) ) ) ) ).

% less_1_simp
tff(fact_1906_word__less__nowrapI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Z: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),K))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),K),Z)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),K)) ) ) ) ) ).

% word_less_nowrapI
tff(fact_1907_plus__minus__not__NULL,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Ab),C2))
         => ( aa(word(A),$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)),Xa),C2) != zero_zero(word(A)) ) ) ) ) ) ).

% plus_minus_not_NULL
tff(fact_1908_word__less__nowrapI_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Z: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),K))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),K),Z)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),K)) ) ) ) ) ).

% word_less_nowrapI'
tff(fact_1909_zdiff__int__split,axiom,
    ! [P: fun(int,$o),Xa: nat,Y: nat] :
      ( aa(int,$o,P,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),Y)))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xa)
         => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),Xa)),aa(nat,int,semiring_1_of_nat(int),Y))) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
         => aa(int,$o,P,zero_zero(int)) ) ) ) ).

% zdiff_int_split
tff(fact_1910_real__of__nat__div2,axiom,
    ! [Na: nat,Xa: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,semiring_1_of_nat(real),Xa))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),Xa)))) ).

% real_of_nat_div2
tff(fact_1911_real__of__nat__div3,axiom,
    ! [Na: nat,Xa: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,semiring_1_of_nat(real),Xa))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),Xa)))),one_one(real)) ).

% real_of_nat_div3
tff(fact_1912_log__divide,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
           => ( aa(real,real,log(A2),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log(A2),Xa)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_1913_mod__add__if__z,axiom,
    ! [Xa: int,Z: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
             => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Y)),Z) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Y)),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Y),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Y)),Z)) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_1914_mod__sub__if__z,axiom,
    ! [Xa: int,Z: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
             => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),Y)),Z) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),Xa),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),Y),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),Y)),Z)) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_1915_real__of__int__div2,axiom,
    ! [Na: int,Xa: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Na)),aa(int,real,ring_1_of_int(real),Xa))),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),Xa)))) ).

% real_of_int_div2
tff(fact_1916_real__of__int__div3,axiom,
    ! [Na: int,Xa: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Na)),aa(int,real,ring_1_of_int(real),Xa))),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),Xa)))),one_one(real)) ).

% real_of_int_div3
tff(fact_1917_word__div__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Y)
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)),Y) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y)),one_one(word(A))) ) ) ) ) ).

% word_div_sub
tff(fact_1918_diff__nat__eq__if,axiom,
    ! [Z: int,Z4: int] :
      aa(nat,nat,aa(nat,fun(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,aa(int,fun(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_1919_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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) ) ).

% word_of_int_2p
tff(fact_1920_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1921_minus__log__eq__powr,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),Y),aa(real,real,log(B2),Xa)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,B2,Y)),Xa)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_1922_neg__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int))),A2))),one_one(int)) ) ) ).

% neg_zmod_mult_2
tff(fact_1923_space__space_H,axiom,
    ! [T: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space,T)),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T)) ).

% space_space'
tff(fact_1924_Tb__T__vebt__buildupi,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),vEBT_V441764108873111860ildupi(Na))),aa(int,int,aa(int,fun(int,int),minus_minus(int),vEBT_VEBT_Tb(Na)),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% Tb_T_vebt_buildupi
tff(fact_1925_Tb__T__vebt__buildupi_H,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),vEBT_V9176841429113362141ildupi(Na)),aa(int,int,aa(int,fun(int,int),minus_minus(int),vEBT_VEBT_Tb(Na)),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% Tb_T_vebt_buildupi'
tff(fact_1926_real__average__minus__second,axiom,
    ! [B2: real,A2: real] : aa(real,real,aa(real,fun(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),B2),A2)),aa(num,real,numeral_numeral(real),bit0(one2)))),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% real_average_minus_second
tff(fact_1927_real__average__minus__first,axiom,
    ! [A2: real,B2: real] : aa(real,real,aa(real,fun(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),A2),B2)),aa(num,real,numeral_numeral(real),bit0(one2)))),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% real_average_minus_first
tff(fact_1928_space__2__pow__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit0(aa(num,num,bit1,one2))))),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)),one_one(real)))) ) ).

% space_2_pow_bound
tff(fact_1929_Tb__Tb_H,axiom,
    ! [T: nat] : vEBT_VEBT_Tb(T) = aa(nat,int,semiring_1_of_nat(int),vEBT_VEBT_Tb2(T)) ).

% Tb_Tb'
tff(fact_1930_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ vEBT_invar_vebt(T,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_1931_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ vEBT_invar_vebt(T,zero_zero(nat)) ).

% valid_0_not
tff(fact_1932_deg__not__0,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% deg_not_0
tff(fact_1933_buildup__gives__valid,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => vEBT_invar_vebt(vEBT_vebt_buildup(Na),Na) ) ).

% buildup_gives_valid
tff(fact_1934_invar__vebt__buildup,axiom,
    ! [Na: nat] :
      ( vEBT_invar_vebt(vEBT_vebt_buildup(Na),Na)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% invar_vebt_buildup
tff(fact_1935_DiffI,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,A3)
     => ( ~ member(A,C2,B3)
       => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) ) ) ).

% DiffI
tff(fact_1936_Diff__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))
    <=> ( member(A,C2,A3)
        & ~ member(A,C2,B3) ) ) ).

% Diff_iff
tff(fact_1937_Diff__idemp,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) ).

% Diff_idemp
tff(fact_1938_insert_H__pres__valid,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => vEBT_invar_vebt(vEBT_VEBT_insert(T,Xa),Na) ) ).

% insert'_pres_valid
tff(fact_1939_space_H__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),U)) ) ) ).

% space'_bound
tff(fact_1940_space__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space,T)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),U)) ) ) ).

% space_bound
tff(fact_1941_vebt__space__linear__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space,T)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ) ).

% vebt_space_linear_bound
tff(fact_1942_cnt__bound_H,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),vEBT_VEBT_cnt(T)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)),one_one(real)))) ) ).

% cnt_bound'
tff(fact_1943_cnt__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),vEBT_VEBT_cnt(T)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2)))))))) ) ).

% cnt_bound
tff(fact_1944_set__diff__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_am(set(A),fun(set(A),fun(A,$o)),A3),B3)) ).

% set_diff_eq
tff(fact_1945_DiffE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))
     => ~ ( member(A,C2,A3)
         => member(A,C2,B3) ) ) ).

% DiffE
tff(fact_1946_DiffD1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))
     => member(A,C2,A3) ) ).

% DiffD1
tff(fact_1947_DiffD2,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))
     => ~ member(A,C2,B3) ) ).

% DiffD2
tff(fact_1948_minus__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) = collect(A,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),minus_minus(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A3)),aTP_Lamp_a(set(A),fun(A,$o),B3))) ).

% minus_set_def
tff(fact_1949_mult__commute__abs,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [C2: A] : aTP_Lamp_an(A,fun(A,A),C2) = aa(A,fun(A,A),times_times(A),C2) ) ).

% mult_commute_abs
tff(fact_1950_time__vebt__succ,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c(T,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)))))) ) ).

% time_vebt_succ
tff(fact_1951_time__vebt__pred,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d(T,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)))))) ) ).

% time_vebt_pred
tff(fact_1952_delete__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_d_e_l_e_t_e(T,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2))))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2)))))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% delete_bound_size_univ
tff(fact_1953_time__vebt__delete,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_d_e_l_e_t_e(T,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2))))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2)))))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)))))) ) ).

% time_vebt_delete
tff(fact_1954_delete__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V1232361888498592333_e_t_e(T,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% delete_bound_size_univ'
tff(fact_1955_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T: vEBT_VEBT] :
      ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Deg) )
     => ( vEBT_invar_vebt(T,Deg)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% height_double_log_univ_size
tff(fact_1956_heigt__uplog__rel,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))) ) ) ).

% heigt_uplog_rel
tff(fact_1957_delete__bound__height_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V1232361888498592333_e_t_e(T,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))) ) ).

% delete_bound_height'
tff(fact_1958_two__powr__height__bound__deg,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ).

% two_powr_height_bound_deg
tff(fact_1959_delete__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_d_e_l_e_t_e(T,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2))))))))) ) ).

% delete_bound_height
tff(fact_1960_succ__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c(T,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% succ_bound_size_univ
tff(fact_1961_pred__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d(T,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% pred_bound_size_univ
tff(fact_1962_succ__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_s_u_c_c(T,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))) ) ).

% succ_bound_height
tff(fact_1963_pred__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_p_r_e_d(T,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))) ) ).

% pred_bound_height
tff(fact_1964_valid__pres__insert,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => vEBT_invar_vebt(vEBT_vebt_insert(T,Xa),Na) ) ) ).

% valid_pres_insert
tff(fact_1965_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xa: nat,Na: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Tree),Xa)
     => ( vEBT_invar_vebt(Tree,Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ) ).

% member_bound
tff(fact_1966_artanh__def,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xa: A] : aa(A,A,artanh(A),Xa) = 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)),Xa)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% artanh_def
tff(fact_1967_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
         => ( aa(nat,$o,vEBT_vebt_member(vEBT_vebt_insert(T,Xa)),Y)
           => ( aa(nat,$o,vEBT_vebt_member(T),Y)
              | ( Xa = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_1968_ln__inj__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( ( aa(real,real,ln_ln(real),Xa) = aa(real,real,ln_ln(real),Y) )
        <=> ( Xa = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_1969_ln__less__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_1970_ln__le__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_1971_ln__less__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xa)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real)) ) ) ).

% ln_less_zero_iff
tff(fact_1972_ln__gt__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa) ) ) ).

% ln_gt_zero_iff
tff(fact_1973_ln__eq__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( ( aa(real,real,ln_ln(real),Xa) = zero_zero(real) )
      <=> ( Xa = one_one(real) ) ) ) ).

% ln_eq_zero_iff
tff(fact_1974_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_1975_ln__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xa)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real)) ) ) ).

% ln_le_zero_iff
tff(fact_1976_ln__ge__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa) ) ) ).

% ln_ge_zero_iff
tff(fact_1977_ln__less__self,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xa)),Xa) ) ).

% ln_less_self
tff(fact_1978_log__def,axiom,
    ! [A2: real,Xa: real] : aa(real,real,log(A2),Xa) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,ln_ln(real),A2)) ).

% log_def
tff(fact_1979_ln__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xa)),Xa) ) ).

% ln_bound
tff(fact_1980_ln__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa)) ) ).

% ln_gt_zero
tff(fact_1981_ln__less__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xa)),zero_zero(real)) ) ) ).

% ln_less_zero
tff(fact_1982_ln__gt__zero__imp__gt__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_1983_ln__ge__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa)) ) ).

% ln_ge_zero
tff(fact_1984_ln__powr,axiom,
    ! [Xa: real,Y: real] :
      ( ( Xa != zero_zero(real) )
     => ( aa(real,real,ln_ln(real),powr(real,Xa,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),aa(real,real,ln_ln(real),Xa)) ) ) ).

% ln_powr
tff(fact_1985_ln__ge__zero__imp__ge__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_1986_ln__add__one__self__le__self,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa) ) ).

% ln_add_one_self_le_self
tff(fact_1987_ln__mult,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_mult
tff(fact_1988_ln__eq__minus__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( ( aa(real,real,ln_ln(real),Xa) = aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),one_one(real)) )
       => ( Xa = one_one(real) ) ) ) ).

% ln_eq_minus_one
tff(fact_1989_ln__div,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_div
tff(fact_1990_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_1991_ln__le__minus__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),one_one(real))) ) ).

% ln_le_minus_one
tff(fact_1992_ln__diff__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,ln_ln(real),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_1993_ln__realpow,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,ln_ln(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,ln_ln(real),Xa)) ) ) ).

% ln_realpow
tff(fact_1994_ln__powr__bound,axiom,
    ! [Xa: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,Xa,A2)),A2)) ) ) ).

% ln_powr_bound
tff(fact_1995_ln__powr__bound2,axiom,
    ! [Xa: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,aa(real,real,ln_ln(real),Xa),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A2,A2)),Xa)) ) ) ).

% ln_powr_bound2
tff(fact_1996_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
         => ( ( B2 != one_one(real) )
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
             => ( aa(real,real,log(A2),Xa) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B2)),aa(real,real,ln_ln(real),A2))),aa(real,real,log(B2),Xa)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_1997_ln__one__plus__pos__lower__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_1998_insert__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),vEBT_VEBT_set_vebt(T)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Xa),bot_bot(set(nat)))) = vEBT_VEBT_set_vebt(vEBT_vebt_insert(T,Xa)) ) ) ) ).

% insert_corr
tff(fact_1999_insert__correct,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),vEBT_set_vebt(T)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Xa),bot_bot(set(nat)))) = vEBT_set_vebt(vEBT_vebt_insert(T,Xa)) ) ) ) ).

% insert_correct
tff(fact_2000_set__vebt__insert,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( vEBT_set_vebt(vEBT_vebt_insert(T,Xa)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),vEBT_set_vebt(T)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Xa),bot_bot(set(nat)))) ) ) ) ).

% set_vebt_insert
tff(fact_2001_htt__vebt__inserti__invar__vebt,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => time_htt(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_inserti(Tia,Xa),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(T,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ) ).

% htt_vebt_inserti_invar_vebt
tff(fact_2002_htt__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : time_htt(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_inserti(Tia,Xa),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(T,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T)))) ).

% htt_vebt_inserti
tff(fact_2003_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_m_e_m_b_e_r(T,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% member_bound_size_univ
tff(fact_2004_VEBT__internal_OTb_Osimps_I1_J,axiom,
    vEBT_VEBT_Tb(zero_zero(nat)) = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) ).

% VEBT_internal.Tb.simps(1)
tff(fact_2005_set__vebt__equal,axiom,
    ! [T_1: vEBT_VEBT,Na: nat,T_2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T_1,Na)
     => ( vEBT_invar_vebt(T_2,Na)
       => ( ( T_1 = T_2 )
        <=> ( vEBT_set_vebt(T_1) = vEBT_set_vebt(T_2) ) ) ) ) ).

% set_vebt_equal
tff(fact_2006_unique__tree,axiom,
    ! [T: vEBT_VEBT,Na: nat,Sa: vEBT_VEBT] :
      ( vEBT_invar_vebt(T,Na)
     => ( vEBT_invar_vebt(Sa,Na)
       => ( ( vEBT_set_vebt(T) = vEBT_set_vebt(Sa) )
         => ( Sa = T ) ) ) ) ).

% unique_tree
tff(fact_2007_uniquetree,axiom,
    ! [T: vEBT_VEBT,Na: nat,Sa: vEBT_VEBT] :
      ( vEBT_invar_vebt(T,Na)
     => ( vEBT_invar_vebt(Sa,Na)
       => ( ( vEBT_VEBT_set_vebt(T) = vEBT_VEBT_set_vebt(Sa) )
         => ( Sa = T ) ) ) ) ).

% uniquetree
tff(fact_2008_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect(nat,vEBT_vebt_member(T)) ).

% set_vebt'_def
tff(fact_2009_set__vebt__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( aa(nat,$o,vEBT_vebt_member(T),Xa)
      <=> member(nat,Xa,vEBT_set_vebt(T)) ) ) ).

% set_vebt_member
tff(fact_2010_set__vebt__buildup,axiom,
    ! [I: nat] : vEBT_set_vebt(vEBT_vebt_buildup(I)) = bot_bot(set(nat)) ).

% set_vebt_buildup
tff(fact_2011_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_2012_buildup__gives__empty,axiom,
    ! [Na: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Na)) = bot_bot(set(nat)) ).

% buildup_gives_empty
tff(fact_2013_succ__member,axiom,
    ! [T: vEBT_VEBT,Xa: nat,Y: nat] :
      ( vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T),Xa,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(T),Y)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
        & ! [Z6: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(T),Z6)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Z6) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Z6) ) ) ) ).

% succ_member
tff(fact_2014_pred__member,axiom,
    ! [T: vEBT_VEBT,Xa: nat,Y: nat] :
      ( vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T),Xa,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(T),Y)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa)
        & ! [Z6: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(T),Z6)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Z6),Xa) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Z6),Y) ) ) ) ).

% pred_member
tff(fact_2015_member__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_e_m_b_e_r(T,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))) ) ).

% member_bound_height
tff(fact_2016_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    vEBT_VEBT_Tb2(zero_zero(nat)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.Tb'.simps(1)
tff(fact_2017_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
     => time_htt(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(Na),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),U)) ) ).

% htt_vebt_buildupi'_univ
tff(fact_2018_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
     => time_htt(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(Na),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),U)) ) ).

% htt_vebt_buildupi_univ
tff(fact_2019_htt__vebt__buildupi_H,axiom,
    ! [Na: nat] : time_htt(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(Na),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na)),vEBT_V441764108873111860ildupi(Na)) ).

% htt_vebt_buildupi'
tff(fact_2020_htt__vebt__buildupi,axiom,
    ! [Na: nat] : time_htt(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(Na),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na)),vEBT_V441764108873111860ildupi(Na)) ).

% htt_vebt_buildupi
tff(fact_2021_inrange,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),vEBT_VEBT_set_vebt(T)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),one_one(nat)))) ) ).

% inrange
tff(fact_2022_insert__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_i_n_s_e_r_t(T,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% insert_bound_size_univ
tff(fact_2023_htt__vebt__deletei,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => time_htt(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_deletei(Tia,Xa),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_delete(T,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,bit0(one2)))))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ) ).

% htt_vebt_deletei
tff(fact_2024_invar__vebt__delete,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => vEBT_invar_vebt(vEBT_vebt_delete(T,Xa),Na) ) ).

% invar_vebt_delete
tff(fact_2025_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( aa(nat,$o,vEBT_vebt_member(vEBT_vebt_delete(T,Xa)),Y)
      <=> ( ( Xa != Y )
          & aa(nat,$o,vEBT_vebt_member(T),Y) ) ) ) ).

% dele_member_cont_corr
tff(fact_2026_set__vebt__delete,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( vEBT_set_vebt(vEBT_vebt_delete(T,Xa)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),vEBT_set_vebt(T)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Xa),bot_bot(set(nat)))) ) ) ).

% set_vebt_delete
tff(fact_2027_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T,Xa)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),vEBT_VEBT_set_vebt(T)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Xa),bot_bot(set(nat)))) ) ) ).

% delete_correct'
tff(fact_2028_delete__correct,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T,Xa)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),vEBT_set_vebt(T)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Xa),bot_bot(set(nat)))) ) ) ).

% delete_correct
tff(fact_2029_insert__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(T,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))))) ) ).

% insert_bound_height
tff(fact_2030_insert_H__correct,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( vEBT_set_vebt(vEBT_VEBT_insert(T,Xa)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),vEBT_set_vebt(T)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Xa),bot_bot(set(nat))))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),one_one(nat)))) ) ) ).

% insert'_correct
tff(fact_2031_T__vebt__buildupi,axiom,
    ! [Na: nat,H: heap_ext(product_unit)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(vEBT_VEBTi,vEBT_V739175172307565963ildupi(Na),H)),vEBT_V441764108873111860ildupi(Na)) ).

% T_vebt_buildupi
tff(fact_2032_atLeastAtMost__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : set_or1337092689740270186AtMost(A,A2,A2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ) ).

% atLeastAtMost_singleton
tff(fact_2033_atLeastAtMost__singleton__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),C2),bot_bot(set(A))) )
        <=> ( ( A2 = B2 )
            & ( B2 = C2 ) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_2034_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_2035_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% atLeastatMost_empty_iff
tff(fact_2036_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A2,B2) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_2037_inf__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( semilattice_inf(A)
     => ! [F2: fun(B,A),G: fun(B,A),Xa: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),inf_inf(fun(B,A)),F2),G),Xa) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F2,Xa)),aa(B,A,G,Xa)) ) ).

% inf_apply
tff(fact_2038_inf__right__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) ) ).

% inf_right_idem
tff(fact_2039_inf_Oright__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ).

% inf.right_idem
tff(fact_2040_inf__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) ) ).

% inf_left_idem
tff(fact_2041_inf_Oleft__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ).

% inf.left_idem
tff(fact_2042_inf__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Xa) = Xa ) ).

% inf_idem
tff(fact_2043_inf_Oidem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),A2) = A2 ) ).

% inf.idem
tff(fact_2044_Int__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
    <=> ( member(A,C2,A3)
        & member(A,C2,B3) ) ) ).

% Int_iff
tff(fact_2045_IntI,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,A3)
     => ( member(A,C2,B3)
       => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) ) ) ).

% IntI
tff(fact_2046_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% inf.bounded_iff
tff(fact_2047_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z) ) ) ) ).

% le_inf_iff
tff(fact_2048_boolean__algebra_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),bot_bot(A)) = bot_bot(A) ) ).

% boolean_algebra.conj_zero_right
tff(fact_2049_boolean__algebra_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),Xa) = bot_bot(A) ) ).

% boolean_algebra.conj_zero_left
tff(fact_2050_inf__bot__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),bot_bot(A)) = bot_bot(A) ) ).

% inf_bot_right
tff(fact_2051_inf__bot__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),Xa) = bot_bot(A) ) ).

% inf_bot_left
tff(fact_2052_sup__inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)) = Xa ) ).

% sup_inf_absorb
tff(fact_2053_inf__sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) = Xa ) ).

% inf_sup_absorb
tff(fact_2054_Int__subset__iff,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),A3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),B3) ) ) ).

% Int_subset_iff
tff(fact_2055_Int__insert__right__if1,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( member(A,A2,A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) ) ) ).

% Int_insert_right_if1
tff(fact_2056_Int__insert__right__if0,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( ~ member(A,A2,A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ) ) ).

% Int_insert_right_if0
tff(fact_2057_insert__inter__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) ).

% insert_inter_insert
tff(fact_2058_Int__insert__left__if1,axiom,
    ! [A: $tType,A2: A,C3: set(A),B3: set(A)] :
      ( member(A,A2,C3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)),C3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) ) ) ).

% Int_insert_left_if1
tff(fact_2059_Int__insert__left__if0,axiom,
    ! [A: $tType,A2: A,C3: set(A),B3: set(A)] :
      ( ~ member(A,A2,C3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3) ) ) ).

% Int_insert_left_if0
tff(fact_2060_Un__Int__eq_I1_J,axiom,
    ! [A: $tType,S: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)),S) = S ).

% Un_Int_eq(1)
tff(fact_2061_Un__Int__eq_I2_J,axiom,
    ! [A: $tType,S: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)),T3) = T3 ).

% Un_Int_eq(2)
tff(fact_2062_Un__Int__eq_I3_J,axiom,
    ! [A: $tType,S: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)) = S ).

% Un_Int_eq(3)
tff(fact_2063_Un__Int__eq_I4_J,axiom,
    ! [A: $tType,T3: set(A),S: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)) = T3 ).

% Un_Int_eq(4)
tff(fact_2064_Int__Un__eq_I1_J,axiom,
    ! [A: $tType,S: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T3)),S) = S ).

% Int_Un_eq(1)
tff(fact_2065_Int__Un__eq_I2_J,axiom,
    ! [A: $tType,S: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T3)),T3) = T3 ).

% Int_Un_eq(2)
tff(fact_2066_Int__Un__eq_I3_J,axiom,
    ! [A: $tType,S: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T3)) = S ).

% Int_Un_eq(3)
tff(fact_2067_Int__Un__eq_I4_J,axiom,
    ! [A: $tType,T3: set(A),S: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),T3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T3)) = T3 ).

% Int_Un_eq(4)
tff(fact_2068_insert__disjoint_I1_J,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)),B3) = bot_bot(set(A)) )
    <=> ( ~ member(A,A2,B3)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) ) ) ) ).

% insert_disjoint(1)
tff(fact_2069_insert__disjoint_I2_J,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)),B3) )
    <=> ( ~ member(A,A2,B3)
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ) ) ) ).

% insert_disjoint(2)
tff(fact_2070_disjoint__insert_I1_J,axiom,
    ! [A: $tType,B3: set(A),A2: A,A3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)) = bot_bot(set(A)) )
    <=> ( ~ member(A,A2,B3)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),A3) = bot_bot(set(A)) ) ) ) ).

% disjoint_insert(1)
tff(fact_2071_disjoint__insert_I2_J,axiom,
    ! [A: $tType,A3: set(A),B2: A,B3: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B3)) )
    <=> ( ~ member(A,B2,A3)
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ) ) ) ).

% disjoint_insert(2)
tff(fact_2072_Diff__disjoint,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)) = bot_bot(set(A)) ).

% Diff_disjoint
tff(fact_2073_Int__left__commute,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)) ).

% Int_left_commute
tff(fact_2074_inf__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_inf(B)
     => ! [F2: fun(A,B),G: fun(A,B),X5: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),inf_inf(fun(A,B)),F2),G),X5) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,X5)),aa(A,B,G,X5)) ) ).

% inf_fun_def
tff(fact_2075_Int__left__absorb,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ).

% Int_left_absorb
tff(fact_2076_Int__commute,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),A3) ).

% Int_commute
tff(fact_2077_Int__absorb,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),A3) = A3 ).

% Int_absorb
tff(fact_2078_Int__assoc,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) ).

% Int_assoc
tff(fact_2079_IntD2,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
     => member(A,C2,B3) ) ).

% IntD2
tff(fact_2080_IntD1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
     => member(A,C2,A3) ) ).

% IntD1
tff(fact_2081_IntE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
     => ~ ( member(A,C2,A3)
         => ~ member(A,C2,B3) ) ) ).

% IntE
tff(fact_2082_inf__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z)) ) ).

% inf_left_commute
tff(fact_2083_inf_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ).

% inf.left_commute
tff(fact_2084_inf__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Xa) ) ).

% inf_commute
tff(fact_2085_inf_Ocommute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),A2) ) ).

% inf.commute
tff(fact_2086_inf__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)) ) ).

% inf_assoc
tff(fact_2087_inf_Oassoc,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ).

% inf.assoc
tff(fact_2088_inf__sup__aci_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Xa) ) ).

% inf_sup_aci(1)
tff(fact_2089_inf__sup__aci_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)) ) ).

% inf_sup_aci(2)
tff(fact_2090_inf__sup__aci_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z)) ) ).

% inf_sup_aci(3)
tff(fact_2091_inf__sup__aci_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) ) ).

% inf_sup_aci(4)
tff(fact_2092_boolean__algebra__cancel_Oinf2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,K: A,B2: A,A2: A] :
          ( ( B3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B3) = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)) ) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_2093_boolean__algebra__cancel_Oinf1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,K: A,A2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),A2) )
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)) ) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_2094_Collect__conj__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),collect(A,P)),collect(A,Q)) ).

% Collect_conj_eq
tff(fact_2095_Int__Collect,axiom,
    ! [A: $tType,Xa: A,A3: set(A),P: fun(A,$o)] :
      ( member(A,Xa,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,P)))
    <=> ( member(A,Xa,A3)
        & aa(A,$o,P,Xa) ) ) ).

% Int_Collect
tff(fact_2096_Int__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_ap(set(A),fun(set(A),fun(A,$o)),A3),B3)) ).

% Int_def
tff(fact_2097_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) ) ) ).

% inf.coboundedI2
tff(fact_2098_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) ) ) ).

% inf.coboundedI1
tff(fact_2099_inf_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb_iff2
tff(fact_2100_inf_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb_iff1
tff(fact_2101_inf_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),B2) ) ).

% inf.cobounded2
tff(fact_2102_inf_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),A2) ) ).

% inf.cobounded1
tff(fact_2103_inf_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ) ) ).

% inf.order_iff
tff(fact_2104_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)) ) ) ) ).

% inf_greatest
tff(fact_2105_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ) ) ).

% inf.boundedI
tff(fact_2106_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% inf.boundedE
tff(fact_2107_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = Y ) ) ) ).

% inf_absorb2
tff(fact_2108_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = Xa ) ) ) ).

% inf_absorb1
tff(fact_2109_inf_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb2
tff(fact_2110_inf_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb1
tff(fact_2111_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = Xa ) ) ) ).

% le_iff_inf
tff(fact_2112_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F2: fun(A,fun(A,A)),Xa: A,Y: A] :
          ( ! [X: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X),Y3)),X)
         => ( ! [X: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X),Y3)),Y3)
           => ( ! [X: A,Y3: A,Z3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),F2,Y3),Z3)) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),F2,Xa),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_2113_inf_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% inf.orderI
tff(fact_2114_inf_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ) ) ).

% inf.orderE
tff(fact_2115_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,Xa: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xa) ) ) ).

% le_infI2
tff(fact_2116_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,Xa: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xa) ) ) ).

% le_infI1
tff(fact_2117_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C2),D2)) ) ) ) ).

% inf_mono
tff(fact_2118_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)) ) ) ) ).

% le_infI
tff(fact_2119_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),B2) ) ) ) ).

% le_infE
tff(fact_2120_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Y) ) ).

% inf_le2
tff(fact_2121_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Xa) ) ).

% inf_le1
tff(fact_2122_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Xa) ) ).

% inf_sup_ord(1)
tff(fact_2123_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Y) ) ).

% inf_sup_ord(2)
tff(fact_2124_inf_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) ) ) ).

% inf.strict_coboundedI2
tff(fact_2125_inf_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) ) ) ).

% inf.strict_coboundedI1
tff(fact_2126_inf_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
tff(fact_2127_inf_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% inf.strict_boundedE
tff(fact_2128_inf_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb4
tff(fact_2129_inf_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb3
tff(fact_2130_less__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,Xa: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xa) ) ) ).

% less_infI2
tff(fact_2131_less__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,Xa: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xa) ) ) ).

% less_infI1
tff(fact_2132_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Z: A,Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)),Xa) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Xa)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z),Xa)) ) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_2133_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Z: A,Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)),Xa) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Xa)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z),Xa)) ) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_2134_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z)) ) ).

% boolean_algebra.disj_conj_distrib
tff(fact_2135_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z)) ) ).

% boolean_algebra.conj_disj_distrib
tff(fact_2136_sup__inf__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Y: A,Z: A,Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)),Xa) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Xa)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z),Xa)) ) ).

% sup_inf_distrib2
tff(fact_2137_sup__inf__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z)) ) ).

% sup_inf_distrib1
tff(fact_2138_inf__sup__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Y: A,Z: A,Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)),Xa) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Xa)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z),Xa)) ) ).

% inf_sup_distrib2
tff(fact_2139_inf__sup__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z)) ) ).

% inf_sup_distrib1
tff(fact_2140_distrib__imp2,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( ! [X: A,Y3: A,Z3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y3),Z3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z3))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z)) ) ) ) ).

% distrib_imp2
tff(fact_2141_distrib__imp1,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( ! [X: A,Y3: A,Z3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y3),Z3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z3))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z)) ) ) ) ).

% distrib_imp1
tff(fact_2142_disjointI,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( ! [X: A] :
          ( member(A,X,A2)
         => ~ member(A,X,B2) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A2),B2) = bot_bot(set(A)) ) ) ).

% disjointI
tff(fact_2143_disjoint__iff__not__equal,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> ! [X4: A] :
          ( member(A,X4,A3)
         => ! [Xa3: A] :
              ( member(A,Xa3,B3)
             => ( X4 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_2144_Int__empty__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),bot_bot(set(A))) = bot_bot(set(A)) ).

% Int_empty_right
tff(fact_2145_Int__empty__left,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),bot_bot(set(A))),B3) = bot_bot(set(A)) ).

% Int_empty_left
tff(fact_2146_disjoint__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> ! [X4: A] :
          ( member(A,X4,A3)
         => ~ member(A,X4,B3) ) ) ).

% disjoint_iff
tff(fact_2147_Int__emptyI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ! [X: A] :
          ( member(A,X,A3)
         => ~ member(A,X,B3) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) ) ) ).

% Int_emptyI
tff(fact_2148_Int__mono,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B3: set(A),D4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),D4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),D4)) ) ) ).

% Int_mono
tff(fact_2149_Int__lower1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),A3) ).

% Int_lower1
tff(fact_2150_Int__lower2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),B3) ).

% Int_lower2
tff(fact_2151_Int__absorb1,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = B3 ) ) ).

% Int_absorb1
tff(fact_2152_Int__absorb2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = A3 ) ) ).

% Int_absorb2
tff(fact_2153_Int__greatest,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),B3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) ) ) ).

% Int_greatest
tff(fact_2154_Int__Collect__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( ! [X: A] :
            ( member(A,X,A3)
           => ( aa(A,$o,P,X)
             => aa(A,$o,Q,X) ) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,P))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),collect(A,Q))) ) ) ).

% Int_Collect_mono
tff(fact_2155_Int__insert__right,axiom,
    ! [A: $tType,A3: set(A),A2: A,B3: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)) = $ite(member(A,A2,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) ).

% Int_insert_right
tff(fact_2156_Int__insert__left,axiom,
    ! [A: $tType,A2: A,B3: set(A),C3: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)),C3) = $ite(member(A,A2,C3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) ).

% Int_insert_left
tff(fact_2157_Diff__Int__distrib2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) ).

% Diff_Int_distrib2
tff(fact_2158_Diff__Int__distrib,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),B3)) ).

% Diff_Int_distrib
tff(fact_2159_Diff__Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ).

% Diff_Diff_Int
tff(fact_2160_Diff__Int2,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)),B3) ).

% Diff_Int2
tff(fact_2161_Int__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),C3)) ).

% Int_Diff
tff(fact_2162_Un__Int__distrib2,axiom,
    ! [A: $tType,B3: set(A),C3: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),A3)) ).

% Un_Int_distrib2
tff(fact_2163_Int__Un__distrib2,axiom,
    ! [A: $tType,B3: set(A),C3: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),A3)) ).

% Int_Un_distrib2
tff(fact_2164_Un__Int__distrib,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C3)) ).

% Un_Int_distrib
tff(fact_2165_Int__Un__distrib,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)) ).

% Int_Un_distrib
tff(fact_2166_Un__Int__crazy,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),A3)) ).

% Un_Int_crazy
tff(fact_2167_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z))) ) ).

% distrib_inf_le
tff(fact_2168_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z))) ) ).

% distrib_sup_le
tff(fact_2169_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na))
     => ( set_or1337092689740270186AtMost(int,M,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na)) = aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na)),set_or1337092689740270186AtMost(int,M,Na)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_2170_range__subset__card,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A),D2: word(A)] :
          ( aa(set(word(A)),$o,aa(set(word(A)),fun(set(word(A)),$o),ord_less_eq(set(word(A))),set_or1337092689740270186AtMost(word(A),A2,B2)),set_or1337092689740270186AtMost(word(A),C2,D2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),D2)
              & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),B2),A2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),D2),C2)) ) ) ) ) ).

% range_subset_card
tff(fact_2171_disjoint__mono,axiom,
    ! [A: $tType,A2: set(A),A4: set(A),B2: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A2),A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),B5)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B5) = bot_bot(set(A)) )
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A2),B2) = bot_bot(set(A)) ) ) ) ) ).

% disjoint_mono
tff(fact_2172_disjoint__alt__simp2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) != A3 )
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) != bot_bot(set(A)) ) ) ).

% disjoint_alt_simp2
tff(fact_2173_disjoint__alt__simp1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) = A3 )
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) ) ) ).

% disjoint_alt_simp1
tff(fact_2174_Diff__triv,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) = A3 ) ) ).

% Diff_triv
tff(fact_2175_Int__Diff__disjoint,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = bot_bot(set(A)) ).

% Int_Diff_disjoint
tff(fact_2176_Un__Int__assoc__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),A3) ) ).

% Un_Int_assoc_eq
tff(fact_2177_Diff__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),C3)) ).

% Diff_Un
tff(fact_2178_Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),C3)) ).

% Diff_Int
tff(fact_2179_Int__Diff__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = A3 ).

% Int_Diff_Un
tff(fact_2180_Un__Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = A3 ).

% Un_Diff_Int
tff(fact_2181_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1337092689740270186AtMost(int,I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),bot_bot(set(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),I),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J))) ).

% simp_from_to
tff(fact_2182_bset_I1_J,axiom,
    ! [D4: int,B3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B3)
                 => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
         => ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)) ) )
     => ( ! [X: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B3)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( aa(int,$o,Q,X)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)) ) )
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B3)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( aa(int,$o,P,X5)
                & aa(int,$o,Q,X5) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D4))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D4)) ) ) ) ) ) ).

% bset(1)
tff(fact_2183_bset_I2_J,axiom,
    ! [D4: int,B3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B3)
                 => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
         => ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)) ) )
     => ( ! [X: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B3)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( aa(int,$o,Q,X)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)) ) )
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B3)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( aa(int,$o,P,X5)
                | aa(int,$o,Q,X5) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D4))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D4)) ) ) ) ) ) ).

% bset(2)
tff(fact_2184_aset_I1_J,axiom,
    ! [D4: int,A3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A3)
                 => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
         => ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)) ) )
     => ( ! [X: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
           => ( aa(int,$o,Q,X)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)) ) )
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A3)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( ( aa(int,$o,P,X5)
                & aa(int,$o,Q,X5) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D4))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D4)) ) ) ) ) ) ).

% aset(1)
tff(fact_2185_aset_I2_J,axiom,
    ! [D4: int,A3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A3)
                 => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
         => ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)) ) )
     => ( ! [X: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
           => ( aa(int,$o,Q,X)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)) ) )
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A3)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( ( aa(int,$o,P,X5)
                | aa(int,$o,Q,X5) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D4))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D4)) ) ) ) ) ) ).

% aset(2)
tff(fact_2186_disjoint__alt__simp3,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),A3)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) != bot_bot(set(A)) ) ) ).

% disjoint_alt_simp3
tff(fact_2187_bset_I3_J,axiom,
    ! [D4: int,T: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),T),one_one(int)),B3)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B3)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( X5 = T )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D4) = T ) ) ) ) ) ).

% bset(3)
tff(fact_2188_bset_I4_J,axiom,
    ! [D4: int,T: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( member(int,T,B3)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B3)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( X5 != T )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D4) != T ) ) ) ) ) ).

% bset(4)
tff(fact_2189_bset_I5_J,axiom,
    ! [D4: int,B3: set(int),T: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B3)
                 => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X5),T)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D4)),T) ) ) ) ).

% bset(5)
tff(fact_2190_bset_I7_J,axiom,
    ! [D4: int,T: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( member(int,T,B3)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B3)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),T),X5)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),T),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D4)) ) ) ) ) ).

% bset(7)
tff(fact_2191_aset_I3_J,axiom,
    ! [D4: int,T: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T),one_one(int)),A3)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A3)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( ( X5 = T )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D4) = T ) ) ) ) ) ).

% aset(3)
tff(fact_2192_aset_I4_J,axiom,
    ! [D4: int,T: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( member(int,T,A3)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A3)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( ( X5 != T )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D4) != T ) ) ) ) ) ).

% aset(4)
tff(fact_2193_aset_I5_J,axiom,
    ! [D4: int,T: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( member(int,T,A3)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A3)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X5),T)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D4)),T) ) ) ) ) ).

% aset(5)
tff(fact_2194_aset_I7_J,axiom,
    ! [D4: int,A3: set(int),T: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A3)
                 => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),T),X5)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),T),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D4)) ) ) ) ).

% aset(7)
tff(fact_2195_periodic__finite__ex,axiom,
    ! [D2: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X: int,K2: int] :
            ( aa(int,$o,P,X)
          <=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2))) )
       => ( ? [X_12: int] : aa(int,$o,P,X_12)
        <=> ? [X4: int] :
              ( member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D2))
              & aa(int,$o,P,X4) ) ) ) ) ).

% periodic_finite_ex
tff(fact_2196_word__subset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),R: word(A),Y: word(A),Sa: word(A)] :
          ( aa(set(word(A)),$o,aa(set(word(A)),fun(set(word(A)),$o),ord_less_eq(set(word(A))),set_or1337092689740270186AtMost(word(A),Xa,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),R)),one_one(word(A))))),set_or1337092689740270186AtMost(word(A),Y,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Sa)),one_one(word(A)))))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),R)),one_one(word(A))))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Sa)),one_one(word(A))))
             => ( ( Sa != zero_zero(word(A)) )
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),R),Sa) ) ) ) ) ) ).

% word_subset_less
tff(fact_2197_bset_I6_J,axiom,
    ! [D4: int,B3: set(int),T: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B3)
                 => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X5),T)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D4)),T) ) ) ) ).

% bset(6)
tff(fact_2198_bset_I8_J,axiom,
    ! [D4: int,T: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),T),one_one(int)),B3)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B3)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T),X5)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D4)) ) ) ) ) ).

% bset(8)
tff(fact_2199_aset_I6_J,axiom,
    ! [D4: int,T: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T),one_one(int)),A3)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A3)
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X5),T)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D4)),T) ) ) ) ) ).

% aset(6)
tff(fact_2200_aset_I8_J,axiom,
    ! [D4: int,A3: set(int),T: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A3)
                 => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T),X5)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D4)) ) ) ) ).

% aset(8)
tff(fact_2201_cpmi,axiom,
    ! [D4: int,P: fun(int,$o),P3: fun(int,$o),B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( ? [Z5: int] :
          ! [X: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Z5)
           => ( aa(int,$o,P,X)
            <=> aa(int,$o,P3,X) ) )
       => ( ! [X: int] :
              ( ! [Xa2: int] :
                  ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D4))
                 => ! [Xb2: int] :
                      ( member(int,Xb2,B3)
                     => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
             => ( aa(int,$o,P,X)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)) ) )
         => ( ! [X: int,K2: int] :
                ( aa(int,$o,P3,X)
              <=> aa(int,$o,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D4))) )
           => ( ? [X_12: int] : aa(int,$o,P,X_12)
            <=> ( ? [X4: int] :
                    ( member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D4))
                    & aa(int,$o,P3,X4) )
                | ? [X4: int] :
                    ( member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D4))
                    & ? [Xa3: int] :
                        ( member(int,Xa3,B3)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa3),X4)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_2202_cppi,axiom,
    ! [D4: int,P: fun(int,$o),P3: fun(int,$o),A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( ? [Z5: int] :
          ! [X: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z5),X)
           => ( aa(int,$o,P,X)
            <=> aa(int,$o,P3,X) ) )
       => ( ! [X: int] :
              ( ! [Xa2: int] :
                  ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D4))
                 => ! [Xb2: int] :
                      ( member(int,Xb2,A3)
                     => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
             => ( aa(int,$o,P,X)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)) ) )
         => ( ! [X: int,K2: int] :
                ( aa(int,$o,P3,X)
              <=> aa(int,$o,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D4))) )
           => ( ? [X_12: int] : aa(int,$o,P,X_12)
            <=> ( ? [X4: int] :
                    ( member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D4))
                    & aa(int,$o,P3,X4) )
                | ? [X4: int] :
                    ( member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D4))
                    & ? [Xa3: int] :
                        ( member(int,Xa3,A3)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa3),X4)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_2203_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_2204_atLeastAtMost__singleton_H,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
         => ( set_or1337092689740270186AtMost(A,A2,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ) ) ) ).

% atLeastAtMost_singleton'
tff(fact_2205_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
              | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
                & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2)
                  | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D2) ) ) )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2206_time__replicate,axiom,
    ! [A: $tType,Xa: heap_Time_Heap(A),C2: nat,Na: nat,H: heap_ext(product_unit)] :
      ( ! [H2: heap_ext(product_unit)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,Xa,H2)),C2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(list(A),vEBT_VEBT_replicatei(A,Na,Xa),H)),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))) ) ).

% time_replicate
tff(fact_2207_TBOUND__buildupi,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => time_TBOUND(vEBT_VEBTi,vEBT_vebt_buildupi(Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ) ).

% TBOUND_buildupi
tff(fact_2208_minNull__delete__time__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( vEBT_VEBT_minNull(vEBT_vebt_delete(T,Xa))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_d_e_l_e_t_e(T,Xa)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))) ) ) ).

% minNull_delete_time_bound
tff(fact_2209_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
         => ( aa(nat,$o,vEBT_V8194947554948674370ptions(T),Xa)
           => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(T,Y)),Xa) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_2210_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(T,Xa)),Xa) ) ) ).

% valid_insert_both_member_options_add
tff(fact_2211_minNull__delete__time__bound_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( vEBT_VEBT_minNull(vEBT_vebt_delete(T,Xa))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V1232361888498592333_e_t_e(T,Xa)),one_one(nat)) ) ) ).

% minNull_delete_time_bound'
tff(fact_2212_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(T)
     => ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(T),X_1) ) ).

% not_min_Null_member
tff(fact_2213_min__Null__member,axiom,
    ! [T: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_minNull(T)
     => ~ aa(nat,$o,vEBT_vebt_member(T),Xa) ) ).

% min_Null_member
tff(fact_2214_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_delete(T,Xa)),Y)
      <=> ( ( Xa != Y )
          & aa(nat,$o,vEBT_V8194947554948674370ptions(T),Y) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_2215_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(T),Xa)
       => aa(nat,$o,vEBT_vebt_member(T),Xa) ) ) ).

% valid_member_both_member_options
tff(fact_2216_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(T),Xa)
      <=> aa(nat,$o,vEBT_vebt_member(T),Xa) ) ) ).

% both_member_options_equiv_member
tff(fact_2217_TBOUND__vebt__buildupi,axiom,
    ! [Na: nat] : time_TBOUND(vEBT_VEBTi,vEBT_V739175172307565963ildupi(Na),vEBT_V441764108873111860ildupi(Na)) ).

% TBOUND_vebt_buildupi
tff(fact_2218_set__vebt__minNull,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( vEBT_VEBT_minNull(T)
      <=> ( vEBT_set_vebt(T) = bot_bot(set(nat)) ) ) ) ).

% set_vebt_minNull
tff(fact_2219_TBOUND__replicate,axiom,
    ! [A: $tType,Xa: heap_Time_Heap(A),C2: nat,Na: nat] :
      ( time_TBOUND(A,Xa,C2)
     => time_TBOUND(list(A),vEBT_VEBT_replicatei(A,Na,Xa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),C2)),Na))) ) ).

% TBOUND_replicate
tff(fact_2220_TBOUND__minNull,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_VEBT_minNull(T)
     => time_TBOUND(vEBT_VEBTi,vEBT_V3964819847710782039nserti(T,Tia,Xa),one_one(nat)) ) ).

% TBOUND_minNull
tff(fact_2221_TBOUND__vebt__deletei,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] :
      time_TBOUND(vEBT_VEBTi,vEBT_V1365221501068881998eletei(T,Tia,Xa),
        $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(T,Xa)),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))))) ).

% TBOUND_vebt_deletei
tff(fact_2222_TBOUND__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] :
      time_TBOUND(vEBT_VEBTi,vEBT_V3964819847710782039nserti(T,Tia,Xa),
        $ite(vEBT_VEBT_minNull(T),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))))) ).

% TBOUND_vebt_inserti
tff(fact_2223_inf__Int__eq,axiom,
    ! [A: $tType,R2: set(A),S: set(A),X5: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),R2)),aTP_Lamp_a(set(A),fun(A,$o),S)),X5)
    <=> member(A,X5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),R2),S)) ) ).

% inf_Int_eq
tff(fact_2224_inf__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = collect(A,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A3)),aTP_Lamp_a(set(A),fun(A,$o),B3))) ).

% inf_set_def
tff(fact_2225_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect(nat,vEBT_V8194947554948674370ptions(T)) ).

% set_vebt_def
tff(fact_2226_insersimp,axiom,
    ! [T: vEBT_VEBT,Na: nat,Y: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(T),X_1)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(T,Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ) ) ).

% insersimp
tff(fact_2227_insertsimp,axiom,
    ! [T: vEBT_VEBT,Na: nat,L: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( vEBT_VEBT_minNull(T)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(T,L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ) ) ).

% insertsimp
tff(fact_2228_buildup__nothing__in__leaf,axiom,
    ! [Na: nat,Xa: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(Na),Xa) ).

% buildup_nothing_in_leaf
tff(fact_2229_succ__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c2(T,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% succ_bound_size_univ'
tff(fact_2230_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),Y))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))))
           => ( archimedean_round(A,Xa) = Y ) ) ) ) ).

% round_unique
tff(fact_2231_pred__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d2(T,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% pred_bound_size_univ'
tff(fact_2232_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(aa(num,num,bit1,bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% vebt_buildupi_rule
tff(fact_2233_tanh__ln__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,tanh(real),aa(real,real,ln_ln(real),Xa)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))) ) ) ).

% tanh_ln_real
tff(fact_2234_is__succ__in__set__def,axiom,
    ! [Xs: set(nat),Xa: nat,Y: nat] :
      ( vEBT_is_succ_in_set(Xs,Xa,Y)
    <=> ( member(nat,Y,Xs)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
        & ! [X4: nat] :
            ( member(nat,X4,Xs)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),X4)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),X4) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_2235_tanh__real__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa(real,real,tanh(real),Xa) = aa(real,real,tanh(real),Y) )
    <=> ( Xa = Y ) ) ).

% tanh_real_eq_iff
tff(fact_2236_inf1I,axiom,
    ! [A: $tType,A3: fun(A,$o),Xa: A,B3: fun(A,$o)] :
      ( aa(A,$o,A3,Xa)
     => ( aa(A,$o,B3,Xa)
       => aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),A3),B3),Xa) ) ) ).

% inf1I
tff(fact_2237_tanh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,tanh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% tanh_0
tff(fact_2238_tanh__real__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa(real,real,tanh(real),Xa) = zero_zero(real) )
    <=> ( Xa = zero_zero(real) ) ) ).

% tanh_real_zero_iff
tff(fact_2239_tanh__real__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),Xa)),aa(real,real,tanh(real),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ).

% tanh_real_less_iff
tff(fact_2240_tanh__real__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tanh(real),Xa)),aa(real,real,tanh(real),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ).

% tanh_real_le_iff
tff(fact_2241_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_2242_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_2243_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_2244_tanh__real__pos__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,tanh(real),Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% tanh_real_pos_iff
tff(fact_2245_tanh__real__neg__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% tanh_real_neg_iff
tff(fact_2246_tanh__real__nonpos__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tanh(real),Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% tanh_real_nonpos_iff
tff(fact_2247_tanh__real__nonneg__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,tanh(real),Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% tanh_real_nonneg_iff
tff(fact_2248_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_2249_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_2250_TBOUND__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : time_TBOUND($o,vEBT_V854960066525838166emberi(T,Tia,Xa),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T)))) ).

% TBOUND_vebt_memberi
tff(fact_2251_TBOUND__vebt__succi,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : time_TBOUND(option(nat),vEBT_VEBT_vebt_succi(T,Tia,Xa),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T)))) ).

% TBOUND_vebt_succi
tff(fact_2252_TBOUND__vebt__predi,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : time_TBOUND(option(nat),vEBT_VEBT_vebt_predi(T,Tia,Xa),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T)))) ).

% TBOUND_vebt_predi
tff(fact_2253_inf1D2,axiom,
    ! [A: $tType,A3: fun(A,$o),B3: fun(A,$o),Xa: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),A3),B3),Xa)
     => aa(A,$o,B3,Xa) ) ).

% inf1D2
tff(fact_2254_inf1D1,axiom,
    ! [A: $tType,A3: fun(A,$o),B3: fun(A,$o),Xa: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),A3),B3),Xa)
     => aa(A,$o,A3,Xa) ) ).

% inf1D1
tff(fact_2255_inf1E,axiom,
    ! [A: $tType,A3: fun(A,$o),B3: fun(A,$o),Xa: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),A3),B3),Xa)
     => ~ ( aa(A,$o,A3,Xa)
         => ~ aa(A,$o,B3,Xa) ) ) ).

% inf1E
tff(fact_2256_artanh__tanh__real,axiom,
    ! [Xa: real] : aa(real,real,artanh(real),aa(real,real,tanh(real),Xa)) = Xa ).

% artanh_tanh_real
tff(fact_2257_tanh__real__lt__1,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),Xa)),one_one(real)) ).

% tanh_real_lt_1
tff(fact_2258_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,Xa)),archimedean_round(A,Y)) ) ) ).

% round_mono
tff(fact_2259_ceiling__ge__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,Xa)),archimedean_ceiling(A,Xa)) ) ).

% ceiling_ge_round
tff(fact_2260_pred__bound__height_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_p_r_e_d2(T,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))) ) ).

% pred_bound_height'
tff(fact_2261_succ_H__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_s_u_c_c2(T,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))) ) ).

% succ'_bound_height
tff(fact_2262_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xa))),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% of_int_round_le
tff(fact_2263_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xa))) ) ).

% of_int_round_ge
tff(fact_2264_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xa))) ) ).

% of_int_round_gt
tff(fact_2265_VEBT__internal_Ovebt__buildupi__rule,axiom,
    ! [Na: nat] : time_htt(vEBT_VEBTi,pure_assn(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)),vEBT_vebt_buildupi(Na),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% VEBT_internal.vebt_buildupi_rule
tff(fact_2266_htt__vebt__memberi__invar__vebt,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => time_htt($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_memberi(Tia,Xa),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_aq(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),T),Tia),Xa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ) ).

% htt_vebt_memberi_invar_vebt
tff(fact_2267_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(Tree,Na)
     => ( aa(nat,$o,vEBT_vebt_member(Tree),Xa)
       => ( vEBT_V5719532721284313246member(Tree,Xa)
          | vEBT_VEBT_membermima(Tree,Xa) ) ) ) ).

% member_valid_both_member_options
tff(fact_2268_htt__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : time_htt($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_memberi(Tia,Xa),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_aq(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),T),Tia),Xa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T)))) ).

% htt_vebt_memberi
tff(fact_2269_TBOUND__minNulli,axiom,
    ! [T: vEBT_VEBTi] : time_TBOUND($o,vEBT_VEBT_minNulli(T),one_one(nat)) ).

% TBOUND_minNulli
tff(fact_2270_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,vEBT_V8194947554948674370ptions(T),Xa)
    <=> ( vEBT_V5719532721284313246member(T,Xa)
        | vEBT_VEBT_membermima(T,Xa) ) ) ).

% both_member_options_def
tff(fact_2271_vebt__deletei__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T: vEBT_VEBT] : refine_Imp_refines(vEBT_VEBTi,vEBT_vebt_deletei(Tia,Xa),vEBT_V1365221501068881998eletei(T,Tia,Xa)) ).

% vebt_deletei_refines
tff(fact_2272_buildup__nothing__in__min__max,axiom,
    ! [Na: nat,Xa: nat] : ~ vEBT_VEBT_membermima(vEBT_vebt_buildup(Na),Xa) ).

% buildup_nothing_in_min_max
tff(fact_2273_refines__replicate,axiom,
    ! [A: $tType,F2: heap_Time_Heap(A),F4: heap_Time_Heap(A),Na: nat] :
      ( refine_Imp_refines(A,F2,F4)
     => refine_Imp_refines(list(A),vEBT_VEBT_replicatei(A,Na,F2),vEBT_VEBT_replicatei(A,Na,F4)) ) ).

% refines_replicate
tff(fact_2274_vebt__buildupi__refines,axiom,
    ! [Na: nat] : refine_Imp_refines(vEBT_VEBTi,vEBT_vebt_buildupi(Na),vEBT_V739175172307565963ildupi(Na)) ).

% vebt_buildupi_refines
tff(fact_2275_vebt__inserti__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T: vEBT_VEBT] : refine_Imp_refines(vEBT_VEBTi,vEBT_vebt_inserti(Tia,Xa),vEBT_V3964819847710782039nserti(T,Tia,Xa)) ).

% vebt_inserti_refines
tff(fact_2276_minNrulli__ruleT,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : time_htt($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_VEBT_minNulli(Tia),aa(vEBT_VEBTi,fun($o,assn),aTP_Lamp_ar(vEBT_VEBT,fun(vEBT_VEBTi,fun($o,assn)),T),Tia),one_one(nat)) ).

% minNrulli_ruleT
tff(fact_2277_refines__case__VEBTi,axiom,
    ! [A: $tType,Tia: vEBT_VEBTi,Ti: vEBT_VEBTi,F1: fun($o,fun($o,heap_Time_Heap(A))),F12: fun($o,fun($o,heap_Time_Heap(A))),F22: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))))),F23: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)))))] :
      ( ( Tia = Ti )
     => ( ! [A5: $o,B6: $o] : refine_Imp_refines(A,aa($o,heap_Time_Heap(A),aa($o,fun($o,heap_Time_Heap(A)),F1,(A5)),(B6)),aa($o,heap_Time_Heap(A),aa($o,fun($o,heap_Time_Heap(A)),F12,(A5)),(B6)))
       => ( ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeArray: array(vEBT_VEBTi),Summary: vEBT_VEBTi] : refine_Imp_refines(A,aa(vEBT_VEBTi,heap_Time_Heap(A),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)))),F22,Info),Deg2),TreeArray),Summary),aa(vEBT_VEBTi,heap_Time_Heap(A),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)))),F23,Info),Deg2),TreeArray),Summary))
         => refine_Imp_refines(A,vEBT_case_VEBTi(heap_Time_Heap(A),F22,F1,Tia),vEBT_case_VEBTi(heap_Time_Heap(A),F23,F12,Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_2278_vebt__memberi__rule,axiom,
    ! [Na: nat,Sa: set(nat),Tia: vEBT_VEBTi,Xa: nat] : time_htt($o,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_memberi(Tia,Xa),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn))),aTP_Lamp_as(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn)))),Na),Sa),Tia),Xa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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_2279_vebt__memberi__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T: vEBT_VEBT] : refine_Imp_refines($o,vEBT_vebt_memberi(Tia,Xa),vEBT_V854960066525838166emberi(T,Tia,Xa)) ).

% vebt_memberi_refines
tff(fact_2280_ln__one__minus__pos__lower__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,uminus_uminus(real),Xa)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),Xa))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_2281_is__pred__in__set__def,axiom,
    ! [Xs: set(nat),Xa: nat,Y: nat] :
      ( vEBT_is_pred_in_set(Xs,Xa,Y)
    <=> ( member(nat,Y,Xs)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa)
        & ! [X4: nat] :
            ( member(nat,X4,Xs)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),Xa)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_2282_arcosh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xa: A] : aa(A,A,arcosh(A),Xa) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),powr(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(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_2283_vebt__memberi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : hoare_hoare_triple($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_V854960066525838166emberi(T,Tia,Xa),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_aq(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),T),Tia),Xa)) ).

% vebt_memberi'_rf_abstr
tff(fact_2284_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_2285_neg__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,uminus_uminus(A),B2) )
        <=> ( A2 = B2 ) ) ) ).

% neg_equal_iff_equal
tff(fact_2286_add_Oinverse__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),A2)) = A2 ) ).

% add.inverse_inverse
tff(fact_2287_verit__minus__simplify_I4_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),B2)) = B2 ) ).

% verit_minus_simplify(4)
tff(fact_2288_uminus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( uminus(A)
     => ! [A3: fun(B,A),Xa: B] : aa(B,A,aa(fun(B,A),fun(B,A),uminus_uminus(fun(B,A)),A3),Xa) = aa(A,A,uminus_uminus(A),aa(B,A,A3,Xa)) ) ).

% uminus_apply
tff(fact_2289_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,uminus_uminus(A),Xa) = aa(A,A,uminus_uminus(A),Y) )
        <=> ( Xa = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
tff(fact_2290_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),Xa)) = Xa ) ).

% boolean_algebra_class.boolean_algebra.double_compl
tff(fact_2291_abs__idempotent,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_idempotent
tff(fact_2292_arsinh__minus__real,axiom,
    ! [Xa: real] : aa(real,real,arsinh(real),aa(real,real,uminus_uminus(real),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,arsinh(real),Xa)) ).

% arsinh_minus_real
tff(fact_2293_neg__equal__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = A2 )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% neg_equal_zero
tff(fact_2294_equal__neg__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% equal_neg_zero
tff(fact_2295_neg__equal__0__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% neg_equal_0_iff_equal
tff(fact_2296_neg__0__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),A2) )
        <=> ( zero_zero(A) = A2 ) ) ) ).

% neg_0_equal_iff_equal
tff(fact_2297_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_2298_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% neg_le_iff_le
tff(fact_2299_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% compl_le_compl_iff
tff(fact_2300_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% compl_less_compl_iff
tff(fact_2301_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% neg_less_iff_less
tff(fact_2302_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_2303_minus__add__distrib,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_add_distrib
tff(fact_2304_minus__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = B2 ) ).

% minus_add_cancel
tff(fact_2305_add__minus__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),B2)) = B2 ) ).

% add_minus_cancel
tff(fact_2306_mult__minus__right,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% mult_minus_right
tff(fact_2307_minus__mult__minus,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) ) ).

% minus_mult_minus
tff(fact_2308_mult__minus__left,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% mult_minus_left
tff(fact_2309_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2) ) ).

% minus_diff_eq
tff(fact_2310_div__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ).

% div_minus_minus
tff(fact_2311_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_2312_abs__eq__0,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0
tff(fact_2313_abs__0__eq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,abs_abs(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_0_eq
tff(fact_2314_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_2315_minNulli__rule,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : hoare_hoare_triple($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_VEBT_minNulli(Tia),aa(vEBT_VEBTi,fun($o,assn),aTP_Lamp_ar(vEBT_VEBT,fun(vEBT_VEBTi,fun($o,assn)),T),Tia)) ).

% minNulli_rule
tff(fact_2316_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_add_abs
tff(fact_2317_mod__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) ) ).

% mod_minus_minus
tff(fact_2318_abs__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),A2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) ) ).

% abs_mult_self_eq
tff(fact_2319_abs__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_divide
tff(fact_2320_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_2321_abs__minus__cancel,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_minus_cancel
tff(fact_2322_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_2323_of__int__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: int] : aa(int,A,ring_1_of_int(A),aa(int,int,abs_abs(int),Xa)) = aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Xa)) ) ).

% of_int_abs
tff(fact_2324_tanh__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,tanh(A),aa(A,A,uminus_uminus(A),Xa)) = aa(A,A,uminus_uminus(A),aa(A,A,tanh(A),Xa)) ) ).

% tanh_minus
tff(fact_2325_tanh__real__abs,axiom,
    ! [Xa: real] : aa(real,real,tanh(real),aa(real,real,abs_abs(real),Xa)) = aa(real,real,abs_abs(real),aa(real,real,tanh(real),Xa)) ).

% tanh_real_abs
tff(fact_2326_vebt__predi__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T: vEBT_VEBT] : refine_Imp_refines(option(nat),vEBT_vebt_predi(Tia,Xa),vEBT_VEBT_vebt_predi(T,Tia,Xa)) ).

% vebt_predi_refines
tff(fact_2327_vebt__succi__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T: vEBT_VEBT] : refine_Imp_refines(option(nat),vEBT_vebt_succi(Tia,Xa),vEBT_VEBT_vebt_succi(T,Tia,Xa)) ).

% vebt_succi_refines
tff(fact_2328_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% neg_less_eq_nonneg
tff(fact_2329_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% less_eq_neg_nonpos
tff(fact_2330_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% neg_le_0_iff_le
tff(fact_2331_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% neg_0_le_iff_le
tff(fact_2332_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% neg_less_0_iff_less
tff(fact_2333_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% neg_0_less_iff_less
tff(fact_2334_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% neg_less_pos
tff(fact_2335_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% less_neg_neg
tff(fact_2336_add_Oright__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),A2)) = zero_zero(A) ) ).

% add.right_inverse
tff(fact_2337_ab__left__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),A2) = zero_zero(A) ) ).

% ab_left_minus
tff(fact_2338_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_2339_verit__minus__simplify_I3_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),B2) = aa(A,A,uminus_uminus(A),B2) ) ).

% verit_minus_simplify(3)
tff(fact_2340_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A2) = aa(A,A,uminus_uminus(A),A2) ) ).

% diff_0
tff(fact_2341_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_2342_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_2343_uminus__add__conv__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2) ) ).

% uminus_add_conv_diff
tff(fact_2344_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) ) ).

% diff_minus_eq_add
tff(fact_2345_div__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),A2) ) ).

% div_minus1_right
tff(fact_2346_divide__minus1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),Xa) ) ).

% divide_minus1
tff(fact_2347_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),zero_zero(A))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_2348_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% abs_le_self_iff
tff(fact_2349_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_nonneg
tff(fact_2350_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A2))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_2351_minus__mod__self1,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% minus_mod_self1
tff(fact_2352_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_2353_inf__compl__bot__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)) = bot_bot(A) ) ).

% inf_compl_bot_left1
tff(fact_2354_inf__compl__bot__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),Y)) = bot_bot(A) ) ).

% inf_compl_bot_left2
tff(fact_2355_inf__compl__bot__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,uminus_uminus(A),Xa))) = bot_bot(A) ) ).

% inf_compl_bot_right
tff(fact_2356_boolean__algebra_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),Xa) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_left
tff(fact_2357_boolean__algebra_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,uminus_uminus(A),Xa)) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_right
tff(fact_2358_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_2359_abs__power__minus,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] : aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Na)) = aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ).

% abs_power_minus
tff(fact_2360_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Y)) ) ).

% boolean_algebra.de_Morgan_disj
tff(fact_2361_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Y)) ) ).

% boolean_algebra.de_Morgan_conj
tff(fact_2362_real__add__minus__iff,axiom,
    ! [Xa: real,A2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,uminus_uminus(real),A2)) = zero_zero(real) )
    <=> ( Xa = A2 ) ) ).

% real_add_minus_iff
tff(fact_2363_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_2364_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_2365_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_2366_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_2367_diff__numeral__special_I12_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% diff_numeral_special(12)
tff(fact_2368_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Na)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Na)),A2)) = A2 ) ).

% left_minus_one_mult_self
tff(fact_2369_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,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Na)) = one_one(A) ) ).

% minus_one_mult_self
tff(fact_2370_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,abs_abs(A),B2)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_2371_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,abs_abs(A),B2))),zero_zero(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_2372_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).

% mod_minus1_right
tff(fact_2373_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,A,abs_abs(A),A2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% abs_of_nonpos
tff(fact_2374_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_2375_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_2376_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Na: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),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_2377_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Na: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(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_2378_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_2379_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_2380_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_2381_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_2382_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_2383_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_2384_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_2385_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_2386_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_2387_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_2388_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_2389_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W: num] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),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),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_2390_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = A2 )
        <=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_2391_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_2392_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B2) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_2393_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_2394_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B2) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_2395_power2__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_minus
tff(fact_2396_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),Na))
        <=> ( ( A2 != zero_zero(A) )
            | ( Na = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_2397_power2__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_abs
tff(fact_2398_abs__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% abs_power2
tff(fact_2399_artanh__minus__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,real,artanh(real),aa(real,real,uminus_uminus(real),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,artanh(real),Xa)) ) ) ).

% artanh_minus_real
tff(fact_2400_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_2401_diff__numeral__special_I10_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% diff_numeral_special(10)
tff(fact_2402_diff__numeral__special_I11_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% diff_numeral_special(11)
tff(fact_2403_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_2404_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_2405_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_2406_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ).

% Power.ring_1_class.power_minus_even
tff(fact_2407_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2))) ) ).

% diff_numeral_special(4)
tff(fact_2408_diff__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),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_2409_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,Xa: num,Na: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa))),Na) )
        <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xa))),Na) ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_2410_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xa: num,Na: nat,Y: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa))),Na) = aa(int,A,ring_1_of_int(A),Y) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xa))),Na) = Y ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_2411_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_2412_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),Xa) ) ) ).

% neg_numeral_less_ceiling
tff(fact_2413_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% ceiling_less_zero
tff(fact_2414_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),Xa) ) ) ).

% zero_le_ceiling
tff(fact_2415_power__minus1__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,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_2416_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_2417_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),Xa) ) ) ).

% neg_numeral_le_ceiling
tff(fact_2418_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,Na: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa))),Na)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xa))),Na)),A2) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_2419_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xa: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa))),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xa))),Na)) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_2420_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xa: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa))),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xa))),Na)) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_2421_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,Na: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa))),Na)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xa))),Na)),A2) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_2422_square__powr__half,axiom,
    ! [Xa: real] : powr(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),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),Xa) ).

% square_powr_half
tff(fact_2423_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,A,abs_abs(A),A2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% abs_of_neg
tff(fact_2424_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X5: A] :
          aa(A,A,abs_abs(A),X5) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),zero_zero(A)),aa(A,A,uminus_uminus(A),X5),X5) ) ).

% abs_if_raw
tff(fact_2425_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A2: A] :
          aa(A,A,abs_abs(A),A2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)),aa(A,A,uminus_uminus(A),A2),A2) ) ).

% abs_if
tff(fact_2426_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2) ) ) ) ).

% abs_leI
tff(fact_2427_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ) ).

% abs_le_D2
tff(fact_2428_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ) ) ).

% abs_le_iff
tff(fact_2429_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_minus_self
tff(fact_2430_minus__equation__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = B2 )
        <=> ( aa(A,A,uminus_uminus(A),B2) = A2 ) ) ) ).

% minus_equation_iff
tff(fact_2431_equation__minus__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% equation_minus_iff
tff(fact_2432_verit__negate__coefficient_I3_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
         => ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,uminus_uminus(A),B2) ) ) ) ).

% verit_negate_coefficient(3)
tff(fact_2433_fun__Compl__def,axiom,
    ! [B: $tType,A: $tType] :
      ( uminus(B)
     => ! [A3: fun(A,B),X5: A] : aa(A,B,aa(fun(A,B),fun(A,B),uminus_uminus(fun(A,B)),A3),X5) = aa(B,B,uminus_uminus(B),aa(A,B,A3,X5)) ) ).

% fun_Compl_def
tff(fact_2434_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ) ) ).

% abs_less_iff
tff(fact_2435_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),A2))),zero_zero(A)) ) ).

% abs_minus_le_zero
tff(fact_2436_abs__real__def,axiom,
    ! [A2: real] :
      aa(real,real,abs_abs(real),A2) = $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real)),aa(real,real,uminus_uminus(real),A2),A2) ).

% abs_real_def
tff(fact_2437_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_self
tff(fact_2438_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% abs_le_D1
tff(fact_2439_abs__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_mult
tff(fact_2440_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) ) ).

% abs_minus_commute
tff(fact_2441_power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] : aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),Na) ) ).

% power_abs
tff(fact_2442_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% le_imp_neg_le
tff(fact_2443_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A2) ) ) ).

% minus_le_iff
tff(fact_2444_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% le_minus_iff
tff(fact_2445_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xa)),Y) ) ) ).

% compl_le_swap2
tff(fact_2446_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),Xa))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_le_swap1
tff(fact_2447_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),Xa)) ) ) ).

% compl_mono
tff(fact_2448_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Y)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Xa)),Y) ) ) ).

% compl_less_swap2
tff(fact_2449_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,uminus_uminus(A),Xa))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_less_swap1
tff(fact_2450_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),A2) ) ) ).

% minus_less_iff
tff(fact_2451_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% less_minus_iff
tff(fact_2452_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% verit_negate_coefficient(2)
tff(fact_2453_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_2454_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_2455_is__num__normalize_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ).

% is_num_normalize(8)
tff(fact_2456_add_Oinverse__distrib__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ).

% add.inverse_distrib_swap
tff(fact_2457_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,K: A,A2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,uminus_uminus(A),A2)) ) ) ) ).

% group_cancel.neg1
tff(fact_2458_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_2459_minus__mult__commute,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_mult_commute
tff(fact_2460_square__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),B2) )
        <=> ( ( A2 = B2 )
            | ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% square_eq_iff
tff(fact_2461_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),B2)),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% minus_diff_commute
tff(fact_2462_minus__diff__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ).

% minus_diff_minus
tff(fact_2463_minus__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_divide_right
tff(fact_2464_minus__divide__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ).

% minus_divide_divide
tff(fact_2465_minus__divide__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% minus_divide_left
tff(fact_2466_div__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% div_minus_right
tff(fact_2467_mod__minus__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2))),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% mod_minus_eq
tff(fact_2468_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,A4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A4),B2) )
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A4)),B2) ) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
tff(fact_2469_mod__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A2)),B2)) ) ).

% mod_minus_right
tff(fact_2470_VEBTi_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),F22: fun($o,fun($o,B)),VEBTi: vEBT_VEBTi] : aa(B,A,H,vEBT_case_VEBTi(B,F1,F22,VEBTi)) = vEBT_case_VEBTi(A,aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),aTP_Lamp_at(fun(B,A),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A))))),H),F1),aa(fun($o,fun($o,B)),fun($o,fun($o,A)),aTP_Lamp_au(fun(B,A),fun(fun($o,fun($o,B)),fun($o,fun($o,A))),H),F22),VEBTi) ).

% VEBTi.case_distrib
tff(fact_2471_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_2472_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_zero
tff(fact_2473_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),zero_zero(A)) ) ).

% abs_not_less_zero
tff(fact_2474_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_pos
tff(fact_2475_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))) ) ).

% abs_triangle_ineq
tff(fact_2476_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),B2)),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2)) ) ) ) ).

% abs_mult_less
tff(fact_2477_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2))) ) ).

% abs_triangle_ineq2_sym
tff(fact_2478_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))) ) ).

% abs_triangle_ineq3
tff(fact_2479_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))) ) ).

% abs_triangle_ineq2
tff(fact_2480_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% nonzero_abs_divide
tff(fact_2481_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_2482_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_2483_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_2484_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_2485_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_2486_add__eq__0__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% add_eq_0_iff
tff(fact_2487_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),A2) = zero_zero(A) ) ).

% ab_group_add_class.ab_left_minus
tff(fact_2488_add_Oinverse__unique,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),A2) = B2 ) ) ) ).

% add.inverse_unique
tff(fact_2489_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) ) ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_2490_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) ) ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_2491_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_2492_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_2493_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_2494_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_2495_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_2496_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_2497_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_2498_numeral__times__minus__swap,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [W: num,Xa: 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),Xa)) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ).

% numeral_times_minus_swap
tff(fact_2499_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_2500_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_2501_square__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [Xa: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa) = one_one(A) )
        <=> ( ( Xa = one_one(A) )
            | ( Xa = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% square_eq_1_iff
tff(fact_2502_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B3: A,K: A,B2: A,A2: A] :
          ( ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_2503_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% diff_conv_add_uminus
tff(fact_2504_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_2505_diff__eq,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,uminus_uminus(A),Y)) ) ).

% diff_eq
tff(fact_2506_inf__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),A2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),B2)) = bot_bot(A) ) ).

% inf_cancel_left1
tff(fact_2507_inf__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),A2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),B2)) = bot_bot(A) ) ).

% inf_cancel_left2
tff(fact_2508_real__minus__mult__self__le,axiom,
    ! [U: real,Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),U),U))),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Xa)) ).

% real_minus_mult_self_le
tff(fact_2509_minus__real__def,axiom,
    ! [Xa: real,Y: real] : aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),Y) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,uminus_uminus(real),Y)) ).

% minus_real_def
tff(fact_2510_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_2511_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [Xa: A] :
          ( ! [E2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xa)),E2) )
         => ( Xa = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_2512_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),Xa) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa)) ) ) ) ).

% abs_mult_pos
tff(fact_2513_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% abs_eq_mult
tff(fact_2514_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),Xa)),Y) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% abs_div_pos
tff(fact_2515_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),Na)) ) ).

% zero_le_power_abs
tff(fact_2516_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,A2: A,R: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),A2))),R)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),R)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R)) ) ) ) ).

% abs_diff_le_iff
tff(fact_2517_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))) ) ).

% abs_triangle_ineq4
tff(fact_2518_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)))) ) ).

% abs_diff_triangle_ineq
tff(fact_2519_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,A2: A,R: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),A2))),R)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),R)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R)) ) ) ) ).

% abs_diff_less_iff
tff(fact_2520_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_2521_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_2522_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_2523_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_2524_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_2525_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_2526_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_2527_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_2528_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_2529_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_2530_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_2531_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_2532_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_2533_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_2534_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_2535_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_2536_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_2537_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_2538_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_2539_mult__1s__ring__1_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))),B2) = aa(A,A,uminus_uminus(A),B2) ) ).

% mult_1s_ring_1(1)
tff(fact_2540_mult__1s__ring__1_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) = aa(A,A,uminus_uminus(A),B2) ) ).

% mult_1s_ring_1(2)
tff(fact_2541_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_2542_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,uminus_uminus(A),B2),A2 = zero_zero(A)) ) ) ).

% eq_minus_divide_eq
tff(fact_2543_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = A2 )
        <=> $ite(C2 != zero_zero(A),aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),A2 = zero_zero(A)) ) ) ).

% minus_divide_eq_eq
tff(fact_2544_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = C2 )
          <=> ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_2545_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( C2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_2546_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ).

% power_minus
tff(fact_2547_power__minus__Bit0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(K))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(K))) ) ).

% power_minus_Bit0
tff(fact_2548_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% inf_shunt
tff(fact_2549_power__minus__Bit1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xa)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K))) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K)))) ) ).

% power_minus_Bit1
tff(fact_2550_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P2: A,Q2: A,R: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P2),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q2),R))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P2),aa(A,A,uminus_uminus(A),Q2))),R) ) ) ).

% sup_neg_inf
tff(fact_2551_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,uminus_uminus(A),Y))),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) ) ) ).

% shunt2
tff(fact_2552_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Y)),Z)) ) ) ).

% shunt1
tff(fact_2553_lemma__interval__lt,axiom,
    ! [A2: real,Xa: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),B2)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [Y2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),Y2))),D3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Y2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),B2) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_2554_powr__minus__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xa: A,A2: A] : powr(A,Xa,aa(A,A,uminus_uminus(A),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),powr(A,Xa,A2)) ) ).

% powr_minus_divide
tff(fact_2555_real__add__less__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,uminus_uminus(real),Xa)) ) ).

% real_add_less_0_iff
tff(fact_2556_real__0__less__add__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),Xa)),Y) ) ).

% real_0_less_add_iff
tff(fact_2557_real__0__le__add__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),Xa)),Y) ) ).

% real_0_le_add_iff
tff(fact_2558_real__add__le__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,uminus_uminus(real),Xa)) ) ).

% real_add_le_0_iff
tff(fact_2559_divide__powr__uminus,axiom,
    ! [A2: real,B2: real,C2: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),powr(real,B2,C2)) = aa(real,real,aa(real,fun(real,real),times_times(real),A2),powr(real,B2,aa(real,real,uminus_uminus(real),C2))) ).

% divide_powr_uminus
tff(fact_2560_tanh__real__gt__neg1,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(real,real,tanh(real),Xa)) ).

% tanh_real_gt_neg1
tff(fact_2561_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,abs_abs(A),Xa))) ) ).

% abs_add_one_gt_zero
tff(fact_2562_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Na))),Xa)
         => ( ( Na = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa) ) ) ) ).

% of_int_leD
tff(fact_2563_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Na))),Xa)
         => ( ( Na = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa) ) ) ) ).

% of_int_lessD
tff(fact_2564_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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),A2),C2)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% less_minus_divide_eq
tff(fact_2565_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
        <=> $ite(
              aa(A,$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),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% minus_divide_less_eq
tff(fact_2566_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_2567_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_2568_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_2569_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_2570_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B2: A,C2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> $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) = B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_2571_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) )
        <=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_2572_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z))),B2) = $ite(Z = zero_zero(A),B2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(3)
tff(fact_2573_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Z))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_2574_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Z))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_2575_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(5)
tff(fact_2576_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z))),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(6)
tff(fact_2577_power2__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
        <=> ( ( Xa = Y )
            | ( Xa = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% power2_eq_iff
tff(fact_2578_lemma__interval,axiom,
    ! [A2: real,Xa: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),B2)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [Y2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),Y2))),D3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Y2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),B2) ) ) ) ) ) ).

% lemma_interval
tff(fact_2579_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: A,M: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),aa(int,A,ring_1_of_int(A),archimedean_round(A,Z))))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),aa(int,A,ring_1_of_int(A),M)))) ) ).

% round_diff_minimal
tff(fact_2580_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xa)),aa(A,A,abs_abs(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_le_square_iff
tff(fact_2581_abs__square__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
        <=> ( aa(A,A,abs_abs(A),Xa) = one_one(A) ) ) ) ).

% abs_square_eq_1
tff(fact_2582_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_2583_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_2584_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_2585_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_2586_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
        <=> $ite(
              aa(A,$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),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% minus_divide_le_eq
tff(fact_2587_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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),A2),C2)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ).

% le_minus_divide_eq
tff(fact_2588_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$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),B2),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)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$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_2589_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_2590_power2__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
        <=> ( ( A2 = one_one(A) )
            | ( A2 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% power2_eq_1_iff
tff(fact_2591_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2592_realpow__square__minus__le,axiom,
    ! [U: real,Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),U),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% realpow_square_minus_le
tff(fact_2593_powr__neg__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Xa) ) ) ).

% powr_neg_one
tff(fact_2594_ln__add__one__self__le__self2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa) ) ).

% ln_add_one_self_le_self2
tff(fact_2595_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xa)),Y) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2596_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xa)),one_one(A)) ) ) ).

% abs_square_le_1
tff(fact_2597_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Xa)),one_one(A)) ) ) ).

% abs_square_less_1
tff(fact_2598_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$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),B2),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)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$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_2599_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_2600_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ) ) ).

% square_le_1
tff(fact_2601_minus__power__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Na)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ).

% minus_power_mult_self
tff(fact_2602_Bernoulli__inequality,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),Xa))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),Na)) ) ).

% Bernoulli_inequality
tff(fact_2603_ln__one__minus__pos__upper__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),Xa))),aa(real,real,uminus_uminus(real),Xa)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_2604_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_2605_log__minus__eq__powr,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log(B2),Xa)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),powr(real,B2,aa(real,real,uminus_uminus(real),Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_2606_powr__neg__numeral,axiom,
    ! [Xa: real,Na: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),Na))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),Na))) ) ) ).

% powr_neg_numeral
tff(fact_2607_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xa))),Xa))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% of_int_round_abs_le
tff(fact_2608_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Na: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(int,A,ring_1_of_int(A),Na)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))
         => ( archimedean_round(A,Xa) = Na ) ) ) ).

% round_unique'
tff(fact_2609_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_2610_arsinh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xa: A] : aa(A,A,arsinh(A),Xa) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),powr(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),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_2611_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_2612_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_2613_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_2614_of__real__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xa: real,Na: nat] : aa(real,A,real_Vector_of_real(A),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(real,A,real_Vector_of_real(A),Xa)),Na) ) ).

% of_real_power
tff(fact_2615_of__real__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xa: real,Y: real] : aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,real_Vector_of_real(A),Xa)),aa(real,A,real_Vector_of_real(A),Y)) ) ).

% of_real_add
tff(fact_2616_of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [Xa: real,Y: real] : aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),Xa)),aa(real,A,real_Vector_of_real(A),Y)) ) ).

% of_real_divide
tff(fact_2617_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B3))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3) ) ).

% Compl_subset_Compl_iff
tff(fact_2618_Compl__anti__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),B3)),aa(set(A),set(A),uminus_uminus(set(A)),A3)) ) ).

% Compl_anti_mono
tff(fact_2619_Compl__disjoint2,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),A3) = bot_bot(set(A)) ).

% Compl_disjoint2
tff(fact_2620_Compl__disjoint,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3)) = bot_bot(set(A)) ).

% Compl_disjoint
tff(fact_2621_Diff__Compl,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ).

% Diff_Compl
tff(fact_2622_Compl__Diff__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = 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)),A3)),B3) ).

% Compl_Diff_eq
tff(fact_2623_max__word__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),W)
        <=> ( W = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% max_word_less_eq_iff
tff(fact_2624_word__minus__one__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Xa)
        <=> ( Xa = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% word_minus_one_le
tff(fact_2625_subset__Compl__singleton,axiom,
    ! [A: $tType,A3: set(A),B2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))))
    <=> ~ member(A,B2,A3) ) ).

% subset_Compl_singleton
tff(fact_2626_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_2627_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_2628_of__real__mult,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xa: real,Y: real] : aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(real,A,real_Vector_of_real(A),Xa)),aa(real,A,real_Vector_of_real(A),Y)) ) ).

% of_real_mult
tff(fact_2629_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_2630_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_2631_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_2632_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_2633_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_2634_int__div__minus__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int))
     => ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = aa(int,int,uminus_uminus(int),A2) )
      <=> ( B2 = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_2635_ceiling__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archimedean_ceiling(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A2)),aa(num,real,numeral_numeral(real),B2))) = 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),A2))),aa(num,int,numeral_numeral(int),B2))) ).

% ceiling_divide_eq_div_numeral
tff(fact_2636_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: 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),A2)),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),A2)),aa(num,int,numeral_numeral(int),B2))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_2637_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_2638_uminus__int__code_I1_J,axiom,
    aa(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).

% uminus_int_code(1)
tff(fact_2639_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_2640_word__neg__numeral__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_neg_numeral_alt
tff(fact_2641_Collect__imp__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_av(fun(A,$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_2642_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_2643_subset__Compl__self__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% subset_Compl_self_eq
tff(fact_2644_Compl__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B3)) ).

% Compl_Un
tff(fact_2645_Compl__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = 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)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B3)) ).

% Compl_Int
tff(fact_2646_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_2647_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_2648_minus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),zero_zero(int)),L) = aa(int,int,uminus_uminus(int),L) ).

% minus_int_code(2)
tff(fact_2649_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(int,int,uminus_uminus(int),L)) != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) != zero_zero(int) ) ) ).

% zmod_zminus2_not_zero
tff(fact_2650_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),K)),L) != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) != zero_zero(int) ) ) ).

% zmod_zminus1_not_zero
tff(fact_2651_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_2652_max__word__not__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) != zero_zero(word(A)) ) ) ).

% max_word_not_0
tff(fact_2653_max__word__not__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Xa) ) ).

% max_word_not_less
tff(fact_2654_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_2655_word__order_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),A2) ) ).

% word_order.extremum_strict
tff(fact_2656_word__order_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( ( A2 != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ) ).

% word_order.not_eq_extremum
tff(fact_2657_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L))),aa(int,int,abs_abs(int),L)) ) ).

% abs_mod_less
tff(fact_2658_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_2659_disjoint__eq__subset__Compl,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),B3)) ) ).

% disjoint_eq_subset_Compl
tff(fact_2660_Compl__insert,axiom,
    ! [A: $tType,Xa: A,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_2661_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_2662_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_2663_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_2664_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_2665_plus__1__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A)))),Xa)
        <=> ( Xa = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% plus_1_less
tff(fact_2666_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_2667_less__x__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( Xa != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))))
          <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Xa)
              | ( Y = Xa ) ) ) ) ) ).

% less_x_plus_1
tff(fact_2668_word__add__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A)))) ) ) ).

% word_add_no_overflow
tff(fact_2669_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),A2)),B2) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2))) ).

% zmod_zminus1_eq_if
tff(fact_2670_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),aa(int,int,uminus_uminus(int),B2)) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)),B2)) ).

% zmod_zminus2_eq_if
tff(fact_2671_no__plus__overflow__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),uminus_uminus(word(A)),Y))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) ) ) ).

% no_plus_overflow_neg
tff(fact_2672_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_2673_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_2674_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_2675_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_2676_verit__less__mono__div__int2,axiom,
    ! [A3: int,B3: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),B3)
     => ( 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),B3),Na)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),Na)) ) ) ).

% verit_less_mono_div_int2
tff(fact_2677_div__eq__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),B2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_2678_word__Suc__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),K: word(A)] :
          ( ( Xa != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A)))),K)
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),K) ) ) ) ).

% word_Suc_le
tff(fact_2679_word__Suc__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),Xa: word(A)] :
          ( ( K != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),one_one(word(A))))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),K) ) ) ) ).

% word_Suc_leq
tff(fact_2680_word__le__make__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( ( Y != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),one_one(word(A)))) ) ) ) ).

% word_le_make_less
tff(fact_2681_ceiling__divide__eq__div,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: int,B2: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A2)),aa(int,A,ring_1_of_int(A),B2))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A2)),B2)) ) ).

% ceiling_divide_eq_div
tff(fact_2682_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_2683_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) ).

% nat_abs_int_diff
tff(fact_2684_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_2685_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_2686_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),K)),L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),L),one_one(int))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int))),L)) ) ) ).

% minus_mod_int_eq
tff(fact_2687_zmod__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),one_one(int))),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_2688_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,uminus_uminus(int),B2)) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),one_one(int))) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_2689_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A2)),B2) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),one_one(int))) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_2690_incr__lemma,axiom,
    ! [D2: int,Z: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),Z))),one_one(int))),D2))) ) ).

% incr_lemma
tff(fact_2691_decr__lemma,axiom,
    ! [D2: int,Xa: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),Z))),one_one(int))),D2))),Z) ) ).

% decr_lemma
tff(fact_2692_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,aa(int,fun(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_2693_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_2694_int__bit__induct,axiom,
    ! [P: fun(int,$o),K: int] :
      ( aa(int,$o,P,zero_zero(int))
     => ( aa(int,$o,P,aa(int,int,uminus_uminus(int),one_one(int)))
       => ( ! [K2: int] :
              ( aa(int,$o,P,K2)
             => ( ( K2 != zero_zero(int) )
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),bit0(one2)))) ) )
         => ( ! [K2: int] :
                ( aa(int,$o,P,K2)
               => ( ( K2 != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),bit0(one2))))) ) )
           => aa(int,$o,P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_2695_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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),one_one(int)) ).

% m1mod2k
tff(fact_2696_nat0__intermed__int__val,axiom,
    ! [Na: nat,F2: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Na)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat)))),aa(nat,int,F2,I3)))),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))
         => ? [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Na)
              & ( aa(nat,int,F2,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_2697_sb__dec__lem_H,axiom,
    ! [K: nat,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)),A2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),A2)) ) ).

% sb_dec_lem'
tff(fact_2698_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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))),one_one(int)) ).

% m1mod22k
tff(fact_2699_sb__dec__lem,axiom,
    ! [K: nat,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),A2))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),A2)) ) ).

% sb_dec_lem
tff(fact_2700_powr__int,axiom,
    ! [Xa: real,I: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(int,real,ring_1_of_int(real),I)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(int,nat,nat2,I)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),I))))) ) ) ).

% powr_int
tff(fact_2701_nonzero__of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [Y: real,Xa: real] :
          ( ( Y != zero_zero(real) )
         => ( aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),Xa)),aa(real,A,real_Vector_of_real(A),Y)) ) ) ) ).

% nonzero_of_real_divide
tff(fact_2702_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,$o)),Xa: A] :
          ( ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
             => aa(A,$o,aa(A,fun(A,$o),P,X),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(A,$o,aa(A,fun(A,$o),P,aa(A,A,abs_abs(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_sqrt_wlog
tff(fact_2703_monoseq__arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => topological_monoseq(real,aTP_Lamp_aw(real,fun(nat,real),Xa)) ) ).

% monoseq_arctan_series
tff(fact_2704_htt__vebt__predi,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => time_htt(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_predi(Tia,Xa),aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_ax(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),T),Tia),Xa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ) ).

% htt_vebt_predi
tff(fact_2705_htt__vebt__succi,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => time_htt(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_succi(Tia,Xa),aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_ay(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),T),Tia),Xa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ) ).

% htt_vebt_succi
tff(fact_2706_dbl__dec__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))) ) ) ).

% dbl_dec_simps(4)
tff(fact_2707_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] :
          aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),Na),A2) = $ite(Na = zero_zero(nat),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))))) ) ).

% signed_take_bit_rec
tff(fact_2708_Compl__eq__Compl__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(set(A),set(A),uminus_uminus(set(A)),B3) )
    <=> ( A3 = B3 ) ) ).

% Compl_eq_Compl_iff
tff(fact_2709_Compl__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),A3))
    <=> ~ member(A,C2,A3) ) ).

% Compl_iff
tff(fact_2710_ComplI,axiom,
    ! [A: $tType,C2: A,A3: set(A)] :
      ( ~ member(A,C2,A3)
     => member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),A3)) ) ).

% ComplI
tff(fact_2711_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),Na),zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_2712_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_2713_signed__take__bit__numeral__of__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(A,A,aa(nat,fun(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_2714_signed__take__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(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_2715_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_2716_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),zero_zero(nat)),A2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% signed_take_bit_0
tff(fact_2717_double__complement,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)) = A3 ).

% double_complement
tff(fact_2718_uminus__set__def,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = collect(A,aa(fun(A,$o),fun(A,$o),uminus_uminus(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A3))) ).

% uminus_set_def
tff(fact_2719_ComplD,axiom,
    ! [A: $tType,C2: A,A3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),A3))
     => ~ member(A,C2,A3) ) ).

% ComplD
tff(fact_2720_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] : collect(A,aTP_Lamp_az(fun(A,$o),fun(A,$o),P)) = aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P)) ).

% Collect_neg_eq
tff(fact_2721_Compl__eq,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = collect(A,aTP_Lamp_ba(set(A),fun(A,$o),A3)) ).

% Compl_eq
tff(fact_2722_signed__take__bit__minus,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),aa(int,int,uminus_uminus(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),K))) = aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),aa(int,int,uminus_uminus(int),K)) ).

% signed_take_bit_minus
tff(fact_2723_signed__take__bit__add,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),K)),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),L))) = aa(int,int,aa(nat,fun(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_2724_signed__take__bit__mult,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),K)),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),L))) = aa(int,int,aa(nat,fun(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_2725_signed__take__bit__diff,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),K)),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),L))) = aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)) ).

% signed_take_bit_diff
tff(fact_2726_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,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% signed_take_bit_int_less_exp
tff(fact_2727_monoseq__realpow,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => topological_monoseq(real,aa(real,fun(nat,real),power_power(real),Xa)) ) ) ).

% monoseq_realpow
tff(fact_2728_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,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_2729_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,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_2730_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,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_2731_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_2732_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,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_2733_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xa: A] : neg_numeral_dbl_dec(A,Xa) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Xa)),one_one(A)) ) ).

% dbl_dec_def
tff(fact_2734_signed__take__bit__int__eq__self,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2735_signed__take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_2736_helpypredd,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_pred(T,Xa) = some(nat,Y) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ) ).

% helpypredd
tff(fact_2737_helpyd,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_succ(T,Xa) = some(nat,Y) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ) ).

% helpyd
tff(fact_2738_pred__empty,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_pred(T,Xa) = none(nat) )
      <=> ( collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_bb(vEBT_VEBT,fun(nat,fun(nat,$o)),T),Xa)) = bot_bot(set(nat)) ) ) ) ).

% pred_empty
tff(fact_2739_succ__empty,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_succ(T,Xa) = none(nat) )
      <=> ( collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_bc(vEBT_VEBT,fun(nat,fun(nat,$o)),T),Xa)) = bot_bot(set(nat)) ) ) ) ).

% succ_empty
tff(fact_2740_vebt__pred_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_VEBT_vebt_predi(T,Tia,Xa),aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_ax(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),T),Tia),Xa)) ) ).

% vebt_pred'_rf_abstr
tff(fact_2741_vebt__succi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_VEBT_vebt_succi(T,Tia,Xa),aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_ay(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),T),Tia),Xa)) ) ).

% vebt_succi'_rf_abstr
tff(fact_2742_power__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xa),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_power,some(nat,Xa)),some(nat,Y)) = some(nat,Z) ) ) ).

% power_shift
tff(fact_2743_pred__correct,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Sx: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_pred(T,Xa) = some(nat,Sx) )
      <=> vEBT_is_pred_in_set(vEBT_set_vebt(T),Xa,Sx) ) ) ).

% pred_correct
tff(fact_2744_set__vebt__pred,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Px: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_pred(T,Xa) = some(nat,Px) )
      <=> vEBT_is_pred_in_set(vEBT_set_vebt(T),Xa,Px) ) ) ).

% set_vebt_pred
tff(fact_2745_succ__correct,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Sx: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_succ(T,Xa) = some(nat,Sx) )
      <=> vEBT_is_succ_in_set(vEBT_set_vebt(T),Xa,Sx) ) ) ).

% succ_correct
tff(fact_2746_pred__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Px: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_pred(T,Xa) = some(nat,Px) )
      <=> vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T),Xa,Px) ) ) ).

% pred_corr
tff(fact_2747_succ__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Sx: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_succ(T,Xa) = some(nat,Sx) )
      <=> vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T),Xa,Sx) ) ) ).

% succ_corr
tff(fact_2748_lesseq__shift,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Y)
    <=> vEBT_VEBT_lesseq(some(nat,Xa),some(nat,Y)) ) ).

% lesseq_shift
tff(fact_2749_sup__Some,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Xa: A,Y: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),some(A,Xa)),some(A,Y)) = some(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% sup_Some
tff(fact_2750_less__option__None__Some__code,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),none(A)),some(A,Xa)) ) ).

% less_option_None_Some_code
tff(fact_2751_less__option__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),some(A,Xa)),some(A,Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% less_option_Some
tff(fact_2752_rel__of__empty,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o)] : rel_of(A,B,aTP_Lamp_bd(A,option(B)),P) = bot_bot(set(product_prod(A,B))) ).

% rel_of_empty
tff(fact_2753_misiz,axiom,
    ! [T: vEBT_VEBT,Na: nat,M: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( some(nat,M) = vEBT_vebt_mint(T) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ) ).

% misiz
tff(fact_2754_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,aa(nat,fun(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),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),pred_numeral(L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_2755_vebt__minNull__mint,axiom,
    ! [T: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(T)
    <=> ( vEBT_vebt_mint(T) = none(nat) ) ) ).

% vebt_minNull_mint
tff(fact_2756_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T) = none(nat) )
     => vEBT_VEBT_minNull(T) ) ).

% minminNull
tff(fact_2757_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(T)
     => ( vEBT_vebt_mint(T) = none(nat) ) ) ).

% minNullmin
tff(fact_2758_mint__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Maxi: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_mint(T) = some(nat,Maxi) )
       => aa(nat,$o,vEBT_vebt_member(T),Maxi) ) ) ).

% mint_member
tff(fact_2759_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,Na: nat,Mini: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_mint(T) = some(nat,Mini) )
       => ( aa(nat,$o,vEBT_vebt_member(T),Xa)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mini),Xa) ) ) ) ).

% mint_corr_help
tff(fact_2760_set__vebt__mint_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_mint(T) = some(nat,Xa) )
      <=> ( member(nat,Xa,vEBT_set_vebt(T))
          & ! [X4: nat] :
              ( member(nat,X4,vEBT_set_vebt(T))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),X4) ) ) ) ) ).

% set_vebt_mint'
tff(fact_2761_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_mint(T) = none(nat) )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot(set(nat)) ) ) ) ).

% mint_corr_help_empty
tff(fact_2762_set__vebt__mint,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_mint(T) = some(nat,Xa) )
      <=> vEBT_VEBT_min_in_set(vEBT_set_vebt(T),Xa) ) ) ).

% set_vebt_mint
tff(fact_2763_mint__sound,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T),Xa)
       => ( vEBT_vebt_mint(T) = some(nat,Xa) ) ) ) ).

% mint_sound
tff(fact_2764_mint__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_mint(T) = some(nat,Xa) )
       => vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T),Xa) ) ) ).

% mint_corr
tff(fact_2765_less__option__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A)] : ~ aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),Xa),none(A)) ) ).

% less_option_None
tff(fact_2766_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_2767_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : pred_numeral(aa(num,num,bit1,K)) = aa(num,nat,numeral_numeral(nat),bit0(K)) ).

% pred_numeral_simps(3)
tff(fact_2768_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,aa(nat,fun(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,aa(nat,fun(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_2769_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,aa(nat,fun(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,aa(nat,fun(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_2770_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(nat,fun(int,int),bit_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_2771_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V2048590022279873568_shift(nat,power_power(nat)) ).

% local.power_def
tff(fact_2772_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K)),one_one(nat)) ).

% pred_numeral_def
tff(fact_2773_bot__option__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( bot_bot(option(A)) = none(A) ) ) ).

% bot_option_def
tff(fact_2774_less__option__None__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),none(A)),some(A,Xa)) ) ).

% less_option_None_Some
tff(fact_2775_less__option__None__is__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),none(A)),Xa)
         => ? [Z3: A] : Xa = some(A,Z3) ) ) ).

% less_option_None_is_Some
tff(fact_2776_vebt__minti__hT,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : time_htt(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_minti(Tia),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_be(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),T),Tia),one_one(nat)) ).

% vebt_minti_hT
tff(fact_2777_vebt__minti__h,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_minti(Tia),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_be(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),T),Tia)) ).

% vebt_minti_h
tff(fact_2778_set__vebt__mint_H_H,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( vEBT_vebt_mint(T) = $ite(vEBT_set_vebt(T) = bot_bot(set(nat)),none(nat),some(nat,lattic643756798350308766er_Min(nat,vEBT_set_vebt(T)))) ) ) ).

% set_vebt_mint''
tff(fact_2779_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_2780_summable__arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => summable(real,aTP_Lamp_bf(real,fun(nat,real),Xa)) ) ).

% summable_arctan_series
tff(fact_2781_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_2782_even__odd__cases,axiom,
    ! [Xa: nat] :
      ( ! [N: nat] : Xa != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)
     => ~ ! [N: nat] : Xa != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,suc,N)) ) ).

% even_odd_cases
tff(fact_2783_TBOUND__vebt__minti,axiom,
    ! [T: vEBT_VEBTi] : time_TBOUND(option(nat),vEBT_vebt_minti(T),one_one(nat)) ).

% TBOUND_vebt_minti
tff(fact_2784_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( aa(nat,nat,suc,X22) = aa(nat,nat,suc,Y22) )
    <=> ( X22 = Y22 ) ) ).

% nat.inject
tff(fact_2785_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_2786_lessI,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,Na)) ).

% lessI
tff(fact_2787_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_2788_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_2789_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_2790_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_2791_diff__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na) ).

% diff_Suc_Suc
tff(fact_2792_Suc__diff__diff,axiom,
    ! [M: nat,Na: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),Na)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),K) ).

% Suc_diff_diff
tff(fact_2793_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_2794_power__0__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(nat,nat,suc,Na)) = zero_zero(A) ) ).

% power_0_Suc
tff(fact_2795_power__Suc0__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% power_Suc0_right
tff(fact_2796_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_2797_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_2798_div__by__Suc__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,zero_zero(nat))) = M ).

% div_by_Suc_0
tff(fact_2799_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_2800_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_2801_nat__power__eq__Suc__0__iff,axiom,
    ! [Xa: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xa),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = zero_zero(nat) )
        | ( Xa = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_2802_power__Suc__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,suc,zero_zero(nat))),Na) = aa(nat,nat,suc,zero_zero(nat)) ).

% power_Suc_0
tff(fact_2803_diff__Suc__1,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Na)),one_one(nat)) = Na ).

% diff_Suc_1
tff(fact_2804_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_2805_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),aa(nat,nat,suc,Na)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_Suc_1
tff(fact_2806_mod__by__Suc__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% mod_by_Suc_0
tff(fact_2807_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_2808_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_2809_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_2810_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_2811_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_2812_Suc__pred,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))) = Na ) ) ).

% Suc_pred
tff(fact_2813_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_2814_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_2815_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))),I) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,J)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_2816_nat__1,axiom,
    aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_2817_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_2818_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_2819_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_2820_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_2821_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_2822_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_2823_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_2824_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_2825_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_2826_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_2827_diff__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),pred_numeral(K)) ).

% diff_Suc_numeral
tff(fact_2828_diff__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),pred_numeral(K)),Na) ).

% diff_numeral_Suc
tff(fact_2829_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_2830_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_2831_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% Suc_1
tff(fact_2832_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_2833_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_2834_Suc__diff__1,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) = Na ) ) ).

% Suc_diff_1
tff(fact_2835_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_2836_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_2837_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),aa(num,num,bit1,one2))),Na)) ).

% div_Suc_eq_div_add3
tff(fact_2838_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_2839_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_2840_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),aa(num,num,bit1,one2))),Na)) ).

% mod_Suc_eq_mod_add3
tff(fact_2841_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),aa(num,num,bit1,one2))),M)),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_2842_signed__take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : aa(int,int,aa(nat,fun(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,aa(nat,fun(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_2843_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_2844_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa(int,int,aa(nat,fun(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,aa(nat,fun(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_2845_signed__take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,suc,Na)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(nat,fun(int,int),bit_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_2846_Suc__inject,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa(nat,nat,suc,Xa) = aa(nat,nat,suc,Y) )
     => ( Xa = Y ) ) ).

% Suc_inject
tff(fact_2847_n__not__Suc__n,axiom,
    ! [Na: nat] : Na != aa(nat,nat,suc,Na) ).

% n_not_Suc_n
tff(fact_2848_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_2849_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero(nat) != aa(nat,nat,suc,X22) ).

% nat.distinct(1)
tff(fact_2850_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_2851_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_2852_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa(nat,nat,suc,X22) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_2853_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_2854_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_2855_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),M: nat,Na: nat] :
      ( ! [X: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,X),zero_zero(nat))
     => ( ! [Y3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,zero_zero(nat)),aa(nat,nat,suc,Y3))
       => ( ! [X: nat,Y3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,X),Y3)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,aa(nat,nat,suc,X)),aa(nat,nat,suc,Y3)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,M),Na) ) ) ) ).

% diff_induct
tff(fact_2856_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_2857_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_2858_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_2859_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_2860_not0__implies__Suc,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => ? [M4: nat] : Na = aa(nat,nat,suc,M4) ) ).

% not0_implies_Suc
tff(fact_2861_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) )
       => ~ ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J3)
             => ( K != aa(nat,nat,suc,J3) ) ) ) ) ).

% Nat.lessE
tff(fact_2862_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_2863_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K)
     => ~ ! [J3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J3)
           => ( K != aa(nat,nat,suc,J3) ) ) ) ).

% Suc_lessE
tff(fact_2864_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_2865_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_2866_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_2867_Ex__less__Suc,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,suc,Na))
          & aa(nat,$o,P,I2) )
    <=> ( aa(nat,$o,P,Na)
        | ? [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
            & aa(nat,$o,P,I2) ) ) ) ).

% Ex_less_Suc
tff(fact_2868_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_2869_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_2870_Nat_OAll__less__Suc,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,suc,Na))
         => aa(nat,$o,P,I2) )
    <=> ( aa(nat,$o,P,Na)
        & ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
           => aa(nat,$o,P,I2) ) ) ) ).

% Nat.All_less_Suc
tff(fact_2871_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)
    <=> ? [M5: nat] :
          ( ( M = aa(nat,nat,suc,M5) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M5) ) ) ).

% Suc_less_eq2
tff(fact_2872_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_2873_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_2874_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_2875_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,I3),aa(nat,nat,suc,I3))
       => ( ! [I3: nat,J3: nat,K2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),K2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),P,I3),J3)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),P,J3),K2)
                   => aa(nat,$o,aa(nat,fun(nat,$o),P,I3),K2) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_2876_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I3: nat] :
            ( ( J = aa(nat,nat,suc,I3) )
           => aa(nat,$o,P,I3) )
       => ( ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J)
             => ( aa(nat,$o,P,aa(nat,nat,suc,I3))
               => aa(nat,$o,P,I3) ) )
         => aa(nat,$o,P,I) ) ) ) ).

% strict_inc_induct
tff(fact_2877_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_2878_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A2: nat] :
      ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),A2) )
     => ( aa(nat,nat,suc,A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,suc,A2)) ) ) ).

% nat_arith.suc1
tff(fact_2879_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_2880_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_2881_transitive__stepwise__le,axiom,
    ! [M: nat,Na: nat,R2: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( ! [X: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R2,X),X)
       => ( ! [X: nat,Y3: nat,Z3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R2,X),Y3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R2,Y3),Z3)
               => aa(nat,$o,aa(nat,fun(nat,$o),R2,X),Z3) ) )
         => ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R2,N),aa(nat,nat,suc,N))
           => aa(nat,$o,aa(nat,fun(nat,$o),R2,M),Na) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_2882_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_2883_full__nat__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M2)),N)
             => aa(nat,$o,P,M2) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Na) ) ).

% full_nat_induct
tff(fact_2884_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_2885_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_2886_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_2887_Suc__le__D,axiom,
    ! [Na: nat,M6: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),M6)
     => ? [M4: nat] : M6 = aa(nat,nat,suc,M4) ) ).

% Suc_le_D
tff(fact_2888_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_2889_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_2890_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_2891_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,aa(nat,fun(nat,nat),minus_minus(nat),K),I)) ) ) ).

% zero_induct_lemma
tff(fact_2892_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_2893_mod__Suc__Suc__eq,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,M))),Na) ).

% mod_Suc_Suc_eq
tff(fact_2894_mod__Suc__eq,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Na) ).

% mod_Suc_eq
tff(fact_2895_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_2896_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_2897_power__Suc,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ).

% power_Suc
tff(fact_2898_power__Suc2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),A2) ) ).

% power_Suc2
tff(fact_2899_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_2900_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_2901_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_2902_Ex__less__Suc2,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,suc,Na))
          & aa(nat,$o,P,I2) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        | ? [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
            & aa(nat,$o,P,aa(nat,nat,suc,I2)) ) ) ) ).

% Ex_less_Suc2
tff(fact_2903_gr0__conv__Suc,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
    <=> ? [M3: nat] : Na = aa(nat,nat,suc,M3) ) ).

% gr0_conv_Suc
tff(fact_2904_All__less__Suc2,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,suc,Na))
         => aa(nat,$o,P,I2) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
           => aa(nat,$o,P,aa(nat,nat,suc,I2)) ) ) ) ).

% All_less_Suc2
tff(fact_2905_gr0__implies__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_implies_Suc
tff(fact_2906_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) )
        | ? [J2: nat] :
            ( ( M = aa(nat,nat,suc,J2) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Na) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_2907_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_2908_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_2909_dbl__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xa: A] : neg_numeral_dbl(A,Xa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Xa) ) ).

% dbl_def
tff(fact_2910_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_2911_less__natE,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ~ ! [Q3: nat] : Na != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)) ) ).

% less_natE
tff(fact_2912_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_2913_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_2914_less__iff__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
    <=> ? [K3: nat] : Na = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3)) ) ).

% less_iff_Suc_add
tff(fact_2915_less__imp__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ? [K2: nat] : Na = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)) ) ).

% less_imp_Suc_add
tff(fact_2916_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_2917_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_2918_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_2919_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_2920_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_2921_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_2922_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_2923_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_2924_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_2925_nat__in__between__eq_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),aa(nat,nat,suc,A2)) )
    <=> ( B2 = A2 ) ) ).

% nat_in_between_eq(2)
tff(fact_2926_nat__in__between__eq_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),aa(nat,nat,suc,A2)) )
    <=> ( B2 = aa(nat,nat,suc,A2) ) ) ).

% nat_in_between_eq(1)
tff(fact_2927_Suc__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,Na))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na) ) ) ).

% Suc_diff_Suc
tff(fact_2928_diff__less__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),aa(nat,nat,suc,M)) ).

% diff_less_Suc
tff(fact_2929_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_2930_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_2931_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_2932_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_2933_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_2934_Suc__diff__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)) ) ) ).

% Suc_diff_le
tff(fact_2935_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),Na) ).

% diff_Suc_eq_diff_pred
tff(fact_2936_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_2937_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_2938_mod__Suc,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Na) = $ite(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)) = Na,zero_zero(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na))) ).

% mod_Suc
tff(fact_2939_mod__induct,axiom,
    ! [P: fun(nat,$o),Na: nat,P2: nat,M: nat] :
      ( aa(nat,$o,P,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),P2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),P2)
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),P2)
               => ( aa(nat,$o,P,N)
                 => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,N)),P2)) ) )
           => aa(nat,$o,P,M) ) ) ) ) ).

% mod_induct
tff(fact_2940_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_2941_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_2942_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_2943_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_2944_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Na))),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,Na)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% power_le_imp_le_base
tff(fact_2945_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Na)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,Na)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_2946_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Na))) ) ) ).

% power_gt1
tff(fact_2947_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_2948_eval__nat__numeral_I3_J,axiom,
    ! [Na: num] : aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Na)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bit0(Na))) ).

% eval_nat_numeral(3)
tff(fact_2949_ex__least__nat__less,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,Na)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),Na)
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),K2)
               => ~ aa(nat,$o,P,I4) )
            & aa(nat,$o,P,aa(nat,nat,suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_2950_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_2951_diff__Suc__less,axiom,
    ! [Na: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,I))),Na) ) ).

% diff_Suc_less
tff(fact_2952_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_2953_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_2954_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_2955_power__gt__expt,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Na),K)) ) ).

% power_gt_expt
tff(fact_2956_nat__one__le__power,axiom,
    ! [I: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Na)) ) ).

% nat_one_le_power
tff(fact_2957_realpow__pos__nth2,axiom,
    ! [A2: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ? [R4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
          & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R4),aa(nat,nat,suc,Na)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_2958_int__ops_I4_J,axiom,
    ! [A2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,A2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),one_one(int)) ).

% int_ops(4)
tff(fact_2959_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_2960_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_2961_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_2962_atLeast0__atMost__Suc,axiom,
    ! [Na: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,Na)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ).

% atLeast0_atMost_Suc
tff(fact_2963_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( set_or1337092689740270186AtMost(nat,M,Na) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_2964_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
     => ( set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,Na)),set_or1337092689740270186AtMost(nat,M,Na)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_2965_atLeastAtMost__insertL,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na)) = set_or1337092689740270186AtMost(nat,M,Na) ) ) ).

% atLeastAtMost_insertL
tff(fact_2966_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Na))),A2) ) ) ) ).

% power_Suc_le_self
tff(fact_2967_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Na))),one_one(A)) ) ) ) ).

% power_Suc_less_one
tff(fact_2968_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_2969_numeral__3__eq__3,axiom,
    aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))) ).

% numeral_3_eq_3
tff(fact_2970_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_2971_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_2972_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),aa(num,num,bit1,one2))),Na) ).

% Suc3_eq_add_3
tff(fact_2973_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),aa(nat,nat,suc,zero_zero(nat)))),M) ) ) ).

% nz_le_conv_less
tff(fact_2974_Suc__pred_H,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( Na = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_2975_Suc__diff__eq__diff__pred,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_2976_div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),Na)) ) ) ) ).

% div_geq
tff(fact_2977_div__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( Na = zero_zero(nat) ) ),
        zero_zero(nat),
        aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),Na)) ) ).

% div_if
tff(fact_2978_add__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = $ite(M = zero_zero(nat),Na,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),Na))) ).

% add_eq_if
tff(fact_2979_Suc__n__minus__m__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),M)
       => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_2980_div__nat__eqI,axiom,
    ! [Na: nat,Q2: 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),Q2)),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,Q2)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_2981_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_2982_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_2983_negD,axiom,
    ! [Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),zero_zero(int))
     => ? [N: nat] : Xa = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% negD
tff(fact_2984_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_2985_Suc__as__int,axiom,
    ! [X5: nat] : aa(nat,nat,suc,X5) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X5)),one_one(int))) ).

% Suc_as_int
tff(fact_2986_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
         => ~ ! [N: nat] : ~ aa(A,$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)))),E) ) ) ).

% nat_approx_posE
tff(fact_2987_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_2988_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_2989_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),aa(num,num,bit1,one2))),M)),Na) ).

% Suc_div_eq_add3_div
tff(fact_2990_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_2991_le__div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),Na)) ) ) ) ).

% le_div_geq
tff(fact_2992_split__div_H,axiom,
    ! [P: fun(nat,$o),M: nat,Na: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na))
    <=> ( ( ( Na = zero_zero(nat) )
          & aa(nat,$o,P,zero_zero(nat)) )
        | ? [Q5: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q5)),M)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,suc,Q5)))
            & aa(nat,$o,P,Q5) ) ) ) ).

% split_div'
tff(fact_2993_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),aa(num,num,bit1,one2))),M)),Na) ).

% Suc_mod_eq_add3_mod
tff(fact_2994_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_2995_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_2996_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_2997_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),aa(num,num,bit1,one2)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
tff(fact_2998_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    vEBT_VEBT_Tb2(aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.Tb'.simps(2)
tff(fact_2999_VEBT__internal_OTb_Osimps_I2_J,axiom,
    vEBT_VEBT_Tb(aa(nat,nat,suc,zero_zero(nat))) = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) ).

% VEBT_internal.Tb.simps(2)
tff(fact_3000_power__odd__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% power_odd_eq
tff(fact_3001_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_3002_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_3003_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_3004_nat__intermed__int__val,axiom,
    ! [M: nat,Na: nat,F2: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),I3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Na) )
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F2,aa(nat,nat,suc,I3))),aa(nat,int,F2,I3)))),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))
           => ? [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),I3)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Na)
                & ( aa(nat,int,F2,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_3005_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3006_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))),zero_zero(A)) ) ) ).

% odd_power_less_zero
tff(fact_3007_power__minus1__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% power_minus1_odd
tff(fact_3008_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),K),M)),K) = aa(nat,int,aa(int,fun(nat,int),power_power(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_3009_signed__take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),K)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,Na)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_3010_nat__ivt__aux,axiom,
    ! [Na: nat,F2: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Na)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F2,aa(nat,nat,suc,I3))),aa(nat,int,F2,I3)))),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))
         => ? [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Na)
              & ( aa(nat,int,F2,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_3011_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),aa(nat,nat,suc,Na)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_3012_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_3013_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),aa(nat,nat,suc,Na)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% unset_bit_Suc
tff(fact_3014_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),aa(nat,nat,suc,Na)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% set_bit_Suc
tff(fact_3015_sb__inc__lem,axiom,
    ! [A2: int,K: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))) ) ).

% sb_inc_lem
tff(fact_3016_sb__inc__lem_H,axiom,
    ! [A2: int,K: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))) ) ).

% sb_inc_lem'
tff(fact_3017_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,Na)))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_3018_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_bg(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_divide_iff
tff(fact_3019_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_bh(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_cmult_iff
tff(fact_3020_Min__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A] : lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = Xa ) ).

% Min_singleton
tff(fact_3021_summable__power__series,axiom,
    ! [F2: fun(nat,real),Z: real] :
      ( ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,I3)),one_one(real))
     => ( ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,I3))
       => ( 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_bi(fun(nat,real),fun(real,fun(nat,real)),F2),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_3022_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,aa(nat,fun(nat,A),aTP_Lamp_bj(fun(nat,A),fun(nat,fun(nat,A)),F2),K))
        <=> summable(A,F2) ) ) ).

% summable_iff_shift
tff(fact_3023_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_bk(nat,A)) ) ).

% summable_zero
tff(fact_3024_summable__complex__of__real,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(complex,aTP_Lamp_bl(fun(nat,real),fun(nat,complex),F2))
    <=> summable(real,F2) ) ).

% summable_complex_of_real
tff(fact_3025_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_bm(nat,fun(fun(nat,A),fun(nat,A)),I),F2)) ) ).

% summable_single
tff(fact_3026_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),B2: nat] :
      ( aa(A,$o,P,K)
     => ( ! [Y3: A] :
            ( aa(A,$o,P,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y3)),B2) )
       => ? [X: A] :
            ( aa(A,$o,P,X)
            & ! [Y2: A] :
                ( aa(A,$o,P,Y2)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y2)),aa(A,nat,F2,X)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_3027_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: A] :
          ( summable(A,aTP_Lamp_bn(A,fun(nat,A),C2))
        <=> ( C2 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_3028_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_bo(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_add
tff(fact_3029_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_bp(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult2
tff(fact_3030_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_bq(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult
tff(fact_3031_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_br(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_diff
tff(fact_3032_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,aTP_Lamp_bs(fun(nat,A),fun(nat,A),F2))
        <=> summable(A,F2) ) ) ).

% summable_Suc_iff
tff(fact_3033_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_bg(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_divide
tff(fact_3034_summable__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => summable(A,aTP_Lamp_bt(fun(nat,A),fun(nat,A),F2)) ) ) ).

% summable_minus
tff(fact_3035_summable__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,aTP_Lamp_bt(fun(nat,A),fun(nat,A),F2))
        <=> summable(A,F2) ) ) ).

% summable_minus_iff
tff(fact_3036_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_bj(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) ) ) ).

% summable_ignore_initial_segment
tff(fact_3037_summable__rabs__cancel,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(real,aTP_Lamp_bu(fun(nat,real),fun(nat,real),F2))
     => summable(real,F2) ) ).

% summable_rabs_cancel
tff(fact_3038_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),Na: nat] :
      ( aa(A,$o,P,K)
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
           => ? [Y2: A] :
                ( aa(A,$o,P,Y2)
                & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y2)),aa(A,nat,F2,X)) ) )
       => ? [Y3: A] :
            ( aa(A,$o,P,Y3)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,K)),Na)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_3039_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_bh(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
         => ( ( C2 != zero_zero(A) )
           => summable(A,F2) ) ) ) ).

% summable_mult_D
tff(fact_3040_summable__zero__power,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aa(A,fun(nat,A),power_power(A),zero_zero(A))) ) ).

% summable_zero_power
tff(fact_3041_summable__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X7: fun(nat,real)] :
          ( summable(real,X7)
         => summable(A,aTP_Lamp_bv(fun(nat,real),fun(nat,A),X7)) ) ) ).

% summable_of_real
tff(fact_3042_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_bw(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_zero_power'
tff(fact_3043_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_bx(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_0_powser
tff(fact_3044_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_by(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_bz(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_split_head
tff(fact_3045_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_ca(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_cb(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% powser_split_head(3)
tff(fact_3046_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_cc(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F2),M),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_bz(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_3047_summable__rabs__comparison__test,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),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_bu(fun(nat,real),fun(nat,real),F2)) ) ) ).

% summable_rabs_comparison_test
tff(fact_3048_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K6: real,Na: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),K6)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H))),K6)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),Na)),aa(nat,A,aa(A,fun(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,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),K6),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),real_V7770717601297561774m_norm(A,H))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_3049_ln__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(num,real,numeral_numeral(real),bit0(one2)))
       => ( aa(real,real,ln_ln(real),Xa) = suminf(real,aTP_Lamp_cd(real,fun(nat,real),Xa)) ) ) ) ).

% ln_series
tff(fact_3050_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( vEBT_V9176841429113362141ildupi(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != one_one(int) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != one_one(int) ) )
         => ~ ! [N: nat] :
                ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
               => ( Y != $ite(aa(nat,$o,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),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(bit0(one2))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
tff(fact_3051_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),aa(nat,nat,suc,Na)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% flip_bit_Suc
tff(fact_3052_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
         => ~ ! [Va2: nat] :
                ( ( Xa = 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(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_3053_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,dvd_dvd(A,A2),B2) ) ) ).

% dvd_add_triv_left_iff
tff(fact_3054_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,dvd_dvd(A,A2),B2) ) ) ).

% dvd_add_triv_right_iff
tff(fact_3055_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,$o,dvd_dvd(A,A2),C2)
           => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2))
            <=> aa(A,$o,dvd_dvd(A,B2),C2) ) ) ) ) ).

% div_dvd_div
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)),aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_3057_flip__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),Na),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% flip_bit_negative_int_iff
tff(fact_3058_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_3059_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_ce(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_3060_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( C2 = zero_zero(A) )
            | aa(A,$o,dvd_dvd(A,A2),B2) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_3061_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | aa(A,$o,dvd_dvd(A,A2),B2) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_3062_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))
          <=> aa(A,$o,dvd_dvd(A,B2),C2) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3063_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2))
          <=> aa(A,$o,dvd_dvd(A,B2),C2) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3064_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),B2))
        <=> aa(A,$o,dvd_dvd(A,A2),B2) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3065_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)))
        <=> aa(A,$o,dvd_dvd(A,A2),B2) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3066_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
           => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A)) ) ) ) ).

% unit_prod
tff(fact_3067_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A2)
         => ( aa(A,$o,dvd_dvd(A,C2),B2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).

% div_add
tff(fact_3068_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),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)),A2)) = A2 ) ) ) ).

% unit_div_1_div_1
tff(fact_3069_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),one_one(A)) ) ) ).

% unit_div_1_unit
tff(fact_3070_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
           => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% unit_div
tff(fact_3071_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)) = B2 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_3072_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),A2) = B2 ) ) ) ).

% dvd_div_mult_self
tff(fact_3073_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A2)
         => ( aa(A,$o,dvd_dvd(A,C2),B2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).

% div_diff
tff(fact_3074_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),A2) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_3075_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_3076_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_3077_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_3078_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_3079_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_3080_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),A2) = B2 ) ) ) ).

% unit_div_mult_self
tff(fact_3081_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) ) ) ) ).

% unit_mult_div_div
tff(fact_3082_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na))
          <=> aa(A,$o,dvd_dvd(A,A2),B2) ) ) ) ).

% pow_divides_pow_iff
tff(fact_3083_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V7770717601297561774m_norm(A,Xa))
        <=> ( Xa != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_3084_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_3085_odd__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> ~ ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ) ).

% odd_add
tff(fact_3086_even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
          <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ) ).

% even_add
tff(fact_3087_even__mult__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ) ).

% even_mult_iff
tff(fact_3088_even__mod__2__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% even_mod_2_iff
tff(fact_3089_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_3090_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_3091_norm__mult__numeral1,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [W: num,A2: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),W)),real_V7770717601297561774m_norm(A,A2)) ) ).

% norm_mult_numeral1
tff(fact_3092_norm__mult__numeral2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A2)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_mult_numeral2
tff(fact_3093_norm__divide__numeral,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A2)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_divide_numeral
tff(fact_3094_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(nat,A)] : suminf(A,aTP_Lamp_cf(fun(nat,A),fun(nat,A),F2)) = aa(nat,A,F2,zero_zero(nat)) ) ).

% powser_zero
tff(fact_3095_summable__geometric__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( summable(A,aa(A,fun(nat,A),power_power(A),C2))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real)) ) ) ).

% summable_geometric_iff
tff(fact_3096_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)))
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% even_plus_one_iff
tff(fact_3097_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))
        <=> 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),A2),B2)) ) ) ).

% even_diff
tff(fact_3098_power__minus__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat,A2: A] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Na) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ) ).

% power_minus_odd
tff(fact_3099_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat,A2: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Na) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na) ) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_3100_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_3101_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_3102_power__even__abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: num,A2: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),aa(num,nat,numeral_numeral(nat),W)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W)) ) ) ) ).

% power_even_abs_numeral
tff(fact_3103_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_2
tff(fact_3104_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(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),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_3105_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_two
tff(fact_3106_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% even_power
tff(fact_3107_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( aa(nat,$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)),A2) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3108_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),zero_zero(A))
        <=> ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq
tff(fact_3109_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A))
        <=> ( ~ aa(nat,$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),A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3110_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,aa(A,fun(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_3111_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,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Na) = one_one(A) ) ) ) ).

% neg_one_even_power
tff(fact_3112_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_3113_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))) = Na ) ) ).

% odd_Suc_minus_one
tff(fact_3114_even__diff__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | aa(nat,$o,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_3115_norm__of__real__add1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xa: real] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,real_Vector_of_real(A),Xa)),one_one(A))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),one_one(real))) ) ).

% norm_of_real_add1
tff(fact_3116_norm__of__real__addn,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xa: real,B2: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,real_Vector_of_real(A),Xa)),aa(num,A,numeral_numeral(A),B2))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(num,real,numeral_numeral(real),B2))) ) ).

% norm_of_real_addn
tff(fact_3117_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))),one_one(A)) = A2 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_3118_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,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A)))
        <=> ( Na = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3119_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W) = zero_zero(nat) )
            | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
              & ( A2 != 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)),A2) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3120_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)) ) ) ).

% odd_two_times_div_two_nat
tff(fact_3121_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A))
        <=> ( aa(nat,$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),A2),zero_zero(A)) )
              | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3122_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_3123_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_3124_suminf__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X7: fun(nat,real)] :
          ( summable(real,X7)
         => ( aa(real,A,real_Vector_of_real(A),suminf(real,X7)) = suminf(A,aTP_Lamp_bv(fun(nat,real),fun(nat,A),X7)) ) ) ) ).

% suminf_of_real
tff(fact_3125_summable__norm,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_cg(fun(nat,A),fun(nat,real),F2))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,suminf(A,F2))),suminf(real,aTP_Lamp_cg(fun(nat,A),fun(nat,real),F2))) ) ) ).

% summable_norm
tff(fact_3126_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
        <=> ( ( A2 = zero_zero(A) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_3127_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
          <=> aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% dvd_add_right_iff
tff(fact_3128_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),C2)
         => ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
          <=> aa(A,$o,dvd_dvd(A,A2),B2) ) ) ) ).

% dvd_add_left_iff
tff(fact_3129_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,$o,dvd_dvd(A,A2),C2)
           => aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ) ).

% dvd_add
tff(fact_3130_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) ) ).

% dvd_triv_right
tff(fact_3131_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)
         => aa(A,$o,dvd_dvd(A,B2),C2) ) ) ).

% dvd_mult_right
tff(fact_3132_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( 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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ).

% mult_dvd_mono
tff(fact_3133_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% dvd_triv_left
tff(fact_3134_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)
         => aa(A,$o,dvd_dvd(A,A2),C2) ) ) ).

% dvd_mult_left
tff(fact_3135_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ).

% dvd_mult2
tff(fact_3136_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),C2)
         => aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ).

% dvd_mult
tff(fact_3137_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
        <=> ? [K3: A] : A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K3) ) ) ).

% dvd_def
tff(fact_3138_dvdI,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [A2: A,B2: A,K: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K) )
         => aa(A,$o,dvd_dvd(A,B2),A2) ) ) ).

% dvdI
tff(fact_3139_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ~ ! [K2: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) ) ) ).

% dvdE
tff(fact_3140_division__decomp,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ? [B8: A,C7: A] :
              ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B8),C7) )
              & aa(A,$o,dvd_dvd(A,B8),B2)
              & aa(A,$o,dvd_dvd(A,C7),C2) ) ) ) ).

% division_decomp
tff(fact_3141_dvd__productE,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [P2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,P2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
         => ~ ! [X: A,Y3: A] :
                ( ( P2 = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y3) )
               => ( aa(A,$o,dvd_dvd(A,X),A2)
                 => ~ aa(A,$o,dvd_dvd(A,Y3),B2) ) ) ) ) ).

% dvd_productE
tff(fact_3142_dvd__diff__commute,axiom,
    ! [A: $tType] :
      ( euclid5891614535332579305n_ring(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))
        <=> aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) ) ) ).

% dvd_diff_commute
tff(fact_3143_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A2)
         => ( aa(A,$o,dvd_dvd(A,C2),B2)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
            <=> ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3144_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
         => ( aa(A,$o,dvd_dvd(A,C2),A2)
           => ( aa(A,$o,dvd_dvd(A,C2),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3145_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,D2),B2)
         => ( aa(A,$o,dvd_dvd(A,B2),A2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),D2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% div_div_div_same
tff(fact_3146_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_3147_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_3148_dvd__power__same,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xa: A,Y: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,Xa),Y)
         => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Na)) ) ) ).

% dvd_power_same
tff(fact_3149_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2))
         => ( aa(A,$o,dvd_dvd(A,C2),B2)
           => aa(A,$o,dvd_dvd(A,C2),A2) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_3150_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B2)
         => ( aa(A,$o,dvd_dvd(A,C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2))
          <=> aa(A,$o,dvd_dvd(A,C2),A2) ) ) ) ).

% dvd_mod_iff
tff(fact_3151_mod__mod__cancel,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B2)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),C2) ) ) ) ).

% mod_mod_cancel
tff(fact_3152_dvd__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [K: A,M: A,Na: A] :
          ( aa(A,$o,dvd_dvd(A,K),M)
         => ( aa(A,$o,dvd_dvd(A,K),Na)
           => aa(A,$o,dvd_dvd(A,K),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),M),Na)) ) ) ) ).

% dvd_mod
tff(fact_3153_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,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)) ) ) ).

% dvd_diff_nat
tff(fact_3154_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aTP_Lamp_ch(A,fun(A,$o),A2))),collect(A,aTP_Lamp_ch(A,fun(A,$o),B2)))
        <=> aa(A,$o,dvd_dvd(A,A2),B2) ) ) ).

% subset_divisors_dvd
tff(fact_3155_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),collect(A,aTP_Lamp_ch(A,fun(A,$o),A2))),collect(A,aTP_Lamp_ch(A,fun(A,$o),B2)))
        <=> ( aa(A,$o,dvd_dvd(A,A2),B2)
            & ~ aa(A,$o,dvd_dvd(A,B2),A2) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3156_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,Sa: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => ( ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),Sa))
          <=> ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),Sa)) ) ) ) ).

% minf(10)
tff(fact_3157_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,Sa: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Z3)
         => ( aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),Sa))
          <=> aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),Sa)) ) ) ) ).

% minf(9)
tff(fact_3158_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,Sa: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => ( ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),Sa))
          <=> ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),Sa)) ) ) ) ).

% pinf(10)
tff(fact_3159_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,Sa: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X5)
         => ( aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),Sa))
          <=> aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),Sa)) ) ) ) ).

% pinf(9)
tff(fact_3160_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3161_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A))
        <=> ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
            & aa(A,$o,dvd_dvd(A,B2),one_one(A)) ) ) ) ).

% is_unit_mult_iff
tff(fact_3162_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_3163_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)
          <=> aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_3164_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
          <=> aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3165_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)
          <=> aa(A,$o,dvd_dvd(A,B2),C2) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3166_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_left_cancel
tff(fact_3167_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_right_cancel
tff(fact_3168_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3169_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3170_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))
          <=> aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% dvd_div_unit_iff
tff(fact_3171_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2)
          <=> aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% div_unit_dvd_iff
tff(fact_3172_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_3173_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,D2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( 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),A2),B2)),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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3174_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)
         => aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ).

% dvd_mult_imp_div
tff(fact_3175_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_3176_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B2)
         => ( aa(A,$o,dvd_dvd(A,B2),A2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% div_div_eq_right
tff(fact_3177_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ) ) ).

% div_mult_swap
tff(fact_3178_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),C2) ) ) ) ).

% dvd_div_mult
tff(fact_3179_norm__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xa: A,Y: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Y)) ) ).

% norm_mult
tff(fact_3180_norm__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),zero_zero(real)) ) ).

% norm_not_less_zero
tff(fact_3181_norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,B2: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)) ) ).

% norm_divide
tff(fact_3182_div__power,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),Na) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na)) ) ) ) ).

% div_power
tff(fact_3183_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% dvd_neg_div
tff(fact_3184_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% dvd_div_neg
tff(fact_3185_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = zero_zero(A) )
         => aa(A,$o,dvd_dvd(A,B2),A2) ) ) ).

% mod_0_imp_dvd
tff(fact_3186_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),A2) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3187_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = zero_zero(A) )
        <=> aa(A,$o,dvd_dvd(A,B2),A2) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_3188_norm__power,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xa: A,Na: nat] : real_V7770717601297561774m_norm(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Xa)),Na) ) ).

% norm_power
tff(fact_3189_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Na: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ).

% le_imp_power_dvd
tff(fact_3190_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat,B2: A,M: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),B2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),B2) ) ) ) ).

% power_le_dvd
tff(fact_3191_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xa: A,Y: A,Na: nat,M: nat] :
          ( aa(A,$o,dvd_dvd(A,Xa),Y)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),M)) ) ) ) ).

% dvd_power_le
tff(fact_3192_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A] : aa(A,$o,dvd_dvd(A,B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2))) ) ).

% dvd_minus_mod
tff(fact_3193_mod__eq__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2) )
        <=> aa(A,$o,dvd_dvd(A,C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ).

% mod_eq_dvd_iff
tff(fact_3194_dvd__pos__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,dvd_dvd(nat,M),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M) ) ) ).

% dvd_pos_nat
tff(fact_3195_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_3196_dvd__minus__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
        | aa(nat,$o,dvd_dvd(nat,M),Na) ) ) ).

% dvd_minus_self
tff(fact_3197_dvd__diffD,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na))
     => ( aa(nat,$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_3198_dvd__diffD1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na))
     => ( aa(nat,$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_3199_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_3200_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D3: nat,X: nat,Y3: nat] :
      ( aa(nat,$o,dvd_dvd(nat,D3),A2)
      & aa(nat,$o,dvd_dvd(nat,D3),B2)
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),D3) )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)),D3) ) ) ) ).

% bezout_add_nat
tff(fact_3201_bezout__lemma__nat,axiom,
    ! [D2: nat,A2: nat,B2: nat,Xa: nat,Y: nat] :
      ( aa(nat,$o,dvd_dvd(nat,D2),A2)
     => ( aa(nat,$o,dvd_dvd(nat,D2),B2)
       => ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),D2) )
            | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y)),D2) ) )
         => ? [X: nat,Y3: nat] :
              ( aa(nat,$o,dvd_dvd(nat,D2),A2)
              & aa(nat,$o,dvd_dvd(nat,D2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))
              & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),Y3)),D2) )
                | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)),D2) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_3202_bezout1__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D3: nat,X: nat,Y3: nat] :
      ( aa(nat,$o,dvd_dvd(nat,D3),A2)
      & aa(nat,$o,dvd_dvd(nat,D3),B2)
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)) = D3 )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)) = D3 ) ) ) ).

% bezout1_nat
tff(fact_3203_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_bo(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_add
tff(fact_3204_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_bp(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ) ).

% suminf_mult2
tff(fact_3205_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_bq(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_3206_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,aa(A,fun(A,A),minus_minus(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_diff
tff(fact_3207_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_bg(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_3208_suminf__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_bt(fun(nat,A),fun(nat,A),F2)) = aa(A,A,uminus_uminus(A),suminf(A,F2)) ) ) ) ).

% suminf_minus
tff(fact_3209_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),M),A2))
        <=> ~ ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            <=> ( M = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_3210_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_3211_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R: real,F2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
         => ( summable(A,F2)
           => ? [N8: nat] :
              ! [N9: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N9)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_bj(fun(nat,A),fun(nat,fun(nat,A)),F2),N9)))),R) ) ) ) ) ).

% suminf_exist_split
tff(fact_3212_summable__norm__cancel,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_cg(fun(nat,A),fun(nat,real),F2))
         => summable(A,F2) ) ) ).

% summable_norm_cancel
tff(fact_3213_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,N))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F2)) ) ) ) ).

% suminf_pos
tff(fact_3214_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,aa(A,fun(nat,A),power_power(A),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),C2)) ) ) ) ).

% suminf_geometric
tff(fact_3215_norm__uminus__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),Xa)),Y)) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) ) ).

% norm_uminus_minus
tff(fact_3216_power__eq__imp__eq__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,Na: nat,Z: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W),Na) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Na) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => ( real_V7770717601297561774m_norm(A,W) = real_V7770717601297561774m_norm(A,Z) ) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_3217_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_3218_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,$o),L: A] :
          ( ? [X4: A] : aa(A,$o,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X4))
        <=> ? [X4: A] :
              ( aa(A,$o,dvd_dvd(A,L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),zero_zero(A)))
              & aa(A,$o,P,X4) ) ) ) ).

% unity_coeff_ex
tff(fact_3219_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [C5: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A2),C5) ) ) ) ).

% unit_dvdE
tff(fact_3220_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)) ) ) ) ).

% nonzero_norm_divide
tff(fact_3221_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3222_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( aa(A,$o,dvd_dvd(A,A2),B2)
             => ( aa(A,$o,dvd_dvd(A,C2),D2)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),D2),C2) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),D2) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3223_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,C2),B2)
           => ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
            <=> aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3224_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,B2),A2)
           => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2)
            <=> aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3225_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,A2),B2)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3226_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D4: A,T: A] :
          ( aa(A,$o,dvd_dvd(A,D2),D4)
         => ! [X5: A,K5: A] :
              ( ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),T))
            <=> ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D4))),T)) ) ) ) ).

% inf_period(4)
tff(fact_3227_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D4: A,T: A] :
          ( aa(A,$o,dvd_dvd(A,D2),D4)
         => ! [X5: A,K5: A] :
              ( aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),T))
            <=> aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D4))),T)) ) ) ) ).

% inf_period(3)
tff(fact_3228_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3229_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_3230_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% unit_div_commute
tff(fact_3231_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,B2),A2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_3232_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_3233_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = C2 )
          <=> ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% unit_eq_div1
tff(fact_3234_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),one_one(A))
        <=> ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
            | ( Na = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_3235_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_3236_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xa: A,R: real,Y: A,Sa: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),R)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Y)),Sa)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),R),Sa)) ) ) ) ).

% norm_mult_less
tff(fact_3237_norm__mult__ineq,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xa: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_mult_ineq
tff(fact_3238_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Y))),E)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),E) ) ) ).

% norm_triangle_lt
tff(fact_3239_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,R: real,Y: A,Sa: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),R)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Y)),Sa)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R),Sa)) ) ) ) ).

% norm_add_less
tff(fact_3240_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Y))),E)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),E) ) ) ).

% norm_triangle_le
tff(fact_3241_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,R: real,B2: A,Sa: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,A2)),R)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),Sa)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R),Sa)) ) ) ) ).

% norm_triangle_mono
tff(fact_3242_norm__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_triangle_ineq
tff(fact_3243_norm__add__leD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),C2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A2)),C2)) ) ) ).

% norm_add_leD
tff(fact_3244_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y))),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z))),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_less
tff(fact_3245_norm__power__ineq,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xa: A,Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Xa)),Na)) ) ).

% norm_power_ineq
tff(fact_3246_norm__diff__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ).

% norm_diff_ineq
tff(fact_3247_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_3248_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_3249_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_3250_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [D3: nat,X: nat,Y3: nat] :
          ( aa(nat,$o,dvd_dvd(nat,D3),A2)
          & aa(nat,$o,dvd_dvd(nat,D3),B2)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),D3) ) ) ) ).

% bezout_add_strong_nat
tff(fact_3251_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na))
    <=> ~ aa(nat,$o,dvd_dvd(nat,Na),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_3252_mod__eq__dvd__iff__nat,axiom,
    ! [Na: nat,M: nat,Q2: 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),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),Q2) )
      <=> aa(nat,$o,dvd_dvd(nat,Q2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_3253_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_3254_summable__rabs,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(real,aTP_Lamp_bu(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_bu(fun(nat,real),fun(nat,real),F2))) ) ).

% summable_rabs
tff(fact_3255_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N7: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
             => aa(real,$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_ci(fun(nat,A),fun(nat,real),F2)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_3256_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,$o),ord_less(A),zero_zero(A)),suminf(A,F2))
            <=> ? [I2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I2)) ) ) ) ) ).

% suminf_pos_iff
tff(fact_3257_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_3258_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_3259_odd__even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2)
           => aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% odd_even_add
tff(fact_3260_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_3261_evenE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ~ ! [B6: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B6) ) ) ).

% evenE
tff(fact_3262_power__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,Na: nat] :
          ( ( aa(nat,A,aa(A,fun(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_3263_bit__eq__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2) )
            & ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ) ).

% bit_eq_rec
tff(fact_3264_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3265_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3266_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [B6: A] :
                  ( ( B6 != zero_zero(A) )
                 => ( aa(A,$o,dvd_dvd(A,B6),one_one(A))
                   => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) = B6 )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B6) = A2 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B6) = one_one(A) )
                         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B6) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3267_even__minus,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% even_minus
tff(fact_3268_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),aa(num,num,bit1,Na))) ) ).

% odd_numeral
tff(fact_3269_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Xa: A,M: nat,Na: nat] :
          ( ( Xa != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na))
          <=> ( aa(A,$o,dvd_dvd(A,Xa),one_one(A))
              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ) ).

% dvd_power_iff
tff(fact_3270_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D2)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)))) ) ).

% norm_diff_triangle_ineq
tff(fact_3271_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_bs(fun(nat,A),fun(nat,A),F2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,F2)),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_3272_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Na: nat,Xa: A] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
            | ( Xa = one_one(A) ) )
         => aa(A,$o,dvd_dvd(A,Xa),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)) ) ) ).

% dvd_power
tff(fact_3273_norm__less__p1,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xa: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),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,Xa))),one_one(A)))) ) ).

% norm_less_p1
tff(fact_3274_even__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),M),A2))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% even_signed_take_bit_iff
tff(fact_3275_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_3276_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_3277_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_3278_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),I)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% power_dvd_imp_le
tff(fact_3279_dvd__minus__add,axiom,
    ! [Q2: nat,Na: nat,R: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q2),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),M))
       => ( aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),Q2))
        <=> aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),M)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_3280_diff__mod__le,axiom,
    ! [A2: nat,D2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),D2)
     => ( aa(nat,$o,dvd_dvd(nat,B2),D2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),B2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),D2),B2)) ) ) ).

% diff_mod_le
tff(fact_3281_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,aa(nat,fun(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_3282_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Xa: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ca(fun(nat,A),fun(A,fun(nat,A)),F2),Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xa))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_ca(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ) ).

% powser_inside
tff(fact_3283_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),one_one(real))
         => summable(A,aa(A,fun(nat,A),power_power(A),Xa)) ) ) ).

% complete_algebra_summable_geometric
tff(fact_3284_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,aa(A,fun(nat,A),power_power(A),C2)) ) ) ).

% summable_geometric
tff(fact_3285_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) = A2 ) ) ) ).

% even_two_times_div_two
tff(fact_3286_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3287_odd__iff__mod__2__eq__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_3288_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( ~ aa(nat,$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),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na)) ) ) ) ).

% power_mono_odd
tff(fact_3289_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X7: fun(A,B)] :
          ( ? [K7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K7)
              & ! [N2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X7,N2))),K7) )
        <=> ? [N10: nat] :
            ! [N2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,X7,N2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N10))) ) ) ).

% lemma_NBseq_def2
tff(fact_3290_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X7: fun(A,B)] :
          ( ? [K7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K7)
              & ! [N2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X7,N2))),K7) )
        <=> ? [N10: nat] :
            ! [N2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X7,N2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N10))) ) ) ).

% lemma_NBseq_def
tff(fact_3291_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_3292_uminus__power__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Na: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Na) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))) ) ).

% uminus_power_if
tff(fact_3293_power__even__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),Na) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na) ) ) ) ).

% power_even_abs
tff(fact_3294_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,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% dvd_power_iff_le
tff(fact_3295_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( aa(A,$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),A2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            | ( M = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_3296_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( aa(A,$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),A2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            & ( M != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_3297_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),Xa: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_bz(fun(nat,A),fun(A,fun(nat,A)),F2),Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xa))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_cj(fun(nat,A),fun(A,fun(nat,real)),F2),Z)) ) ) ) ).

% powser_insidea
tff(fact_3298_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_ca(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_ca(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_cb(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_3299_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_ca(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_cb(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ca(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_3300_square__norm__one,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xa: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
         => ( real_V7770717601297561774m_norm(A,Xa) = one_one(real) ) ) ) ).

% square_norm_one
tff(fact_3301_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ~ ! [B6: A] : A2 != 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))),B6)),one_one(A)) ) ) ).

% oddE
tff(fact_3302_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = $ite(aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2),zero_zero(A),one_one(A)) ) ).

% mod2_eq_if
tff(fact_3303_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) ) )
         => ~ ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
             => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_3304_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))
        <=> ( aa(nat,$o,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)),A2) ) ) ) ) ).

% zero_le_power_eq
tff(fact_3305_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: A] :
          ( ~ aa(nat,$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,aa(A,fun(nat,A),power_power(A),A2),Na))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ).

% zero_le_odd_power
tff(fact_3306_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: A] :
          ( aa(nat,$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,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ).

% zero_le_even_power
tff(fact_3307_minus__one__power__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),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_3308_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,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),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,aa(A,fun(A,A),minus_minus(A),Z),W)))) ) ) ) ).

% norm_power_diff
tff(fact_3309_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( aa(nat,$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),A2)),aa(A,A,abs_abs(A),B2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na)) ) ) ) ).

% power_mono_even
tff(fact_3310_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R: real,R0: real,A2: fun(nat,A),M7: real] :
          ( aa(real,$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,A2,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),R0),N))),M7)
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_ck(real,fun(fun(nat,A),fun(nat,real)),R),A2)) ) ) ) ) ).

% Abel_lemma
tff(fact_3311_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))
        <=> ( ( Na = zero_zero(nat) )
            | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
              & ( A2 != 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)),A2) ) ) ) ) ).

% zero_less_power_eq
tff(fact_3312_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_3313_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,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% even_mask_div_iff'
tff(fact_3314_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
            & ( ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
              | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3315_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% even_mod_4_div_2
tff(fact_3316_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,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% even_mask_div_iff
tff(fact_3317_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),aa(num,num,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,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% odd_mod_4_div_2
tff(fact_3318_Bernoulli__inequality__even,axiom,
    ! [Na: nat,Xa: 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)),Xa))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),Na)) ) ).

% Bernoulli_inequality_even
tff(fact_3319_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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
tff(fact_3320_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( aa(A,$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),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
            | ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
              & aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3321_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V441764108873111860ildupi(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(nat,nat,suc,zero_zero(nat)) ) )
         => ~ ! [N: nat] :
                ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
               => ( Y != $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_3322_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),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% VEBT_internal.Tb'.simps(3)
tff(fact_3323_VEBT__internal_OTb_H_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_VEBT_Tb2(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
         => ~ ! [N: nat] :
                ( ( Xa = 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),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
tff(fact_3324_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),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% VEBT_internal.Tb.simps(3)
tff(fact_3325_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),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(bit0(one2))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
tff(fact_3326_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),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_3327_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(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_3328_VEBT__internal_OTb_Oelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( vEBT_VEBT_Tb(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) ) )
         => ~ ! [N: nat] :
                ( ( Xa = 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),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
tff(fact_3329_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
         => ~ ! [Va2: nat] :
                ( ( Xa = 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),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_3330_div2__even__ext__nat,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),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,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Xa)
        <=> aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Y) )
       => ( Xa = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_3331_arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,real,arctan,Xa) = suminf(real,aTP_Lamp_bf(real,fun(nat,real),Xa)) ) ) ).

% arctan_series
tff(fact_3332_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_cl(nat,real)) ).

% pi_series
tff(fact_3333_httI__TBOUND,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),T: nat] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( time_TBOUND(A,C2,T)
       => time_htt(A,P,C2,Q,T) ) ) ).

% httI_TBOUND
tff(fact_3334_norm__pre__pure__iff__htt,axiom,
    ! [A: $tType,P: assn,B2: $o,F2: heap_Time_Heap(A),Q: fun(A,assn),T: nat] :
      ( time_htt(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B2))),F2,Q,T)
    <=> ( (B2)
       => time_htt(A,P,F2,Q,T) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_3335_vebt__inserti_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : hoare_hoare_triple(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_V3964819847710782039nserti(T,Tia,Xa),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(T,Xa))) ).

% vebt_inserti'_rf_abstr
tff(fact_3336_deleti_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => hoare_hoare_triple(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_V1365221501068881998eletei(T,Tia,Xa),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_delete(T,Xa))) ) ).

% deleti'_rf_abstr
tff(fact_3337_builupicorr,axiom,
    ! [Na: nat] : hoare_hoare_triple(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(Na),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na))) ).

% builupicorr
tff(fact_3338_builupi_Hcorr,axiom,
    ! [Na: nat] : hoare_hoare_triple(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(Na),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na))) ).

% builupi'corr
tff(fact_3339_arctan__eq__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa(real,real,arctan,Xa) = zero_zero(real) )
    <=> ( Xa = zero_zero(real) ) ) ).

% arctan_eq_zero_iff
tff(fact_3340_arctan__zero__zero,axiom,
    aa(real,real,arctan,zero_zero(real)) = zero_zero(real) ).

% arctan_zero_zero
tff(fact_3341_zero__less__arctan__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,arctan,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% zero_less_arctan_iff
tff(fact_3342_arctan__less__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% arctan_less_zero_iff
tff(fact_3343_zero__le__arctan__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,arctan,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% zero_le_arctan_iff
tff(fact_3344_arctan__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arctan,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% arctan_le_zero_iff
tff(fact_3345_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_3346_zdvd1__eq,axiom,
    ! [Xa: int] :
      ( aa(int,$o,dvd_dvd(int,Xa),one_one(int))
    <=> ( aa(int,int,abs_abs(int),Xa) = one_one(int) ) ) ).

% zdvd1_eq
tff(fact_3347_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_3348_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_3349_arctan__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa(real,real,arctan,Xa) = aa(real,real,arctan,Y) )
    <=> ( Xa = Y ) ) ).

% arctan_eq_iff
tff(fact_3350_pi__neq__zero,axiom,
    pi != zero_zero(real) ).

% pi_neq_zero
tff(fact_3351_arctan__monotone,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,Xa)),aa(real,real,arctan,Y)) ) ).

% arctan_monotone
tff(fact_3352_arctan__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,Xa)),aa(real,real,arctan,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ).

% arctan_less_iff
tff(fact_3353_arctan__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arctan,Xa)),aa(real,real,arctan,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ).

% arctan_le_iff
tff(fact_3354_arctan__monotone_H,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arctan,Xa)),aa(real,real,arctan,Y)) ) ).

% arctan_monotone'
tff(fact_3355_zdvd__zdiffD,axiom,
    ! [K: int,M: int,Na: int] :
      ( aa(int,$o,dvd_dvd(int,K),aa(int,int,aa(int,fun(int,int),minus_minus(int),M),Na))
     => ( aa(int,$o,dvd_dvd(int,K),Na)
       => aa(int,$o,dvd_dvd(int,K),M) ) ) ).

% zdvd_zdiffD
tff(fact_3356_zdvd__antisym__abs,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,dvd_dvd(int,A2),B2)
     => ( aa(int,$o,dvd_dvd(int,B2),A2)
       => ( aa(int,int,abs_abs(int),A2) = aa(int,int,abs_abs(int),B2) ) ) ) ).

% zdvd_antisym_abs
tff(fact_3357_arctan__minus,axiom,
    ! [Xa: real] : aa(real,real,arctan,aa(real,real,uminus_uminus(real),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,arctan,Xa)) ).

% arctan_minus
tff(fact_3358_pi__not__less__zero,axiom,
    ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),zero_zero(real)) ).

% pi_not_less_zero
tff(fact_3359_pi__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),pi) ).

% pi_gt_zero
tff(fact_3360_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_3361_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_3362_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_3363_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_3364_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_3365_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_3366_zdvd__period,axiom,
    ! [A2: int,D2: int,Xa: int,T: int,C2: int] :
      ( aa(int,$o,dvd_dvd(int,A2),D2)
     => ( aa(int,$o,dvd_dvd(int,A2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),T))
      <=> aa(int,$o,dvd_dvd(int,A2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),aa(int,int,aa(int,fun(int,int),times_times(int),C2),D2))),T)) ) ) ).

% zdvd_period
tff(fact_3367_abs__div,axiom,
    ! [Y: int,Xa: int] :
      ( aa(int,$o,dvd_dvd(int,Y),Xa)
     => ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),Y)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),Xa)),aa(int,int,abs_abs(int),Y)) ) ) ).

% abs_div
tff(fact_3368_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_3369_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_3370_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_3371_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_3372_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_3373_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_3374_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_3375_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),aa(num,num,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),aa(num,num,bit1,aa(num,num,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),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,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_3376_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,aa(real,fun(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),aa(num,num,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),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))))))) ).

% machin
tff(fact_3377_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_3378_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_3379_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_3380_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_3381_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_3382_int__div__sub__1,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),M)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Na),one_one(int))),M) = $ite(aa(int,$o,dvd_dvd(int,M),Na),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),M)),one_one(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),M)) ) ) ).

% int_div_sub_1
tff(fact_3383_aset_I10_J,axiom,
    ! [D2: int,D4: int,A3: set(int),T: int] :
      ( aa(int,$o,dvd_dvd(int,D2),D4)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A3)
                 => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( ~ aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),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),X5),D4)),T)) ) ) ) ).

% aset(10)
tff(fact_3384_aset_I9_J,axiom,
    ! [D2: int,D4: int,A3: set(int),T: int] :
      ( aa(int,$o,dvd_dvd(int,D2),D4)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A3)
                 => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),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),X5),D4)),T)) ) ) ) ).

% aset(9)
tff(fact_3385_bset_I10_J,axiom,
    ! [D2: int,D4: int,B3: set(int),T: int] :
      ( aa(int,$o,dvd_dvd(int,D2),D4)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B3)
                 => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( ~ aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),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),minus_minus(int),X5),D4)),T)) ) ) ) ).

% bset(10)
tff(fact_3386_bset_I9_J,axiom,
    ! [D2: int,D4: int,B3: set(int),T: int] :
      ( aa(int,$o,dvd_dvd(int,D2),D4)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B3)
                 => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),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),minus_minus(int),X5),D4)),T)) ) ) ) ).

% bset(9)
tff(fact_3387_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L))
    <=> aa(int,$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_3388_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_3389_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_3390_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_3391_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_3392_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_3393_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_3394_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_3395_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_3396_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_3397_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_3398_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_3399_htt__htD,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),T: nat] :
      ( time_htt(A,P,C2,Q,T)
     => hoare_hoare_triple(A,P,C2,Q) ) ).

% htt_htD
tff(fact_3400_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_3401_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_3402_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_3403_htt__refine,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),T: nat,C8: heap_Time_Heap(A)] :
      ( time_htt(A,P,C2,Q,T)
     => ( refine_Imp_refines(A,C8,C2)
       => time_htt(A,P,C8,Q,T) ) ) ).

% htt_refine
tff(fact_3404_arctan__add,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,arctan,Xa)),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),Xa),Y)),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y)))) ) ) ) ).

% arctan_add
tff(fact_3405_arctan__double,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,Xa)) = 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))),Xa)),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% arctan_double
tff(fact_3406_norm__pre__pure__iff__htt_H,axiom,
    ! [A: $tType,B2: $o,P: assn,F2: heap_Time_Heap(A),Q: fun(A,assn),T: nat] :
      ( time_htt(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn((B2))),P),F2,Q,T)
    <=> ( (B2)
       => time_htt(A,P,F2,Q,T) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_3407_lowi__hT,axiom,
    ! [Xa: nat,Na: nat] : time_htt(nat,one_one(assn),vEBT_VEBT_lowi(Xa,Na),aa(nat,fun(nat,assn),aTP_Lamp_cm(nat,fun(nat,fun(nat,assn)),Xa),Na),one_one(nat)) ).

% lowi_hT
tff(fact_3408_highi__hT,axiom,
    ! [Xa: nat,Na: nat] : time_htt(nat,one_one(assn),vEBT_VEBT_highi(Xa,Na),aa(nat,fun(nat,assn),aTP_Lamp_cn(nat,fun(nat,fun(nat,assn)),Xa),Na),one_one(nat)) ).

% highi_hT
tff(fact_3409_sin__cos__npi,axiom,
    ! [Na: nat] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Na) ).

% sin_cos_npi
tff(fact_3410_lowi__h,axiom,
    ! [Xa: nat,Na: nat] : hoare_hoare_triple(nat,one_one(assn),vEBT_VEBT_lowi(Xa,Na),aa(nat,fun(nat,assn),aTP_Lamp_cm(nat,fun(nat,fun(nat,assn)),Xa),Na)) ).

% lowi_h
tff(fact_3411_highi__h,axiom,
    ! [Xa: nat,Na: nat] : hoare_hoare_triple(nat,one_one(assn),vEBT_VEBT_highi(Xa,Na),aa(nat,fun(nat,assn),aTP_Lamp_cn(nat,fun(nat,fun(nat,assn)),Xa),Na)) ).

% highi_h
tff(fact_3412_TBOUND__lowi,axiom,
    ! [Xa: nat,Na: nat] : time_TBOUND(nat,vEBT_VEBT_lowi(Xa,Na),one_one(nat)) ).

% TBOUND_lowi
tff(fact_3413_TBOUND__highi,axiom,
    ! [Xa: nat,Na: nat] : time_TBOUND(nat,vEBT_VEBT_highi(Xa,Na),one_one(nat)) ).

% TBOUND_highi
tff(fact_3414_sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,sin(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sin_zero
tff(fact_3415_sin__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,A,sin(A),aa(A,A,uminus_uminus(A),Xa)) = aa(A,A,uminus_uminus(A),aa(A,A,sin(A),Xa)) ) ).

% sin_minus
tff(fact_3416_sin__pi,axiom,
    aa(real,real,sin(real),pi) = zero_zero(real) ).

% sin_pi
tff(fact_3417_sin__pi__minus,axiom,
    ! [Xa: real] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),Xa)) = aa(real,real,sin(real),Xa) ).

% sin_pi_minus
tff(fact_3418_sin__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,sin(A),aa(real,A,real_Vector_of_real(A),pi)) = zero_zero(A) ) ) ).

% sin_of_real_pi
tff(fact_3419_sin__periodic__pi2,axiom,
    ! [Xa: real] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),pi),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,sin(real),Xa)) ).

% sin_periodic_pi2
tff(fact_3420_sin__periodic__pi,axiom,
    ! [Xa: real] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),pi)) = aa(real,real,uminus_uminus(real),aa(real,real,sin(real),Xa)) ).

% sin_periodic_pi
tff(fact_3421_sin__minus__pi,axiom,
    ! [Xa: real] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),pi)) = aa(real,real,uminus_uminus(real),aa(real,real,sin(real),Xa)) ).

% sin_minus_pi
tff(fact_3422_sin__npi,axiom,
    ! [Na: nat] : aa(real,real,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_3423_sin__npi2,axiom,
    ! [Na: nat] : aa(real,real,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_3424_sin__npi__int,axiom,
    ! [Na: int] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),Na))) = zero_zero(real) ).

% sin_npi_int
tff(fact_3425_sin__two__pi,axiom,
    aa(real,real,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_3426_sin__pi__half,axiom,
    aa(real,real,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_3427_sin__periodic,axiom,
    ! [Xa: real] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = aa(real,real,sin(real),Xa) ).

% sin_periodic
tff(fact_3428_sin__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,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_3429_sin__2npi,axiom,
    ! [Na: nat] : aa(real,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))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)) = zero_zero(real) ).

% sin_2npi
tff(fact_3430_sin__2pi__minus,axiom,
    ! [Xa: real] : aa(real,real,sin(real),aa(real,real,aa(real,fun(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)),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,sin(real),Xa)) ).

% sin_2pi_minus
tff(fact_3431_sin__int__2pin,axiom,
    ! [Na: int] : aa(real,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)),aa(int,real,ring_1_of_int(real),Na))) = zero_zero(real) ).

% sin_int_2pin
tff(fact_3432_sin__3over2__pi,axiom,
    aa(real,real,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),aa(num,num,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_3433_sin__of__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: real] : aa(A,A,sin(A),aa(real,A,real_Vector_of_real(A),Xa)) = aa(real,A,real_Vector_of_real(A),aa(real,real,sin(real),Xa)) ) ).

% sin_of_real
tff(fact_3434_sin__x__le__x,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sin(real),Xa)),Xa) ) ).

% sin_x_le_x
tff(fact_3435_sin__le__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sin(real),Xa)),one_one(real)) ).

% sin_le_one
tff(fact_3436_abs__sin__x__le__abs__x,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,sin(real),Xa))),aa(real,real,abs_abs(real),Xa)) ).

% abs_sin_x_le_abs_x
tff(fact_3437_sin__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [M: int,Xa: real] : aa(A,A,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),Xa))) = aa(real,A,real_Vector_of_real(A),aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),M)),Xa))) ) ).

% sin_int_times_real
tff(fact_3438_sin__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),pi)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sin(real),Xa)) ) ) ).

% sin_gt_zero
tff(fact_3439_sin__x__ge__neg__x,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),Xa)),aa(real,real,sin(real),Xa)) ) ).

% sin_x_ge_neg_x
tff(fact_3440_sin__ge__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sin(real),Xa)) ) ) ).

% sin_ge_zero
tff(fact_3441_sin__ge__minus__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(real,real,sin(real),Xa)) ).

% sin_ge_minus_one
tff(fact_3442_abs__sin__le__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,sin(real),Xa))),one_one(real)) ).

% abs_sin_le_one
tff(fact_3443_sin__eq__0__pi,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),pi)
       => ( ( aa(real,real,sin(real),Xa) = zero_zero(real) )
         => ( Xa = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_3444_sin__zero__pi__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),pi)
     => ( ( aa(real,real,sin(real),Xa) = zero_zero(real) )
      <=> ( Xa = zero_zero(real) ) ) ) ).

% sin_zero_pi_iff
tff(fact_3445_sin__zero__iff__int2,axiom,
    ! [Xa: real] :
      ( ( aa(real,real,sin(real),Xa) = zero_zero(real) )
    <=> ? [I2: int] : Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I2)),pi) ) ).

% sin_zero_iff_int2
tff(fact_3446_sin__gt__zero__02,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sin(real),Xa)) ) ) ).

% sin_gt_zero_02
tff(fact_3447_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)),aa(real,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_3448_sin__gt__zero2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sin(real),Xa)) ) ) ).

% sin_gt_zero2
tff(fact_3449_sin__lt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sin(real),Xa)),zero_zero(real)) ) ) ).

% sin_lt_zero
tff(fact_3450_sin__30,axiom,
    aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(aa(num,num,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_3451_sin__inj__pi,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( ( aa(real,real,sin(real),Xa) = aa(real,real,sin(real),Y) )
             => ( Xa = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_3452_sin__mono__le__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sin(real),Xa)),aa(real,real,sin(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_3453_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sin(real),Y)),aa(real,real,sin(real),Xa)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_3454_sin__le__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),pi),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sin(real),Xa)),zero_zero(real)) ) ) ).

% sin_le_zero
tff(fact_3455_sin__less__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sin(real),Xa)),zero_zero(real)) ) ) ).

% sin_less_zero
tff(fact_3456_sin__mono__less__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sin(real),Xa)),aa(real,real,sin(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_3457_sin__monotone__2pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sin(real),Y)),aa(real,real,sin(real),Xa)) ) ) ) ).

% sin_monotone_2pi
tff(fact_3458_sin__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ? [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( aa(real,real,sin(real),X) = Y )
            & ! [Y2: 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))))),Y2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
                  & ( aa(real,real,sin(real),Y2) = Y ) )
               => ( Y2 = X ) ) ) ) ) ).

% sin_total
tff(fact_3459_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)),aa(real,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_3460_sin__zero__iff__int,axiom,
    ! [Xa: real] :
      ( ( aa(real,real,sin(real),Xa) = zero_zero(real) )
    <=> ? [I2: int] :
          ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),I2)
          & ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I2)),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_3461_sin__zero__lemma,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( ( aa(real,real,sin(real),Xa) = zero_zero(real) )
       => ? [N: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
            & ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_3462_sin__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa(real,real,sin(real),Xa) = zero_zero(real) )
    <=> ( ? [N2: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)
            & ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N2: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)
            & ( Xa = 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_3463_lowi__def,axiom,
    ! [Xa: nat,Na: nat] : vEBT_VEBT_lowi(Xa,Na) = heap_Time_return(nat,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% lowi_def
tff(fact_3464_highi__def,axiom,
    ! [Xa: nat,Na: nat] : vEBT_VEBT_highi(Xa,Na) = heap_Time_return(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% highi_def
tff(fact_3465_sincos__total__2pi,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
     => ~ ! [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( ( Xa = aa(real,real,cos(real),T4) )
               => ( Y != aa(real,real,sin(real),T4) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_3466_cos__pi__eq__zero,axiom,
    ! [M: nat] : aa(real,real,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_3467_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_3468_highsimp,axiom,
    ! [Xa: nat,Na: nat] : heap_Time_return(nat,vEBT_VEBT_high(Xa,Na)) = vEBT_VEBT_highi(Xa,Na) ).

% highsimp
tff(fact_3469_lowsimp,axiom,
    ! [Xa: nat,Na: nat] : heap_Time_return(nat,vEBT_VEBT_low(Xa,Na)) = vEBT_VEBT_lowi(Xa,Na) ).

% lowsimp
tff(fact_3470_cos__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,A,cos(A),aa(A,A,uminus_uminus(A),Xa)) = aa(A,A,cos(A),Xa) ) ).

% cos_minus
tff(fact_3471_cos__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,cos(A),zero_zero(A)) = one_one(A) ) ) ).

% cos_zero
tff(fact_3472_cos__pi,axiom,
    aa(real,real,cos(real),pi) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% cos_pi
tff(fact_3473_cos__periodic__pi,axiom,
    ! [Xa: real] : aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),pi)) = aa(real,real,uminus_uminus(real),aa(real,real,cos(real),Xa)) ).

% cos_periodic_pi
tff(fact_3474_cos__periodic__pi2,axiom,
    ! [Xa: real] : aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),pi),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,cos(real),Xa)) ).

% cos_periodic_pi2
tff(fact_3475_cos__pi__minus,axiom,
    ! [Xa: real] : aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,cos(real),Xa)) ).

% cos_pi_minus
tff(fact_3476_cos__minus__pi,axiom,
    ! [Xa: real] : aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),pi)) = aa(real,real,uminus_uminus(real),aa(real,real,cos(real),Xa)) ).

% cos_minus_pi
tff(fact_3477_sin__cos__squared__add3,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: 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,cos(A),Xa)),aa(A,A,cos(A),Xa))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sin(A),Xa)),aa(A,A,sin(A),Xa))) = one_one(A) ) ).

% sin_cos_squared_add3
tff(fact_3478_cos__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,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_3479_cos__pi__half,axiom,
    aa(real,real,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_3480_cos__two__pi,axiom,
    aa(real,real,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_3481_cos__periodic,axiom,
    ! [Xa: real] : aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = aa(real,real,cos(real),Xa) ).

% cos_periodic
tff(fact_3482_cos__2pi__minus,axiom,
    ! [Xa: real] : aa(real,real,cos(real),aa(real,real,aa(real,fun(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)),Xa)) = aa(real,real,cos(real),Xa) ).

% cos_2pi_minus
tff(fact_3483_cos__npi2,axiom,
    ! [Na: nat] : aa(real,real,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,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Na) ).

% cos_npi2
tff(fact_3484_cos__npi,axiom,
    ! [Na: nat] : aa(real,real,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,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Na) ).

% cos_npi
tff(fact_3485_sin__cos__squared__add2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cos(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sin(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% sin_cos_squared_add2
tff(fact_3486_sin__cos__squared__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sin(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cos(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% sin_cos_squared_add
tff(fact_3487_cos__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,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_3488_cos__2npi,axiom,
    ! [Na: nat] : aa(real,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))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)) = one_one(real) ).

% cos_2npi
tff(fact_3489_cos__int__2pin,axiom,
    ! [Na: int] : aa(real,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)),aa(int,real,ring_1_of_int(real),Na))) = one_one(real) ).

% cos_int_2pin
tff(fact_3490_cos__3over2__pi,axiom,
    aa(real,real,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),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) = zero_zero(real) ).

% cos_3over2_pi
tff(fact_3491_cos__npi__int,axiom,
    ! [Na: int] :
      aa(real,real,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_3492_cos__of__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: real] : aa(A,A,cos(A),aa(real,A,real_Vector_of_real(A),Xa)) = aa(real,A,real_Vector_of_real(A),aa(real,real,cos(real),Xa)) ) ).

% cos_of_real
tff(fact_3493_cos__le__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,cos(real),Xa)),one_one(real)) ).

% cos_le_one
tff(fact_3494_polar__Ex,axiom,
    ! [Xa: real,Y: real] :
    ? [R4: real,A5: real] :
      ( ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),R4),aa(real,real,cos(real),A5)) )
      & ( Y = aa(real,real,aa(real,fun(real,real),times_times(real),R4),aa(real,real,sin(real),A5)) ) ) ).

% polar_Ex
tff(fact_3495_cos__arctan__not__zero,axiom,
    ! [Xa: real] : aa(real,real,cos(real),aa(real,real,arctan,Xa)) != zero_zero(real) ).

% cos_arctan_not_zero
tff(fact_3496_cos__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [M: int,Xa: real] : aa(A,A,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),Xa))) = aa(real,A,real_Vector_of_real(A),aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),M)),Xa))) ) ).

% cos_int_times_real
tff(fact_3497_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( aa(A,A,cos(A),Xa) = one_one(A) )
         => ( aa(A,A,sin(A),Xa) = zero_zero(A) ) ) ) ).

% cos_one_sin_zero
tff(fact_3498_sin__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sin(A),Xa)),aa(A,A,cos(A),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),Xa)),aa(A,A,sin(A),Y))) ) ).

% sin_add
tff(fact_3499_sin__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sin(A),Xa)),aa(A,A,cos(A),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),Xa)),aa(A,A,sin(A),Y))) ) ).

% sin_diff
tff(fact_3500_cos__inj__pi,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( ( aa(real,real,cos(real),Xa) = aa(real,real,cos(real),Y) )
             => ( Xa = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_3501_cos__mono__le__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,cos(real),Xa)),aa(real,real,cos(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_3502_cos__monotone__0__pi__le,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,cos(real),Xa)),aa(real,real,cos(real),Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_3503_cos__ge__minus__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(real,real,cos(real),Xa)) ).

% cos_ge_minus_one
tff(fact_3504_abs__cos__le__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,cos(real),Xa))),one_one(real)) ).

% abs_cos_le_one
tff(fact_3505_cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),Xa)),aa(A,A,cos(A),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sin(A),Xa)),aa(A,A,sin(A),Y))) ) ).

% cos_add
tff(fact_3506_cos__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)) = 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,cos(A),Xa)),aa(A,A,cos(A),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sin(A),Xa)),aa(A,A,sin(A),Y))) ) ).

% cos_diff
tff(fact_3507_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( aa(A,A,sin(A),Xa) = zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,cos(A),Xa)) = one_one(real) ) ) ) ).

% sin_zero_norm_cos_one
tff(fact_3508_cos__two__neq__zero,axiom,
    aa(real,real,cos(real),aa(num,real,numeral_numeral(real),bit0(one2))) != zero_zero(real) ).

% cos_two_neq_zero
tff(fact_3509_cos__monotone__0__pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cos(real),Xa)),aa(real,real,cos(real),Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_3510_cos__mono__less__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cos(real),Xa)),aa(real,real,cos(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_3511_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,cos(real),Y)),aa(real,real,cos(real),Xa)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_3512_sin__zero__abs__cos__one,axiom,
    ! [Xa: real] :
      ( ( aa(real,real,sin(real),Xa) = zero_zero(real) )
     => ( aa(real,real,abs_abs(real),aa(real,real,cos(real),Xa)) = one_one(real) ) ) ).

% sin_zero_abs_cos_one
tff(fact_3513_sin__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,sin(A),Xa))),aa(A,A,cos(A),Xa)) ) ).

% sin_double
tff(fact_3514_cos__two__less__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cos(real),aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_less_zero
tff(fact_3515_cos__two__le__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,cos(real),aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_le_zero
tff(fact_3516_cos__is__zero,axiom,
    ? [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(num,real,numeral_numeral(real),bit0(one2)))
      & ( aa(real,real,cos(real),X) = zero_zero(real) )
      & ! [Y2: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),aa(num,real,numeral_numeral(real),bit0(one2)))
            & ( aa(real,real,cos(real),Y2) = zero_zero(real) ) )
         => ( Y2 = X ) ) ) ).

% cos_is_zero
tff(fact_3517_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cos(real),Y)),aa(real,real,cos(real),Xa)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_3518_cos__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ? [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),pi)
            & ( aa(real,real,cos(real),X) = Y )
            & ! [Y2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),pi)
                  & ( aa(real,real,cos(real),Y2) = Y ) )
               => ( Y2 = X ) ) ) ) ) ).

% cos_total
tff(fact_3519_sincos__principal__value,axiom,
    ! [Xa: real] :
    ? [Y3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Y3)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),pi)
      & ( aa(real,real,sin(real),Y3) = aa(real,real,sin(real),Xa) )
      & ( aa(real,real,cos(real),Y3) = aa(real,real,cos(real),Xa) ) ) ).

% sincos_principal_value
tff(fact_3520_sin__cos__le1,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sin(real),Xa)),aa(real,real,sin(real),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,cos(real),Xa)),aa(real,real,cos(real),Y))))),one_one(real)) ).

% sin_cos_le1
tff(fact_3521_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),aa(A,A,cos(A),W)),aa(A,A,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),aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z))),aa(A,A,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_3522_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),aa(A,A,cos(A),W)),aa(A,A,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))),aa(A,A,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)))))),aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_plus_cos
tff(fact_3523_sin__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sin(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cos(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% sin_squared_eq
tff(fact_3524_cos__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cos(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sin(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cos_squared_eq
tff(fact_3525_cos__double__less__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),Xa))),one_one(real)) ) ) ).

% cos_double_less_one
tff(fact_3526_cos__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,cos(real),Xa)) ) ) ).

% cos_gt_zero
tff(fact_3527_cos__60,axiom,
    aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,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_3528_cos__one__2pi__int,axiom,
    ! [Xa: real] :
      ( ( aa(real,real,cos(real),Xa) = one_one(real) )
    <=> ? [X4: int] : Xa = 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),X4)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi) ) ).

% cos_one_2pi_int
tff(fact_3529_cos__double__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A] : aa(A,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,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cos(A),W)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),one_one(A)) ) ).

% cos_double_cos
tff(fact_3530_cos__treble__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))),Xa)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cos(A),Xa)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))),aa(A,A,cos(A),Xa))) ) ).

% cos_treble_cos
tff(fact_3531_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),aa(A,A,sin(A),W)),aa(A,A,sin(A),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z))),aa(A,A,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_3532_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),aa(A,A,sin(A),W)),aa(A,A,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),aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sin_times_cos
tff(fact_3533_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),aa(A,A,cos(A),W)),aa(A,A,sin(A),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cos_times_sin
tff(fact_3534_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),aa(A,A,sin(A),W)),aa(A,A,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))),aa(A,A,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)))))),aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_plus_sin
tff(fact_3535_sin__diff__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,sin(A),W)),aa(A,A,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))),aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),aa(A,A,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_3536_cos__diff__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,cos(A),W)),aa(A,A,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))),aa(A,A,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)))))),aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),W)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_diff_cos
tff(fact_3537_cos__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cos(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sin(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cos_double
tff(fact_3538_sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,sin(A),Xa) = aa(A,A,cos(A),aa(A,A,aa(A,fun(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)))),Xa)) ) ).

% sin_cos_eq
tff(fact_3539_cos__sin__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,cos(A),Xa) = aa(A,A,sin(A),aa(A,A,aa(A,fun(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)))),Xa)) ) ).

% cos_sin_eq
tff(fact_3540_cos__gt__zero__pi,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,cos(real),Xa)) ) ) ).

% cos_gt_zero_pi
tff(fact_3541_cos__ge__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,cos(real),Xa)) ) ) ).

% cos_ge_zero
tff(fact_3542_cos__one__2pi,axiom,
    ! [Xa: real] :
      ( ( aa(real,real,cos(real),Xa) = one_one(real) )
    <=> ( ? [X4: nat] : Xa = 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),X4)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)
        | ? [X4: nat] : Xa = 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),X4)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_3543_cos__double__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A] : aa(A,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,aa(A,fun(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,aa(A,fun(nat,A),power_power(A),aa(A,A,sin(A),W)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% cos_double_sin
tff(fact_3544_minus__sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,uminus_uminus(A),aa(A,A,sin(A),Xa)) = aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),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_3545_sincos__total__pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),pi)
            & ( Xa = aa(real,real,cos(real),T4) )
            & ( Y = aa(real,real,sin(real),T4) ) ) ) ) ).

% sincos_total_pi
tff(fact_3546_sin__expansion__lemma,axiom,
    ! [Xa: real,M: nat] : aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M))),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) = aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% sin_expansion_lemma
tff(fact_3547_cos__zero__iff__int,axiom,
    ! [Xa: real] :
      ( ( aa(real,real,cos(real),Xa) = zero_zero(real) )
    <=> ? [I2: int] :
          ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),I2)
          & ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I2)),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_3548_cos__zero__lemma,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( ( aa(real,real,cos(real),Xa) = zero_zero(real) )
       => ? [N: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
            & ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_3549_cos__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa(real,real,cos(real),Xa) = zero_zero(real) )
    <=> ( ? [N2: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)
            & ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N2: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)
            & ( Xa = 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_3550_cos__expansion__lemma,axiom,
    ! [Xa: real,M: nat] : aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M))),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) = aa(real,real,uminus_uminus(real),aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% cos_expansion_lemma
tff(fact_3551_sincos__total__pi__half,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
         => ? [T4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
              & ( Xa = aa(real,real,cos(real),T4) )
              & ( Y = aa(real,real,sin(real),T4) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_3552_sincos__total__2pi__le,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
     => ? [T4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
          & ( Xa = aa(real,real,cos(real),T4) )
          & ( Y = aa(real,real,sin(real),T4) ) ) ) ).

% sincos_total_2pi_le
tff(fact_3553_return__sp__rule,axiom,
    ! [A: $tType,P: assn,Xa: A] : hoare_hoare_triple(A,P,heap_Time_return(A,Xa),aa(A,fun(A,assn),aTP_Lamp_co(assn,fun(A,fun(A,assn)),P),Xa)) ).

% return_sp_rule
tff(fact_3554_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( aa(A,A,cos(A),Xa) != zero_zero(A) )
         => ( ( aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)) != zero_zero(A) )
           => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,tan(A),Xa))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% tan_double
tff(fact_3555_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_3556_obtain__set__succ,axiom,
    ! [Xa: nat,Z: nat,A3: set(nat),B3: set(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Z)
     => ( vEBT_VEBT_max_in_set(A3,Z)
       => ( finite_finite2(nat,B3)
         => ( ( A3 = B3 )
           => ? [X_1: nat] : vEBT_is_succ_in_set(A3,Xa,X_1) ) ) ) ) ).

% obtain_set_succ
tff(fact_3557_obtain__set__pred,axiom,
    ! [Z: nat,Xa: nat,A3: set(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Z),Xa)
     => ( vEBT_VEBT_min_in_set(A3,Z)
       => ( finite_finite2(nat,A3)
         => ? [X_1: nat] : vEBT_is_pred_in_set(A3,Xa,X_1) ) ) ) ).

% obtain_set_pred
tff(fact_3558_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T,Na)
     => finite_finite2(nat,vEBT_VEBT_set_vebt(T)) ) ).

% set_vebt_finite
tff(fact_3559_pred__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_1: nat] : vEBT_is_pred_in_set(Xs,A2,X_1)
     => ( finite_finite2(nat,Xs)
       => ~ ? [X5: nat] :
              ( member(nat,X5,Xs)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X5),A2) ) ) ) ).

% pred_none_empty
tff(fact_3560_succ__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_1: nat] : vEBT_is_succ_in_set(Xs,A2,X_1)
     => ( finite_finite2(nat,Xs)
       => ~ ? [X5: nat] :
              ( member(nat,X5,Xs)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),X5) ) ) ) ).

% succ_none_empty
tff(fact_3561_false__rule,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),Q: fun(A,assn)] : hoare_hoare_triple(A,bot_bot(assn),C2,Q) ).

% false_rule
tff(fact_3562_tan__pi,axiom,
    aa(real,real,tan(real),pi) = zero_zero(real) ).

% tan_pi
tff(fact_3563_tan__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,tan(A),zero_zero(A)) = zero_zero(A) ) ) ).

% tan_zero
tff(fact_3564_tan__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,tan(A),aa(A,A,uminus_uminus(A),Xa)) = aa(A,A,uminus_uminus(A),aa(A,A,tan(A),Xa)) ) ).

% tan_minus
tff(fact_3565_tan__periodic__pi,axiom,
    ! [Xa: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),pi)) = aa(real,real,tan(real),Xa) ).

% tan_periodic_pi
tff(fact_3566_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ finite_finite2(A,set_or1337092689740270186AtMost(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Icc_iff
tff(fact_3567_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_3568_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( finite_finite2(nat,collect(nat,P))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cp(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F2)) ) ) ).

% summable_If_finite
tff(fact_3569_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F2: fun(nat,A)] :
          ( finite_finite2(nat,A3)
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cq(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F2)) ) ) ).

% summable_If_finite_set
tff(fact_3570_Min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),lattic643756798350308766er_Min(A,A3))
            <=> ! [X4: A] :
                  ( member(A,X4,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X4) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_3571_Min__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),lattic643756798350308766er_Min(A,A3))
            <=> ! [X4: A] :
                  ( member(A,X4,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X4) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_3572_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_3573_tan__periodic__n,axiom,
    ! [Xa: real,Na: num] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),Na)),pi))) = aa(real,real,tan(real),Xa) ).

% tan_periodic_n
tff(fact_3574_tan__periodic__nat,axiom,
    ! [Xa: real,Na: nat] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),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),Xa) ).

% tan_periodic_nat
tff(fact_3575_tan__periodic__int,axiom,
    ! [Xa: real,I: int] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),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),Xa) ).

% tan_periodic_int
tff(fact_3576_tan__periodic,axiom,
    ! [Xa: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),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),Xa) ).

% tan_periodic
tff(fact_3577_tan__arctan,axiom,
    ! [Y: real] : aa(real,real,tan(real),aa(real,real,arctan,Y)) = Y ).

% tan_arctan
tff(fact_3578_bounded__nat__set__is__finite,axiom,
    ! [N3: set(nat),Na: nat] :
      ( ! [X: nat] :
          ( member(nat,X,N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Na) )
     => finite_finite2(nat,N3) ) ).

% bounded_nat_set_is_finite
tff(fact_3579_finite__nat__set__iff__bounded,axiom,
    ! [N3: set(nat)] :
      ( finite_finite2(nat,N3)
    <=> ? [M3: nat] :
        ! [X4: nat] :
          ( member(nat,X4,N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),M3) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_3580_finite__if__eq__beyond__finite,axiom,
    ! [A: $tType,S: set(A),S3: set(A)] :
      ( finite_finite2(A,S)
     => finite_finite2(set(A),collect(set(A),aa(set(A),fun(set(A),$o),aTP_Lamp_cr(set(A),fun(set(A),fun(set(A),$o)),S),S3))) ) ).

% finite_if_eq_beyond_finite
tff(fact_3581_tan__of__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: real] : aa(real,A,real_Vector_of_real(A),aa(real,real,tan(real),Xa)) = aa(A,A,tan(A),aa(real,A,real_Vector_of_real(A),Xa)) ) ).

% tan_of_real
tff(fact_3582_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,$o),I: nat] : finite_finite2(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_cs(fun(nat,$o),fun(nat,fun(nat,$o)),P),I))) ).

% finite_M_bounded_by_nat
tff(fact_3583_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))
     => finite_finite2(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ct(fun(nat,nat),fun(nat,fun(nat,$o)),F2),U))) ) ).

% finite_less_ub
tff(fact_3584_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X7: set(A)] :
          ( ( X7 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,X7)
               => ? [Xa2: A] :
                    ( member(A,Xa2,X7)
                    & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Xa2) ) )
           => ~ finite_finite2(A,X7) ) ) ) ).

% infinite_growing
tff(fact_3585_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S: set(A)] :
          ( finite_finite2(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ? [X: A] :
                ( member(A,X,S)
                & ~ ? [Xa2: A] :
                      ( member(A,Xa2,S)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa2),X) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_3586_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ finite_finite2(A,set_or1337092689740270186AtMost(A,A2,B2)) ) ) ).

% infinite_Icc
tff(fact_3587_Min__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => member(A,lattic643756798350308766er_Min(A,A3),A3) ) ) ) ).

% Min_in
tff(fact_3588_finite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),P: fun(set(A),$o),F2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [X: A,S4: set(A)] :
                  ( finite_finite2(A,S4)
                 => ( ! [Y2: A] :
                        ( member(A,Y2,S4)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y2)),aa(A,B,F2,X)) )
                   => ( aa(set(A),$o,P,S4)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S4)) ) ) )
             => aa(set(A),$o,P,S) ) ) ) ) ).

% finite_ranking_induct
tff(fact_3589_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),P: fun(set(A),$o)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B6: A,A7: set(A)] :
                  ( finite_finite2(A,A7)
                 => ( ! [X5: A] :
                        ( member(A,X5,A7)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),B6) )
                   => ( aa(set(A),$o,P,A7)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B6),A7)) ) ) )
             => aa(set(A),$o,P,A3) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_3590_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),P: fun(set(A),$o)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B6: A,A7: set(A)] :
                  ( finite_finite2(A,A7)
                 => ( ! [X5: A] :
                        ( member(A,X5,A7)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B6),X5) )
                   => ( aa(set(A),$o,P,A7)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B6),A7)) ) ) )
             => aa(set(A),$o,P,A3) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_3591_Min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( member(A,A5,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A5) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),lattic643756798350308766er_Min(A,A3)) ) ) ) ) ).

% Min.boundedI
tff(fact_3592_Min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),lattic643756798350308766er_Min(A,A3))
             => ! [A8: A] :
                  ( member(A,A8,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A8) ) ) ) ) ) ).

% Min.boundedE
tff(fact_3593_eq__Min__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( M = lattic643756798350308766er_Min(A,A3) )
            <=> ( member(A,M,A3)
                & ! [X4: A] :
                    ( member(A,X4,A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),X4) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_3594_Min__le__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798350308766er_Min(A,A3)),Xa)
            <=> ? [X4: A] :
                  ( member(A,X4,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa) ) ) ) ) ) ).

% Min_le_iff
tff(fact_3595_Min__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( lattic643756798350308766er_Min(A,A3) = M )
            <=> ( member(A,M,A3)
                & ! [X4: A] :
                    ( member(A,X4,A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),X4) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_3596_Min__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),lattic643756798350308766er_Min(A,A3)),Xa)
            <=> ? [X4: A] :
                  ( member(A,X4,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Xa) ) ) ) ) ) ).

% Min_less_iff
tff(fact_3597_Min__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( finite_finite2(A,A3)
         => ( ! [B6: A] :
                ( member(A,B6,A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B6) )
           => ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)) = A2 ) ) ) ) ).

% Min_insert2
tff(fact_3598_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => finite_finite2(nat,collect(nat,aTP_Lamp_cu(nat,fun(nat,$o),M))) ) ).

% finite_divisors_nat
tff(fact_3599_tan__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X5: A] : aa(A,A,tan(A),X5) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,sin(A),X5)),aa(A,A,cos(A),X5)) ) ).

% tan_def
tff(fact_3600_Min__antimono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M7: set(A),N3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M7),N3)
         => ( ( M7 != bot_bot(set(A)) )
           => ( finite_finite2(A,N3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798350308766er_Min(A,N3)),lattic643756798350308766er_Min(A,M7)) ) ) ) ) ).

% Min_antimono
tff(fact_3601_Min_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( finite_finite2(A,B3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798350308766er_Min(A,B3)),lattic643756798350308766er_Min(A,A3)) ) ) ) ) ).

% Min.subset_imp
tff(fact_3602_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => finite_finite2(A,collect(A,aTP_Lamp_cv(nat,fun(A,$o),Na))) ) ) ).

% finite_roots_unity
tff(fact_3603_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_3604_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ? [X: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,tan(real),X)) ) ) ).

% lemma_tan_total
tff(fact_3605_tan__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,tan(real),Xa)) ) ) ).

% tan_gt_zero
tff(fact_3606_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),X) = Y ) ) ).

% lemma_tan_total1
tff(fact_3607_tan__mono__lt__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Xa)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_3608_tan__monotone_H,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),Xa)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_3609_tan__monotone,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),Xa)) ) ) ) ).

% tan_monotone
tff(fact_3610_tan__total,axiom,
    ! [Y: real] :
    ? [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),X) = Y )
      & ! [Y2: 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))))),Y2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),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),Y2) = Y ) )
         => ( Y2 = X ) ) ) ).

% tan_total
tff(fact_3611_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_3612_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,aa(real,fun(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_3613_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,cos(A),Xa) != zero_zero(A) )
         => ( ( aa(A,A,cos(A),Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),Xa)),aa(A,A,cos(A),Y))) ) ) ) ) ).

% add_tan_eq
tff(fact_3614_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ? [X: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
          & ( aa(real,real,tan(real),X) = Y ) ) ) ).

% tan_total_pos
tff(fact_3615_tan__pos__pi2__le,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,tan(real),Xa)) ) ) ).

% tan_pos_pi2_le
tff(fact_3616_tan__less__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Xa)),zero_zero(real)) ) ) ).

% tan_less_zero
tff(fact_3617_tan__mono__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tan(real),Xa)),aa(real,real,tan(real),Y)) ) ) ) ).

% tan_mono_le
tff(fact_3618_tan__mono__le__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tan(real),Xa)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_3619_tan__bound__pi2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,tan(real),Xa))),one_one(real)) ) ).

% tan_bound_pi2
tff(fact_3620_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_3621_arctan__tan,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,real,arctan,aa(real,real,tan(real),Xa)) = Xa ) ) ) ).

% arctan_tan
tff(fact_3622_arctan__unique,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( ( aa(real,real,tan(real),Xa) = Y )
         => ( aa(real,real,arctan,Y) = Xa ) ) ) ) ).

% arctan_unique
tff(fact_3623_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,cos(A),Xa) != zero_zero(A) )
         => ( ( aa(A,A,cos(A),Y) != zero_zero(A) )
           => ( ( aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Y))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_add
tff(fact_3624_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,cos(A),Xa) != zero_zero(A) )
         => ( ( aa(A,A,cos(A),Y) != zero_zero(A) )
           => ( ( aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Y))),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3625_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,cos(A),Xa) != zero_zero(A) )
         => ( ( aa(A,A,cos(A),Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Y))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),Xa)),aa(A,A,cos(A),Y))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_3626_tan__total__pi4,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ? [Z3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))),Z3)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
          & ( aa(real,real,tan(real),Z3) = Xa ) ) ) ).

% tan_total_pi4
tff(fact_3627_frame__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),R2: assn] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),R2),C2,aa(assn,fun(A,assn),aTP_Lamp_cw(fun(A,assn),fun(assn,fun(A,assn)),Q),R2)) ) ).

% frame_rule
tff(fact_3628_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,tan(A),Xa) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa))),one_one(A))) ) ).

% tan_half
tff(fact_3629_finite__Diff__insert,axiom,
    ! [A: $tType,A3: set(A),A2: A,B3: set(A)] :
      ( finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3)))
    <=> finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) ) ).

% finite_Diff_insert
tff(fact_3630_finite__Collect__le__nat,axiom,
    ! [K: nat] : finite_finite2(nat,collect(nat,aTP_Lamp_cx(nat,fun(nat,$o),K))) ).

% finite_Collect_le_nat
tff(fact_3631_finite__Collect__less__nat,axiom,
    ! [K: nat] : finite_finite2(nat,collect(nat,aTP_Lamp_cy(nat,fun(nat,$o),K))) ).

% finite_Collect_less_nat
tff(fact_3632_finite__Collect__subsets,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => finite_finite2(set(A),collect(set(A),aTP_Lamp_cz(set(A),fun(set(A),$o),A3))) ) ).

% finite_Collect_subsets
tff(fact_3633_finite__induct__select,axiom,
    ! [A: $tType,S: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,S)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [T5: set(A)] :
              ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),T5),S)
             => ( aa(set(A),$o,P,T5)
               => ? [X5: A] :
                    ( member(A,X5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T5))
                    & aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),T5)) ) ) )
         => aa(set(A),$o,P,S) ) ) ) ).

% finite_induct_select
tff(fact_3634_finite__Collect__conjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ( finite_finite2(A,collect(A,P))
        | finite_finite2(A,collect(A,Q)) )
     => finite_finite2(A,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ).

% finite_Collect_conjI
tff(fact_3635_finite__Collect__disjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( finite_finite2(A,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ae(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( finite_finite2(A,collect(A,P))
        & finite_finite2(A,collect(A,Q)) ) ) ).

% finite_Collect_disjI
tff(fact_3636_finite__interval__int1,axiom,
    ! [A2: int,B2: int] : finite_finite2(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_da(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int1
tff(fact_3637_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : finite_finite2(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_db(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int4
tff(fact_3638_finite__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( finite_finite2(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3))
    <=> finite_finite2(A,A3) ) ).

% finite_insert
tff(fact_3639_finite__Un,axiom,
    ! [A: $tType,F3: set(A),G2: set(A)] :
      ( finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F3),G2))
    <=> ( finite_finite2(A,F3)
        & finite_finite2(A,G2) ) ) ).

% finite_Un
tff(fact_3640_finite__nth__roots,axiom,
    ! [Na: nat,C2: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => finite_finite2(complex,collect(complex,aa(complex,fun(complex,$o),aTP_Lamp_dc(nat,fun(complex,fun(complex,$o)),Na),C2))) ) ).

% finite_nth_roots
tff(fact_3641_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : finite_finite2(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_dd(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int2
tff(fact_3642_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : finite_finite2(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_de(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int3
tff(fact_3643_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero(int) )
     => finite_finite2(int,collect(int,aTP_Lamp_df(int,fun(int,$o),I))) ) ).

% finite_divisors_int
tff(fact_3644_not__finite__existsD,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ finite_finite2(A,collect(A,P))
     => ? [X_1: A] : aa(A,$o,P,X_1) ) ).

% not_finite_existsD
tff(fact_3645_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B),R2: fun(A,fun(B,$o))] :
      ( ~ finite_finite2(A,A3)
     => ( finite_finite2(B,B3)
       => ( ! [X: A] :
              ( member(A,X,A3)
             => ? [Xa2: B] :
                  ( member(B,Xa2,B3)
                  & aa(B,$o,aa(A,fun(B,$o),R2,X),Xa2) ) )
         => ? [X: B] :
              ( member(B,X,B3)
              & ~ finite_finite2(A,collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_dg(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),A3),R2),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3646_finite_OemptyI,axiom,
    ! [A: $tType] : finite_finite2(A,bot_bot(set(A))) ).

% finite.emptyI
tff(fact_3647_infinite__imp__nonempty,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ finite_finite2(A,S)
     => ( S != bot_bot(set(A)) ) ) ).

% infinite_imp_nonempty
tff(fact_3648_finite_OinsertI,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( finite_finite2(A,A3)
     => finite_finite2(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)) ) ).

% finite.insertI
tff(fact_3649_finite__UnI,axiom,
    ! [A: $tType,F3: set(A),G2: set(A)] :
      ( finite_finite2(A,F3)
     => ( finite_finite2(A,G2)
       => finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F3),G2)) ) ) ).

% finite_UnI
tff(fact_3650_Un__infinite,axiom,
    ! [A: $tType,S: set(A),T3: set(A)] :
      ( ~ finite_finite2(A,S)
     => ~ finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)) ) ).

% Un_infinite
tff(fact_3651_infinite__Un,axiom,
    ! [A: $tType,S: set(A),T3: set(A)] :
      ( ~ finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3))
    <=> ( ~ finite_finite2(A,S)
        | ~ finite_finite2(A,T3) ) ) ).

% infinite_Un
tff(fact_3652_finite__psubset__induct,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,A3)
     => ( ! [A7: set(A)] :
            ( finite_finite2(A,A7)
           => ( ! [B9: set(A)] :
                  ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B9),A7)
                 => aa(set(A),$o,P,B9) )
             => aa(set(A),$o,P,A7) ) )
       => aa(set(A),$o,P,A3) ) ) ).

% finite_psubset_induct
tff(fact_3653_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X: A] :
                ( member(A,X,A3)
                & ! [Xa2: A] :
                    ( member(A,Xa2,A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa2),X)
                     => ( X = Xa2 ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_3654_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X: A] :
                ( member(A,X,A3)
                & ! [Xa2: A] :
                    ( member(A,Xa2,A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa2)
                     => ( X = Xa2 ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_3655_infinite__finite__induct,axiom,
    ! [A: $tType,P: fun(set(A),$o),A3: set(A)] :
      ( ! [A7: set(A)] :
          ( ~ finite_finite2(A,A7)
         => aa(set(A),$o,P,A7) )
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [X: A,F5: set(A)] :
              ( finite_finite2(A,F5)
             => ( ~ member(A,X,F5)
               => ( aa(set(A),$o,P,F5)
                 => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),F5)) ) ) )
         => aa(set(A),$o,P,A3) ) ) ) ).

% infinite_finite_induct
tff(fact_3656_finite__ne__induct,axiom,
    ! [A: $tType,F3: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,F3)
     => ( ( F3 != bot_bot(set(A)) )
       => ( ! [X: A] : aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))
         => ( ! [X: A,F5: set(A)] :
                ( finite_finite2(A,F5)
               => ( ( F5 != bot_bot(set(A)) )
                 => ( ~ member(A,X,F5)
                   => ( aa(set(A),$o,P,F5)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),F5)) ) ) ) )
           => aa(set(A),$o,P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_3657_finite__induct,axiom,
    ! [A: $tType,F3: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,F3)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [X: A,F5: set(A)] :
              ( finite_finite2(A,F5)
             => ( ~ member(A,X,F5)
               => ( aa(set(A),$o,P,F5)
                 => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),F5)) ) ) )
         => aa(set(A),$o,P,F3) ) ) ) ).

% finite_induct
tff(fact_3658_finite_Osimps,axiom,
    ! [A: $tType,A2: set(A)] :
      ( finite_finite2(A,A2)
    <=> ( ( A2 = bot_bot(set(A)) )
        | ? [A9: set(A),A10: A] :
            ( ( A2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A10),A9) )
            & finite_finite2(A,A9) ) ) ) ).

% finite.simps
tff(fact_3659_finite_Ocases,axiom,
    ! [A: $tType,A2: set(A)] :
      ( finite_finite2(A,A2)
     => ( ( A2 != bot_bot(set(A)) )
       => ~ ! [A7: set(A)] :
              ( ? [A5: A] : A2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A5),A7)
             => ~ finite_finite2(A,A7) ) ) ) ).

% finite.cases
tff(fact_3660_finite__subset__induct_H,axiom,
    ! [A: $tType,F3: set(A),A3: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F3),A3)
       => ( aa(set(A),$o,P,bot_bot(set(A)))
         => ( ! [A5: A,F5: set(A)] :
                ( finite_finite2(A,F5)
               => ( member(A,A5,A3)
                 => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F5),A3)
                   => ( ~ member(A,A5,F5)
                     => ( aa(set(A),$o,P,F5)
                       => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A5),F5)) ) ) ) ) )
           => aa(set(A),$o,P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_3661_finite__subset__induct,axiom,
    ! [A: $tType,F3: set(A),A3: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F3),A3)
       => ( aa(set(A),$o,P,bot_bot(set(A)))
         => ( ! [A5: A,F5: set(A)] :
                ( finite_finite2(A,F5)
               => ( member(A,A5,A3)
                 => ( ~ member(A,A5,F5)
                   => ( aa(set(A),$o,P,F5)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A5),F5)) ) ) ) )
           => aa(set(A),$o,P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_3662_infinite__remove,axiom,
    ! [A: $tType,S: set(A),A2: A] :
      ( ~ finite_finite2(A,S)
     => ~ finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) ) ).

% infinite_remove
tff(fact_3663_infinite__coinduct,axiom,
    ! [A: $tType,X7: fun(set(A),$o),A3: set(A)] :
      ( aa(set(A),$o,X7,A3)
     => ( ! [A7: set(A)] :
            ( aa(set(A),$o,X7,A7)
           => ? [X5: A] :
                ( member(A,X5,A7)
                & ( aa(set(A),$o,X7,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),bot_bot(set(A)))))
                  | ~ finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),bot_bot(set(A))))) ) ) )
       => ~ finite_finite2(A,A3) ) ) ).

% infinite_coinduct
tff(fact_3664_finite__empty__induct,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,A3)
     => ( aa(set(A),$o,P,A3)
       => ( ! [A5: A,A7: set(A)] :
              ( finite_finite2(A,A7)
             => ( member(A,A5,A7)
               => ( aa(set(A),$o,P,A7)
                 => aa(set(A),$o,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A5),bot_bot(set(A))))) ) ) )
         => aa(set(A),$o,P,bot_bot(set(A))) ) ) ) ).

% finite_empty_induct
tff(fact_3665_finite__remove__induct,axiom,
    ! [A: $tType,B3: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,B3)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [A7: set(A)] :
              ( finite_finite2(A,A7)
             => ( ( A7 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A7),B3)
                 => ( ! [X5: A] :
                        ( member(A,X5,A7)
                       => aa(set(A),$o,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A7) ) ) ) )
         => aa(set(A),$o,P,B3) ) ) ) ).

% finite_remove_induct
tff(fact_3666_remove__induct,axiom,
    ! [A: $tType,P: fun(set(A),$o),B3: set(A)] :
      ( aa(set(A),$o,P,bot_bot(set(A)))
     => ( ( ~ finite_finite2(A,B3)
         => aa(set(A),$o,P,B3) )
       => ( ! [A7: set(A)] :
              ( finite_finite2(A,A7)
             => ( ( A7 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A7),B3)
                 => ( ! [X5: A] :
                        ( member(A,X5,A7)
                       => aa(set(A),$o,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A7) ) ) ) )
         => aa(set(A),$o,P,B3) ) ) ) ).

% remove_induct
tff(fact_3667_assn__basic__inequalities_I3_J,axiom,
    bot_bot(assn) != one_one(assn) ).

% assn_basic_inequalities(3)
tff(fact_3668_pure__false,axiom,
    pure_assn($false) = bot_bot(assn) ).

% pure_false
tff(fact_3669_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( pure_assn((P)) = bot_bot(assn) )
    <=> ~ (P) ) ).

% pure_assn_eq_false_iff
tff(fact_3670_star__false__left,axiom,
    ! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),bot_bot(assn)),P) = bot_bot(assn) ).

% star_false_left
tff(fact_3671_star__false__right,axiom,
    ! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),bot_bot(assn)) = bot_bot(assn) ).

% star_false_right
tff(fact_3672_set__encode__insert,axiom,
    ! [A3: set(nat),Na: nat] :
      ( finite_finite2(nat,A3)
     => ( ~ member(nat,Na,A3)
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Na),A3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(set(nat),nat,nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_3673_diff__preserves__multiset,axiom,
    ! [A: $tType,M7: fun(A,nat),N3: fun(A,nat)] :
      ( finite_finite2(A,collect(A,aTP_Lamp_dh(fun(A,nat),fun(A,$o),M7)))
     => finite_finite2(A,collect(A,aa(fun(A,nat),fun(A,$o),aTP_Lamp_di(fun(A,nat),fun(fun(A,nat),fun(A,$o)),M7),N3))) ) ).

% diff_preserves_multiset
tff(fact_3674_add__mset__in__multiset,axiom,
    ! [A: $tType,M7: fun(A,nat),A2: A] :
      ( finite_finite2(A,collect(A,aTP_Lamp_dh(fun(A,nat),fun(A,$o),M7)))
     => finite_finite2(A,collect(A,aa(A,fun(A,$o),aTP_Lamp_dj(fun(A,nat),fun(A,fun(A,$o)),M7),A2))) ) ).

% add_mset_in_multiset
tff(fact_3675_sin__tan,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( aa(real,real,sin(real),Xa) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,tan(real),Xa)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,tan(real),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% sin_tan
tff(fact_3676_real__sqrt__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa(real,real,sqrt,Xa) = aa(real,real,sqrt,Y) )
    <=> ( Xa = Y ) ) ).

% real_sqrt_eq_iff
tff(fact_3677_real__sqrt__zero,axiom,
    aa(real,real,sqrt,zero_zero(real)) = zero_zero(real) ).

% real_sqrt_zero
tff(fact_3678_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [Xa: real] :
      ( ( aa(real,real,sqrt,Xa) = zero_zero(real) )
    <=> ( Xa = zero_zero(real) ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_3679_real__sqrt__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ).

% real_sqrt_less_iff
tff(fact_3680_real__sqrt__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ).

% real_sqrt_le_iff
tff(fact_3681_real__sqrt__eq__1__iff,axiom,
    ! [Xa: real] :
      ( ( aa(real,real,sqrt,Xa) = one_one(real) )
    <=> ( Xa = one_one(real) ) ) ).

% real_sqrt_eq_1_iff
tff(fact_3682_real__sqrt__one,axiom,
    aa(real,real,sqrt,one_one(real)) = one_one(real) ).

% real_sqrt_one
tff(fact_3683_real__sqrt__lt__0__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_3684_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_3685_real__sqrt__le__0__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% real_sqrt_le_0_iff
tff(fact_3686_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_3687_real__sqrt__lt__1__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xa)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_3688_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Y) ) ).

% real_sqrt_gt_1_iff
tff(fact_3689_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_3690_real__sqrt__le__1__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real)) ) ).

% real_sqrt_le_1_iff
tff(fact_3691_real__sqrt__abs2,axiom,
    ! [Xa: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Xa)) = aa(real,real,abs_abs(real),Xa) ).

% real_sqrt_abs2
tff(fact_3692_real__sqrt__mult__self,axiom,
    ! [A2: real] : aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,A2)),aa(real,real,sqrt,A2)) = aa(real,real,abs_abs(real),A2) ).

% real_sqrt_mult_self
tff(fact_3693_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_3694_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_3695_real__sqrt__abs,axiom,
    ! [Xa: real] : aa(real,real,sqrt,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(real,real,abs_abs(real),Xa) ).

% real_sqrt_abs
tff(fact_3696_real__sqrt__pow2__iff,axiom,
    ! [Xa: real] :
      ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xa )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% real_sqrt_pow2_iff
tff(fact_3697_real__sqrt__pow2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xa ) ) ).

% real_sqrt_pow2
tff(fact_3698_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [Xa: real,Y: real,Xaa: real,Ya: real] : aa(nat,real,aa(real,fun(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,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xaa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),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,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xaa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_3699_real__sqrt__power,axiom,
    ! [Xa: real,K: nat] : aa(real,real,sqrt,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),K)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xa)),K) ).

% real_sqrt_power
tff(fact_3700_real__sqrt__minus,axiom,
    ! [Xa: real] : aa(real,real,sqrt,aa(real,real,uminus_uminus(real),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,sqrt,Xa)) ).

% real_sqrt_minus
tff(fact_3701_real__sqrt__mult,axiom,
    ! [Xa: real,Y: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Y)) ).

% real_sqrt_mult
tff(fact_3702_real__sqrt__divide,axiom,
    ! [Xa: real,Y: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Y)) ).

% real_sqrt_divide
tff(fact_3703_real__sqrt__le__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_3704_real__sqrt__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_3705_real__sqrt__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sqrt,Xa)) ) ).

% real_sqrt_gt_zero
tff(fact_3706_real__sqrt__eq__zero__cancel,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( ( aa(real,real,sqrt,Xa) = zero_zero(real) )
       => ( Xa = zero_zero(real) ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_3707_real__sqrt__ge__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,Xa)) ) ).

% real_sqrt_ge_zero
tff(fact_3708_real__sqrt__ge__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,Xa)) ) ).

% real_sqrt_ge_one
tff(fact_3709_real__div__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),aa(real,real,sqrt,Xa)) = aa(real,real,sqrt,Xa) ) ) ).

% real_div_sqrt
tff(fact_3710_sqrt__add__le__add__sqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_3711_le__real__sqrt__sumsq,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Xa)),aa(real,real,aa(real,fun(real,real),times_times(real),Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_3712_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_3713_real__less__rsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_3714_real__le__rsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_3715_sqrt__le__D,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% sqrt_le_D
tff(fact_3716_real__sqrt__unique,axiom,
    ! [Y: real,Xa: real] :
      ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xa )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,real,sqrt,Xa) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_3717_real__le__lsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_3718_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_3719_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) = Y )
     => ( Xa = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_3720_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) = Xa )
     => ( Y = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_3721_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),C2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),B2),D2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),C2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_3722_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_3723_real__sqrt__sum__squares__ge1,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_3724_sqrt__ge__absD,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,sqrt,Y))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y) ) ).

% sqrt_ge_absD
tff(fact_3725_cos__45,axiom,
    aa(real,real,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_3726_sin__45,axiom,
    aa(real,real,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_3727_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),aa(num,num,bit1,one2)))) = aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))) ).

% tan_60
tff(fact_3728_real__less__lsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xa)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_3729_sqrt__sum__squares__le__sum,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_3730_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,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)) = aa(nat,real,aa(real,fun(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_3731_real__sqrt__ge__abs1,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_ge_abs1
tff(fact_3732_real__sqrt__ge__abs2,axiom,
    ! [Y: real,Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_ge_abs2
tff(fact_3733_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,abs_abs(real),Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_3734_ln__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,ln_ln(real),aa(real,real,sqrt,Xa)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Xa)),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% ln_sqrt
tff(fact_3735_cos__30,axiom,
    aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% cos_30
tff(fact_3736_sin__60,axiom,
    aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sin_60
tff(fact_3737_real__sqrt__power__even,axiom,
    ! [Na: nat,Xa: 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)),Xa)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xa)),Na) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_3738_arsinh__real__aux,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ).

% arsinh_real_aux
tff(fact_3739_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xa: real,Y: real,Xaa: 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,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xaa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_3740_arith__geo__mean__sqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_3741_powr__half__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(real,real,sqrt,Xa) ) ) ).

% powr_half_sqrt
tff(fact_3742_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(aa(num,num,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),aa(num,num,bit1,one2)))) ).

% tan_30
tff(fact_3743_arsinh__real__def,axiom,
    ! [Xa: real] : aa(real,real,arsinh(real),Xa) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ).

% arsinh_real_def
tff(fact_3744_cos__x__y__le__one,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),one_one(real)) ).

% cos_x_y_le_one
tff(fact_3745_real__sqrt__sum__squares__less,axiom,
    ! [Xa: real,U: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_3746_arcosh__real__def,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => ( aa(real,real,arcosh(real),Xa) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ) ) ).

% arcosh_real_def
tff(fact_3747_cos__arctan,axiom,
    ! [Xa: real] : aa(real,real,cos(real),aa(real,real,arctan,Xa)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% cos_arctan
tff(fact_3748_sin__arctan,axiom,
    ! [Xa: real] : aa(real,real,sin(real),aa(real,real,arctan,Xa)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% sin_arctan
tff(fact_3749_sqrt__sum__squares__half__less,axiom,
    ! [Xa: real,U: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_3750_sin__cos__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sin(real),Xa))
     => ( aa(real,real,sin(real),Xa) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,cos(real),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_cos_sqrt
tff(fact_3751_arctan__half,axiom,
    ! [Xa: real] : aa(real,real,arctan,Xa) = 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),Xa),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,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% arctan_half
tff(fact_3752_even__set__encode__iff,axiom,
    ! [A3: set(nat)] :
      ( finite_finite2(nat,A3)
     => ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(nat),nat,nat_set_encode,A3))
      <=> ~ member(nat,zero_zero(nat),A3) ) ) ).

% even_set_encode_iff
tff(fact_3753_filter__preserves__multiset,axiom,
    ! [A: $tType,M7: fun(A,nat),P: fun(A,$o)] :
      ( finite_finite2(A,collect(A,aTP_Lamp_dh(fun(A,nat),fun(A,$o),M7)))
     => finite_finite2(A,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dk(fun(A,nat),fun(fun(A,$o),fun(A,$o)),M7),P))) ) ).

% filter_preserves_multiset
tff(fact_3754_cos__tan,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( aa(real,real,cos(real),Xa) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,tan(real),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% cos_tan
tff(fact_3755_infinite__int__iff__unbounded,axiom,
    ! [S: set(int)] :
      ( ~ finite_finite2(int,S)
    <=> ! [M3: int] :
        ? [N2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),M3),aa(int,int,abs_abs(int),N2))
          & member(int,N2,S) ) ) ).

% infinite_int_iff_unbounded
tff(fact_3756_cos__arcsin,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( aa(real,real,cos(real),aa(real,real,arcsin,Xa)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% cos_arcsin
tff(fact_3757_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),Xa: fun(A,B),Y: fun(A,B)] :
          ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_dl(set(A),fun(fun(A,B),fun(A,$o)),I5),Xa)))
         => ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_dl(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_dm(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xa),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_3758_sum_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),Xa: fun(A,B),Y: fun(A,B)] :
          ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_dn(set(A),fun(fun(A,B),fun(A,$o)),I5),Xa)))
         => ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_dn(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_do(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xa),Y))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_3759_arcsin__0,axiom,
    aa(real,real,arcsin,zero_zero(real)) = zero_zero(real) ).

% arcsin_0
tff(fact_3760_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))
       => ( aa(real,real,sin(real),aa(real,real,arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_3761_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_3762_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_3763_arcsin__le__arcsin,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Xa)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_3764_arcsin__minus,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( aa(real,real,arcsin,aa(real,real,uminus_uminus(real),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,arcsin,Xa)) ) ) ) ).

% arcsin_minus
tff(fact_3765_arcsin__le__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Xa)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_3766_arcsin__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( ( aa(real,real,arcsin,Xa) = aa(real,real,arcsin,Y) )
        <=> ( Xa = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_3767_arcsin__less__arcsin,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Xa)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_3768_arcsin__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Xa)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_3769_cos__arcsin__nonzero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => ( aa(real,real,cos(real),aa(real,real,arcsin,Xa)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_3770_finite__transitivity__chain,axiom,
    ! [A: $tType,A3: set(A),R2: fun(A,fun(A,$o))] :
      ( finite_finite2(A,A3)
     => ( ! [X: A] : ~ aa(A,$o,aa(A,fun(A,$o),R2,X),X)
       => ( ! [X: A,Y3: A,Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),R2,X),Y3)
             => ( aa(A,$o,aa(A,fun(A,$o),R2,Y3),Z3)
               => aa(A,$o,aa(A,fun(A,$o),R2,X),Z3) ) )
         => ( ! [X: A] :
                ( member(A,X,A3)
               => ? [Y2: A] :
                    ( member(A,Y2,A3)
                    & aa(A,$o,aa(A,fun(A,$o),R2,X),Y2) ) )
           => ( A3 = bot_bot(set(A)) ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_3771_unbounded__k__infinite,axiom,
    ! [K: nat,S: set(nat)] :
      ( ! [M4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),M4)
         => ? [N9: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N9)
              & member(nat,N9,S) ) )
     => ~ finite_finite2(nat,S) ) ).

% unbounded_k_infinite
tff(fact_3772_infinite__nat__iff__unbounded,axiom,
    ! [S: set(nat)] :
      ( ~ finite_finite2(nat,S)
    <=> ! [M3: nat] :
        ? [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N2)
          & member(nat,N2,S) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_3773_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_3774_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_3775_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_3776_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_3777_arcsin__sin,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,real,arcsin,aa(real,real,sin(real),Xa)) = Xa ) ) ) ).

% arcsin_sin
tff(fact_3778_le__arcsin__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,arcsin,Xa))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sin(real),Y)),Xa) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_3779_arcsin__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Xa)),Y)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,sin(real),Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_3780_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)
          & ( aa(real,real,sin(real),aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_3781_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))))
          & ( aa(real,real,sin(real),aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_3782_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xa: A,M: nat,Na: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xa)),set_or1337092689740270186AtMost(nat,M,Na)) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M),
            zero_zero(A),
            $ite(Xa = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))),M)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,Na)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa))) ) ) ).

% sum_gp
tff(fact_3783_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))
     => ( aa(real,real,sin(real),aa(real,real,arccos,Y)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_arccos_abs
tff(fact_3784_sin__arccos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( aa(real,real,sin(real),aa(real,real,arccos,Xa)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% sin_arccos
tff(fact_3785_dbl__inc__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_inc(A,one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2)) ) ) ).

% dbl_inc_simps(3)
tff(fact_3786_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_dp(B,A)),A3) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_3787_of__nat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [F2: fun(B,nat),A3: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,F2),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_dq(fun(B,nat),fun(B,A),F2)),A3) ) ).

% of_nat_sum
tff(fact_3788_of__int__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [F2: fun(B,int),A3: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,groups7311177749621191930dd_sum(B,int,F2),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_dr(fun(B,int),fun(B,A),F2)),A3) ) ).

% of_int_sum
tff(fact_3789_abs__sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(A,A,abs_abs(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_ds(fun(B,A),fun(B,A),F2)),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_ds(fun(B,A),fun(B,A),F2)),A3) ) ).

% abs_sum_abs
tff(fact_3790_of__real__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [F2: fun(B,real),Sa: set(B)] : aa(real,A,real_Vector_of_real(A),aa(set(B),real,groups7311177749621191930dd_sum(B,real,F2),Sa)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_dt(fun(B,real),fun(B,A),F2)),Sa) ) ).

% of_real_sum
tff(fact_3791_sum_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty
tff(fact_3792_arccos__1,axiom,
    aa(real,real,arccos,one_one(real)) = zero_zero(real) ).

% arccos_1
tff(fact_3793_sum_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),A2: A,B2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_du(A,fun(fun(A,B),fun(A,B)),A2),B2)),S) = $ite(member(A,A2,S),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta'
tff(fact_3794_sum_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),A2: A,B2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_dv(A,fun(fun(A,B),fun(A,B)),A2),B2)),S) = $ite(member(A,A2,S),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta
tff(fact_3795_sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(A,$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),A3))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_ds(fun(B,A),fun(B,A),F2)),A3)) ) ).

% sum_abs
tff(fact_3796_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_3797_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_3798_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),aa(num,num,bit1,K)) ) ).

% dbl_inc_simps(5)
tff(fact_3799_sum_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),Xa: A,G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ~ member(A,Xa,A3)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xa)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)) ) ) ) ) ).

% sum.insert
tff(fact_3800_arccos__minus__1,axiom,
    aa(real,real,arccos,aa(real,real,uminus_uminus(real),one_one(real))) = pi ).

% arccos_minus_1
tff(fact_3801_sum__abs__ge__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_ds(fun(B,A),fun(B,A),F2)),A3)) ) ).

% sum_abs_ge_zero
tff(fact_3802_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_3803_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_3804_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))
       => ( aa(real,real,cos(real),aa(real,real,arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_3805_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_3806_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: fun(nat,A),A3: set(nat)] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_dw(fun(nat,A),fun(nat,A),C2)),A3) = $ite(
            ( finite_finite2(nat,A3)
            & member(nat,zero_zero(nat),A3) ),
            aa(nat,A,C2,zero_zero(nat)),
            zero_zero(A) ) ) ).

% sum_zero_power
tff(fact_3807_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_3808_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: fun(nat,A),D2: fun(nat,A),A3: set(nat)] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dx(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),A3) = $ite(
            ( finite_finite2(nat,A3)
            & member(nat,zero_zero(nat),A3) ),
            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_3809_sum_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,fun(C,A)),B3: set(C),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(set(C),fun(B,A),aTP_Lamp_dy(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B3)),A3) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(set(B),fun(C,A),aTP_Lamp_ea(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A3)),B3) ) ).

% sum.swap
tff(fact_3810_sum__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [K6: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,K6)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),aa(A,B,G,I3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),K6)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),K6)) ) ) ).

% sum_mono
tff(fact_3811_norm__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3))),aa(set(B),real,groups7311177749621191930dd_sum(B,real,aTP_Lamp_eb(fun(B,A),fun(B,real),F2)),A3)) ) ).

% norm_sum
tff(fact_3812_sum_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),H: fun(B,A),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ec(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),A3)) ) ).

% sum.distrib
tff(fact_3813_sum__product,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),A3: set(B),G: fun(C,A),B3: set(C)] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)),aa(set(C),A,groups7311177749621191930dd_sum(C,A,G),B3)) = 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_ee(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B3)),A3) ) ).

% sum_product
tff(fact_3814_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),A3: set(B),R: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)),R) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_ef(fun(B,A),fun(A,fun(B,A)),F2),R)),A3) ) ).

% sum_distrib_right
tff(fact_3815_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R: A,F2: fun(B,A),A3: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_eg(A,fun(fun(B,A),fun(B,A)),R),F2)),A3) ) ).

% sum_distrib_left
tff(fact_3816_sum__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),G: fun(B,A),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_eh(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)) ) ).

% sum_subtractf
tff(fact_3817_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),A3: set(B),R: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)),R) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_ei(fun(B,A),fun(A,fun(B,A)),F2),R)),A3) ) ).

% sum_divide_distrib
tff(fact_3818_sum__negf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_ej(fun(B,A),fun(B,A),F2)),A3) = aa(A,A,uminus_uminus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)) ) ).

% sum_negf
tff(fact_3819_sum_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A3: set(A),B3: set(B),G: fun(A,fun(B,C)),R2: fun(A,fun(B,$o))] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(B,B3)
           => ( 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_el(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B3),G),R2)),A3) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_en(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A3),G),R2)),B3) ) ) ) ) ).

% sum.swap_restrict
tff(fact_3820_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),A2: A,A3: 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_eo(fun(B,A),fun(A,fun(B,A)),F2),A2)),A3)),A2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)),A2) ) ).

% mod_sum_eq
tff(fact_3821_summable__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F2: fun(A,fun(nat,B))] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => summable(B,aa(A,fun(nat,B),F2,I3)) )
         => summable(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_eq(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F2)) ) ) ).

% summable_sum
tff(fact_3822_sum_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_al(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_er(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A3) ) ) ) ).

% sum.inter_filter
tff(fact_3823_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_es(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_3824_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_et(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_3825_suminf__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F2: fun(A,fun(nat,B))] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => summable(B,aa(A,fun(nat,B),F2,I3)) )
         => ( suminf(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_eq(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F2)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aTP_Lamp_eu(fun(A,fun(nat,B)),fun(A,B),F2)),I5) ) ) ) ).

% suminf_sum
tff(fact_3826_sum__strict__mono__ex1,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ! [X: A] :
                ( member(A,X,A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X)) )
           => ( ? [X5: A] :
                  ( member(A,X5,A3)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X5)),aa(A,B,G,X5)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_3827_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R2: fun(A,fun(A,$o)),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R2,zero_zero(A)),zero_zero(A))
         => ( ! [X1: A,Y1: A,X2: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R2,X1),X2)
                  & aa(A,$o,aa(A,fun(A,$o),R2,Y1),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y23)) )
           => ( finite_finite2(B,S)
             => ( ! [X: B] :
                    ( member(B,X,S)
                   => aa(A,$o,aa(A,fun(A,$o),R2,aa(B,A,H,X)),aa(B,A,G,X)) )
               => aa(A,$o,aa(A,fun(A,$o),R2,aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),S)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),S)) ) ) ) ) ) ).

% sum.related
tff(fact_3828_sum__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X: A] :
                  ( member(A,X,A3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,G,X)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)) ) ) ) ) ).

% sum_strict_mono
tff(fact_3829_sum_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),Xa: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) = $ite(member(A,Xa,A3),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xa)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3))) ) ) ) ).

% sum.insert_if
tff(fact_3830_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [Sa: set(A),F2: fun(A,B),I: A] :
          ( finite_finite2(A,Sa)
         => ( ! [I3: A] :
                ( member(A,I3,Sa)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I3)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),Sa) = zero_zero(B) )
             => ( member(A,I,Sa)
               => ( aa(A,B,F2,I) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_3831_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [Sa: set(A),F2: fun(A,B),B3: B,I: A] :
          ( finite_finite2(A,Sa)
         => ( ! [I3: A] :
                ( member(A,I3,Sa)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I3)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),Sa) = B3 )
             => ( member(A,I,Sa)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),B3) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_3832_sum_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),B3: set(A)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(set(A),fun(A,B),aTP_Lamp_ev(fun(A,B),fun(set(A),fun(A,B)),G),B3)),A3) ) ) ) ).

% sum.inter_restrict
tff(fact_3833_sum_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),collect(A,aTP_Lamp_ew(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_3834_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,M: nat,I5: set(nat)] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ex(A,fun(nat,fun(nat,A)),Xa),M)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xa)),I5)) ) ).

% sum_power_add
tff(fact_3835_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_ey(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M)),set_or1337092689740270186AtMost(nat,Na,M)) ) ).

% sum.atLeastAtMost_rev
tff(fact_3836_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N3: set(nat),F2: fun(nat,A)] :
          ( 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_3837_arccos__le__arccos,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),aa(real,real,arccos,Xa)) ) ) ) ).

% arccos_le_arccos
tff(fact_3838_arccos__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real)) )
     => ( ( aa(real,real,arccos,Xa) = aa(real,real,arccos,Y) )
      <=> ( Xa = Y ) ) ) ).

% arccos_eq_iff
tff(fact_3839_arccos__le__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Xa)),aa(real,real,arccos,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa) ) ) ) ).

% arccos_le_mono
tff(fact_3840_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),I: A,F2: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,I))
             => ( ! [I3: A] :
                    ( member(A,I3,I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I3)) )
               => aa(B,$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_3841_sum__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( member(A,I3,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,I3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),I5)) ) ) ) ) ).

% sum_pos
tff(fact_3842_sum_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B3: set(A),A3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
         => ( finite_finite2(A,A3)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) = 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),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),B3)) ) ) ) ) ).

% sum.subset_diff
tff(fact_3843_sum_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),B3: set(A)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) = 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),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))) ) ) ) ).

% sum.Int_Diff
tff(fact_3844_sum_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B3)
           => ( 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),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),B3)) ) ) ) ) ).

% sum.union_inter
tff(fact_3845_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_3846_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_3847_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_3848_dbl__inc__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xa: A] : neg_numeral_dbl_inc(A,Xa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Xa)),one_one(A)) ) ).

% dbl_inc_def
tff(fact_3849_sum_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),P: fun(A,$o),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( 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_ez(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H),G)),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,P)))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P))))) ) ) ) ).

% sum.If_cases
tff(fact_3850_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_es(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_3851_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_fa(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,aa(nat,nat,suc,Na))),aa(nat,A,F2,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_3852_sum_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B3)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
                 => ( aa(A,B,G,X) = zero_zero(B) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),B3)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_3853_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),Xa: A,G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( member(A,Xa,A3)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xa)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) ) ) ) ) ).

% sum.remove
tff(fact_3854_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),Xa: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xa)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) ) ) ) ).

% sum.insert_remove
tff(fact_3855_sum__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A3: set(A),B3: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B3)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),B3))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ) ).

% sum_Un
tff(fact_3856_sum__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A3: set(A),F2: fun(A,B),A2: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = $ite(member(A,A2,A3),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(A,B,F2,A2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)) ) ) ) ).

% sum_diff1
tff(fact_3857_sum_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B3)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),B3)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_3858_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_3859_arccos__less__arccos,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Y)),aa(real,real,arccos,Xa)) ) ) ) ).

% arccos_less_arccos
tff(fact_3860_sum_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B3)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ) ).

% sum.union_diff2
tff(fact_3861_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ).

% sum_Un2
tff(fact_3862_arccos__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Xa)),aa(real,real,arccos,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa) ) ) ) ).

% arccos_less_mono
tff(fact_3863_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_3864_arccos__cos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => ( aa(real,real,arccos,aa(real,real,cos(real),Xa)) = Xa ) ) ) ).

% arccos_cos
tff(fact_3865_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_3866_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))
     => ( aa(real,real,cos(real),aa(real,real,arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_3867_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,aa(real,real,cos(real),Theta)) = aa(real,real,abs_abs(real),Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_3868_sum__div__partition,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A3: set(A),F2: fun(A,B),B2: B] :
          ( finite_finite2(A,A3)
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(B,fun(A,B),aTP_Lamp_fb(fun(A,B),fun(B,fun(A,B)),F2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,aa(B,fun(A,$o),aTP_Lamp_fc(fun(A,B),fun(B,fun(A,$o)),F2),B2))))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,aa(B,fun(A,$o),aTP_Lamp_fd(fun(A,B),fun(B,fun(A,$o)),F2),B2))))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_3869_sum_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),A2: A,B2: fun(A,B),C2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_fe(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C2)),S) = $ite(member(A,A2,S),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,B2,A2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_3870_set__encode__def,axiom,
    nat_set_encode = groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% set_encode_def
tff(fact_3871_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B3: set(A),A3: set(A),B2: A,F2: fun(A,B)] :
          ( finite_finite2(A,B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
           => ( member(A,B2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3))
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,B2))
               => ( ! [X: A] :
                      ( member(A,X,B3)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X)) )
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),B3)) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_3872_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: A,A3: set(A),F2: fun(A,B)] :
          ( member(A,I,A3)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I),bot_bot(set(A)))))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X)) )
           => ( finite_finite2(A,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)) ) ) ) ) ).

% member_le_sum
tff(fact_3873_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),Xa: fun(A,B),A2: fun(A,B),B2: B,Delta: B] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,Xa,I3)) )
         => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,Xa),I5) = one_one(B) )
           => ( ! [I3: A] :
                  ( member(A,I3,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A2,I3)),B2))),Delta) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ff(fun(A,B),fun(fun(A,B),fun(A,B)),Xa),A2)),I5)),B2))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_3874_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_3875_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_3876_sin__arccos__nonzero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => ( aa(real,real,sin(real),aa(real,real,arccos,Xa)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_3877_arccos__cos2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Xa)
       => ( aa(real,real,arccos,aa(real,real,cos(real),Xa)) = aa(real,real,uminus_uminus(real),Xa) ) ) ) ).

% arccos_cos2
tff(fact_3878_arccos__minus,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),Xa)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),aa(real,real,arccos,Xa)) ) ) ) ).

% arccos_minus
tff(fact_3879_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_fg(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,aa(A,fun(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_3880_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_fh(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,Na)),aa(nat,A,F2,M)) ) ) ) ).

% sum_telescope''
tff(fact_3881_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),E: real] :
          ( summable(A,F2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
           => ~ ! [N8: nat] :
                  ~ ! [M2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),M2)
                     => ! [N9: nat] : aa(real,$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,M2,N9)))),E) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_3882_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2)))),collect(nat,aTP_Lamp_cy(nat,fun(nat,$o),Na))) ) ).

% mask_eq_sum_exp
tff(fact_3883_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Na: nat,Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xa)),set_or1337092689740270186AtMost(nat,M,Na))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,Na))) ) ) ) ).

% sum_gp_multiplied
tff(fact_3884_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_fi(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.in_pairs
tff(fact_3885_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)
          & ( aa(real,real,cos(real),aa(real,real,arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_3886_arccos__minus__abs,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),Xa)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),aa(real,real,arccos,Xa)) ) ) ).

% arccos_minus_abs
tff(fact_3887_mask__eq__sum__exp__nat,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(nat,nat,suc,zero_zero(nat))) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))),collect(nat,aTP_Lamp_cy(nat,fun(nat,$o),Na))) ).

% mask_eq_sum_exp_nat
tff(fact_3888_gauss__sum__nat,axiom,
    ! [Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fj(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_3889_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,D2: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fk(A,fun(A,fun(nat,A)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),D2))) ) ).

% double_arith_series
tff(fact_3890_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_3891_arith__series__nat,axiom,
    ! [A2: nat,D2: nat,Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fl(nat,fun(nat,fun(nat,nat)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Na)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),D2)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% arith_series_nat
tff(fact_3892_Sum__Icc__nat,axiom,
    ! [M: nat,Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fj(nat,nat)),set_or1337092689740270186AtMost(nat,M,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),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,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Icc_nat
tff(fact_3893_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_3894_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_3895_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,D2: A,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fm(A,fun(A,fun(nat,A)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),D2)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% arith_series
tff(fact_3896_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_3897_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xa: A,M: nat,Na: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xa)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) = $ite(Xa = one_one(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,Na))))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa))) ) ).

% sum_gp_offset
tff(fact_3898_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_3899_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] : aa(real,real,arccos,aa(real,real,cos(real),Theta)) != aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Theta),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),K2)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_3900_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z: A,Na: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),Na)),aa(nat,A,aa(A,fun(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,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),times_times(A),H),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fo(A,fun(A,fun(nat,fun(nat,A))),H),Z),Na)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ) ).

% lemma_termdiff2
tff(fact_3901_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_fp(A,fun(nat,A),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% geometric_deriv_sums
tff(fact_3902_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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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_3903_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_fq(A,A),Na,zero_zero(A)) ) ).

% of_nat_code
tff(fact_3904_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_lessThan(A,K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),K) ) ) ).

% lessThan_iff
tff(fact_3905_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => aa(A,$o,sums(A,aTP_Lamp_bk(nat,A)),zero_zero(A)) ) ).

% sums_zero
tff(fact_3906_lessThan__0,axiom,
    set_ord_lessThan(nat,zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_3907_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_3908_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),K),bot_bot(set(A)))),set_ord_lessThan(A,K)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),K),bot_bot(set(A))) ) ).

% single_Diff_lessThan
tff(fact_3909_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A),Xa: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_bx(fun(nat,A),fun(nat,A),A2)),Xa)
        <=> ( aa(nat,A,A2,zero_zero(nat)) = Xa ) ) ) ).

% powser_sums_zero_iff
tff(fact_3910_int__sum,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A)] : aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3)) = aa(set(A),int,groups7311177749621191930dd_sum(A,int,aTP_Lamp_fr(fun(A,nat),fun(A,int),F2)),A3) ).

% int_sum
tff(fact_3911_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),Na: A] :
          ( ! [X: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Q,X)),aa(A,nat,P,X))
         => ( aa(nat,nat,aa(nat,fun(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_fs(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),set_ord_lessThan(A,Na)) ) ) ) ).

% sum_diff_distrib
tff(fact_3912_sums__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F2: fun(A,fun(nat,B)),Xa: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => aa(B,$o,sums(B,aa(A,fun(nat,B),F2,I3)),aa(A,B,Xa,I3)) )
         => aa(B,$o,sums(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_eq(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,Xa),I5)) ) ) ).

% sums_sum
tff(fact_3913_lessThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xa: A] : set_ord_lessThan(A,Xa) != bot_bot(set(A)) ) ).

% lessThan_non_empty
tff(fact_3914_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_bm(nat,fun(fun(nat,A),fun(nat,A)),I),F2)),aa(nat,A,F2,I)) ) ).

% sums_single
tff(fact_3915_pochhammer__of__nat,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xa: nat,Na: nat] : comm_s3205402744901411588hammer(A,aa(nat,A,semiring_1_of_nat(A),Xa),Na) = aa(nat,A,semiring_1_of_nat(A),comm_s3205402744901411588hammer(nat,Xa,Na)) ) ).

% pochhammer_of_nat
tff(fact_3916_pochhammer__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & comm_semiring_1(A) )
     => ! [Xa: real,Na: nat] : comm_s3205402744901411588hammer(A,aa(real,A,real_Vector_of_real(A),Xa),Na) = aa(real,A,real_Vector_of_real(A),comm_s3205402744901411588hammer(real,Xa,Na)) ) ).

% pochhammer_of_real
tff(fact_3917_sums__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X7: fun(nat,real),A2: real] :
          ( aa(real,$o,sums(real,X7),A2)
         => aa(A,$o,sums(A,aTP_Lamp_bv(fun(nat,real),fun(nat,A),X7)),aa(real,A,real_Vector_of_real(A),A2)) ) ) ).

% sums_of_real
tff(fact_3918_sums__of__real__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,real),C2: real] :
          ( aa(A,$o,sums(A,aTP_Lamp_ft(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_3919_sums__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),A2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => aa(A,$o,sums(A,aTP_Lamp_bt(fun(nat,A),fun(nat,A),F2)),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% sums_minus
tff(fact_3920_sums__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_bg(fun(nat,A),fun(A,fun(nat,A)),F2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)) ) ) ).

% sums_divide
tff(fact_3921_sums__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => ( aa(A,$o,sums(A,G),B2)
           => aa(A,$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),minus_minus(A),A2),B2)) ) ) ) ).

% sums_diff
tff(fact_3922_sums__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_bp(fun(nat,A),fun(A,fun(nat,A)),F2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ).

% sums_mult2
tff(fact_3923_sums__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_bq(fun(nat,A),fun(A,fun(nat,A)),F2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) ) ) ).

% sums_mult
tff(fact_3924_sums__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => ( aa(A,$o,sums(A,G),B2)
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bo(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% sums_add
tff(fact_3925_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_lessThan(A,U) = collect(A,aTP_Lamp_fu(A,fun(A,$o),U)) ) ).

% lessThan_def
tff(fact_3926_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Na: nat,Sa: A] :
          ( aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_bj(fun(nat,A),fun(nat,fun(nat,A)),F2),Na)),Sa)
        <=> aa(A,$o,sums(A,F2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Sa),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Na)))) ) ) ).

% sums_iff_shift
tff(fact_3927_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Sa: A,Na: nat] :
          ( aa(A,$o,sums(A,F2),Sa)
         => aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_bj(fun(nat,A),fun(nat,fun(nat,A)),F2),Na)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Sa),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Na)))) ) ) ).

% sums_split_initial_segment
tff(fact_3928_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Na: nat,Sa: A] :
          ( aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_bj(fun(nat,A),fun(nat,fun(nat,A)),F2),Na)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Sa),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Na))))
        <=> aa(A,$o,sums(A,F2),Sa) ) ) ).

% sums_iff_shift'
tff(fact_3929_Iio__eq__empty__iff,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & order_bot(A) )
     => ! [Na: A] :
          ( ( set_ord_lessThan(A,Na) = bot_bot(set(A)) )
        <=> ( Na = bot_bot(A) ) ) ) ).

% Iio_eq_empty_iff
tff(fact_3930_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_ord_lessThan(A,M)),set_ord_lessThan(A,Na))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),Na) ) ) ).

% lessThan_strict_subset_iff
tff(fact_3931_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_fv(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_3932_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_fw(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_3933_lessThan__empty__iff,axiom,
    ! [Na: nat] :
      ( ( set_ord_lessThan(nat,Na) = bot_bot(set(nat)) )
    <=> ( Na = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_3934_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xa,Na)) ) ) ).

% pochhammer_pos
tff(fact_3935_lessThan__Suc,axiom,
    ! [K: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),K),set_ord_lessThan(nat,K)) ).

% lessThan_Suc
tff(fact_3936_sum__subtractf__nat,axiom,
    ! [A: $tType,A3: set(A),G: fun(A,nat),F2: fun(A,nat)] :
      ( ! [X: A] :
          ( member(A,X,A3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,G,X)),aa(A,nat,F2,X)) )
     => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_fx(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F2)),A3) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,G),A3)) ) ) ).

% sum_subtractf_nat
tff(fact_3937_sum__SucD,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A),Na: nat] :
      ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3) = aa(nat,nat,suc,Na) )
     => ? [X: A] :
          ( member(A,X,A3)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X)) ) ) ).

% sum_SucD
tff(fact_3938_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A),A2: A] :
          ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bh(A,fun(fun(nat,A),fun(nat,A)),C2),F2)),A2)
         => ( ( C2 != zero_zero(A) )
           => aa(A,$o,sums(A,F2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)) ) ) ) ).

% sums_mult_D
tff(fact_3939_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Sa: A] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(A,$o,sums(A,aTP_Lamp_bs(fun(nat,A),fun(nat,A),F2)),Sa)
           => aa(A,$o,sums(A,F2),Sa) ) ) ) ).

% sums_Suc_imp
tff(fact_3940_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Sa: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_bs(fun(nat,A),fun(nat,A),F2)),Sa)
        <=> aa(A,$o,sums(A,F2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Sa),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_3941_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_fy(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_3942_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Na: nat,F2: fun(nat,A),Sa: A] :
          ( ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Na)
             => ( aa(nat,A,F2,I3) = zero_zero(A) ) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fz(nat,fun(fun(nat,A),fun(nat,A)),Na),F2)),Sa)
          <=> aa(A,$o,sums(A,F2),Sa) ) ) ) ).

% sums_zero_iff_shift
tff(fact_3943_ivl__disj__int__one_I4_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,L)),set_or1337092689740270186AtMost(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(4)
tff(fact_3944_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xa,Na)) ) ) ).

% pochhammer_nonneg
tff(fact_3945_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_3946_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_3947_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N3: set(nat),F2: fun(nat,A)] :
          ( finite_finite2(nat,N3)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N3)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => aa(A,$o,sums(A,F2),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),N3)) ) ) ) ).

% sums_finite
tff(fact_3948_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( finite_finite2(nat,collect(nat,P))
         => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cp(fun(nat,$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_3949_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F2: fun(nat,A)] :
          ( finite_finite2(nat,A3)
         => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cq(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),A3)) ) ) ).

% sums_If_finite_set
tff(fact_3950_lessThan__nat__numeral,axiom,
    ! [K: num] : set_ord_lessThan(nat,aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),pred_numeral(K)),set_ord_lessThan(nat,pred_numeral(K))) ).

% lessThan_nat_numeral
tff(fact_3951_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_ga(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_3952_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_gb(complex,complex)),collect(complex,aa(complex,fun(complex,$o),aTP_Lamp_dc(nat,fun(complex,fun(complex,$o)),Na),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_3953_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_gb(complex,complex)),collect(complex,aTP_Lamp_gc(nat,fun(complex,$o),Na))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_3954_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_gd(nat,fun(A,fun(nat,A)),M),Z)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),M)) ) ).

% powser_sums_if
tff(fact_3955_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A)] : aa(A,$o,sums(A,aTP_Lamp_bx(fun(nat,A),fun(nat,A),A2)),aa(nat,A,A2,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_3956_sum__diff1__nat,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A),A2: A] :
      aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = $ite(member(A,A2,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3)),aa(A,nat,F2,A2)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3)) ).

% sum_diff1_nat
tff(fact_3957_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K: A,Xa: A] :
          aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,K)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),K),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))),bot_bot(set(A))) ) ).

% Iio_Int_singleton
tff(fact_3958_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S: A,A3: set(nat),S5: A,F2: fun(nat,A)] :
          ( aa(A,$o,sums(A,G),S)
         => ( finite_finite2(nat,A3)
           => ( ( S5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ge(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G),F2)),A3)) )
             => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_gf(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A3),F2)),S5) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_3959_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)),Na)) ) ).

% pochhammer_rec
tff(fact_3960_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A2,Na)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),Na))) ) ).

% pochhammer_Suc
tff(fact_3961_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_3962_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_3963_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_3964_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Na: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Na) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),Na)
              & ( A2 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_3965_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_3966_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_3967_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_es(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Na))) ) ).

% sum.lessThan_Suc_shift
tff(fact_3968_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_gg(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,M)) ) ).

% sum_lessThan_telescope'
tff(fact_3969_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_fa(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,M)),aa(nat,A,F2,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_3970_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [F2: fun(nat,A),Na: nat,R: A] : aa(A,A,aa(A,fun(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_gh(fun(nat,A),fun(A,fun(nat,A)),F2),R)),set_ord_lessThan(nat,Na)) ) ).

% sumr_diff_mult_const2
tff(fact_3971_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Xa: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,N))),Xa)
           => summable(A,F2) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_3972_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_es(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Na)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_3973_sum__Un__nat,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),F2: fun(A,nat)] :
      ( finite_finite2(A,A3)
     => ( finite_finite2(A,B3)
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),B3))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ).

% sum_Un_nat
tff(fact_3974_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,Na: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xa)),set_ord_lessThan(nat,Na))) ) ).

% one_diff_power_eq
tff(fact_3975_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,Na: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),one_one(A))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xa)),set_ord_lessThan(nat,Na))) ) ).

% power_diff_1_eq
tff(fact_3976_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xa: A,Na: nat] :
          ( ( Xa != one_one(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xa)),set_ord_lessThan(nat,Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_3977_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Na: nat,Z: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( comm_s3205402744901411588hammer(A,Z,Na) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,M)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))) ) ) ) ).

% pochhammer_product
tff(fact_3978_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_bj(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_3979_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_bj(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) = aa(A,A,aa(A,fun(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_3980_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Na: nat] :
          ( summable(A,F2)
         => ( ! [M4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,M4)) )
           => aa(A,$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_3981_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_gi(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_gj(A,fun(A,fun(nat,fun(nat,A))),Z),H),M)),set_ord_lessThan(nat,M)) ) ).

% lemma_termdiff1
tff(fact_3982_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xa: A,Na: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xa)),set_ord_lessThan(nat,Na)) = $ite(Xa = 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,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa))) ) ).

% sum_gp_strict
tff(fact_3983_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,Na: nat,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,Na))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_gk(A,fun(nat,fun(A,fun(nat,A))),Xa),Na),Y)),set_ord_lessThan(nat,aa(nat,nat,suc,Na)))) ) ).

% diff_power_eq_sum
tff(fact_3984_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,Na: nat,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_gl(A,fun(nat,fun(A,fun(nat,A))),Xa),Na),Y)),set_ord_lessThan(nat,Na))) ) ).

% power_diff_sumr2
tff(fact_3985_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),R),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_3986_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,aa(A,fun(nat,A),power_power(A),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),C2))) ) ) ).

% geometric_sums
tff(fact_3987_power__half__series,axiom,
    aa(real,$o,sums(real,aTP_Lamp_gm(nat,real)),one_one(real)) ).

% power_half_series
tff(fact_3988_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,F2: fun(nat,A),K6: A,K: nat] :
          ( ! [P4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P4),Na)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,P4)),K6) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),K6)
           => aa(A,$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,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),K6)) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_3989_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Na: nat,I: nat] :
          ( summable(A,F2)
         => ( ! [M4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,M4)) )
           => ( aa(nat,$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_3990_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,Na: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gn(A,fun(nat,fun(nat,A)),Xa),Na)),set_ord_lessThan(nat,Na))) ) ).

% one_diff_power_eq'
tff(fact_3991_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).

% pochhammer_minus
tff(fact_3992_pochhammer__minus_H,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K)) ) ).

% pochhammer_minus'
tff(fact_3993_sums__if_H,axiom,
    ! [G: fun(nat,real),Xa: real] :
      ( aa(real,$o,sums(real,G),Xa)
     => aa(real,$o,sums(real,aTP_Lamp_go(fun(nat,real),fun(nat,real),G)),Xa) ) ).

% sums_if'
tff(fact_3994_sums__if,axiom,
    ! [G: fun(nat,real),Xa: real,F2: fun(nat,real),Y: real] :
      ( aa(real,$o,sums(real,G),Xa)
     => ( aa(real,$o,sums(real,F2),Y)
       => aa(real,$o,sums(real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_gp(fun(nat,real),fun(fun(nat,real),fun(nat,real)),G),F2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)) ) ) ).

% sums_if
tff(fact_3995_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_gq(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_gr(fun(nat,real),fun(nat,real),F2)),set_ord_lessThan(nat,Na))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gs(fun(nat,real),fun(nat,real),G)),set_ord_lessThan(nat,Na))) ).

% sum_split_even_odd
tff(fact_3996_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_gt(int,int)),set_or1337092689740270186AtMost(int,M,Na)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(int,int,aa(int,fun(int,int),plus_plus(int),Na),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),M),aa(int,int,aa(int,fun(int,int),minus_minus(int),M),one_one(int))))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ) ).

% Sum_Icc_int
tff(fact_3997_sum__pos__lt__pair,axiom,
    ! [F2: fun(nat,real),K: nat] :
      ( summable(real,F2)
     => ( ! [D3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,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)))),D3)))),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)))),D3)),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_3998_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_es(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_3999_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_gu(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_4000_sumr__cos__zero__one,axiom,
    ! [Na: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gv(nat,real)),set_ord_lessThan(nat,aa(nat,nat,suc,Na))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_4001_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] :
          comm_s3205402744901411588hammer(A,A2,Na) = $ite(Na = zero_zero(nat),one_one(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_gw(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)),one_one(A))) ) ).

% pochhammer_code
tff(fact_4002_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B)] : groups7121269368397514597t_prod(B,A,aTP_Lamp_gx(B,A),A3) = one_one(A) ) ).

% prod.neutral_const
tff(fact_4003_of__nat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,nat),A3: set(B)] : aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(B,nat,F2,A3)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_gy(fun(B,nat),fun(B,A),F2),A3) ) ).

% of_nat_prod
tff(fact_4004_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [F2: fun(B,int),A3: set(B)] : aa(int,A,ring_1_of_int(A),groups7121269368397514597t_prod(B,int,F2,A3)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_gz(fun(B,int),fun(B,A),F2),A3) ) ).

% of_int_prod
tff(fact_4005_of__real__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2191834092415804123ebra_1(A) )
     => ! [F2: fun(B,real),Sa: set(B)] : aa(real,A,real_Vector_of_real(A),groups7121269368397514597t_prod(B,real,F2,Sa)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_ha(fun(B,real),fun(B,A),F2),Sa) ) ).

% of_real_prod
tff(fact_4006_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_4007_cos__coeff__0,axiom,
    aa(nat,real,cos_coeff,zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_4008_prod_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A2: A,B2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_hb(A,fun(fun(A,B),fun(A,B)),A2),B2),S) = $ite(member(A,A2,S),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta
tff(fact_4009_prod_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A2: A,B2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_hc(A,fun(fun(A,B),fun(A,B)),A2),B2),S) = $ite(member(A,A2,S),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta'
tff(fact_4010_prod_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),Xa: A,G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ~ member(A,Xa,A3)
           => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xa)),groups7121269368397514597t_prod(A,B,G,A3)) ) ) ) ) ).

% prod.insert
tff(fact_4011_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_4012_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_4013_abs__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(A,A,abs_abs(A),groups7121269368397514597t_prod(B,A,F2,A3)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_hd(fun(B,A),fun(B,A),F2),A3) ) ).

% abs_prod
tff(fact_4014_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),G: fun(B,A),A3: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_he(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(B,A,F2,A3)),groups7121269368397514597t_prod(B,A,G,A3)) ) ).

% prod_dividef
tff(fact_4015_prod_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,fun(C,A)),B3: set(C),A3: set(B)] : groups7121269368397514597t_prod(B,A,aa(set(C),fun(B,A),aTP_Lamp_hf(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B3),A3) = groups7121269368397514597t_prod(C,A,aa(set(B),fun(C,A),aTP_Lamp_hh(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A3),B3) ) ).

% prod.swap
tff(fact_4016_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),H: fun(B,A),A3: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_hi(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G,A3)),groups7121269368397514597t_prod(B,A,H,A3)) ) ).

% prod.distrib
tff(fact_4017_prod__power__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A3: set(B),Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),groups7121269368397514597t_prod(B,A,F2,A3)),Na) = groups7121269368397514597t_prod(B,A,aa(nat,fun(B,A),aTP_Lamp_hj(fun(B,A),fun(nat,fun(B,A)),F2),Na),A3) ) ).

% prod_power_distrib
tff(fact_4018_prod_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A3: set(A),B3: set(B),G: fun(A,fun(B,C)),R2: fun(A,fun(B,$o))] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(B,B3)
           => ( 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_hk(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B3),G),R2),A3) = groups7121269368397514597t_prod(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_hm(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A3),G),R2),B3) ) ) ) ) ).

% prod.swap_restrict
tff(fact_4019_norm__prod__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [F2: fun(B,A),A3: set(B)] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,groups7121269368397514597t_prod(B,A,F2,A3))),groups7121269368397514597t_prod(B,real,aTP_Lamp_hn(fun(B,A),fun(B,real),F2),A3)) ) ).

% norm_prod_le
tff(fact_4020_prod__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [F2: fun(A,B),A3: set(A)] : groups7121269368397514597t_prod(A,real,aTP_Lamp_ho(fun(A,B),fun(A,real),F2),A3) = real_V7770717601297561774m_norm(B,groups7121269368397514597t_prod(A,B,F2,A3)) ) ).

% prod_norm
tff(fact_4021_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),A2: A,A3: set(B)] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),groups7121269368397514597t_prod(B,A,aa(A,fun(B,A),aTP_Lamp_eo(fun(B,A),fun(A,fun(B,A)),F2),A2),A3)),A2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),groups7121269368397514597t_prod(B,A,F2,A3)),A2) ) ).

% mod_prod_eq
tff(fact_4022_prod__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,X)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),groups7121269368397514597t_prod(A,B,F2,A3)) ) ) ).

% prod_pos
tff(fact_4023_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F2: fun(nat,A),A2: nat,B2: nat] : groups7121269368397514597t_prod(nat,A,F2,set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_hp(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_4024_prod_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( finite_finite2(A,A3)
         => ( groups7121269368397514597t_prod(A,B,G,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_al(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) = groups7121269368397514597t_prod(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_hq(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P),A3) ) ) ) ).

% prod.inter_filter
tff(fact_4025_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_hr(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_4026_power__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C2: A,F2: fun(B,nat),A3: set(B)] : aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,F2),A3)) = groups7121269368397514597t_prod(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_hs(A,fun(fun(B,nat),fun(B,A)),C2),F2),A3) ) ).

% power_sum
tff(fact_4027_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_ht(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_4028_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R2: fun(A,fun(A,$o)),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R2,one_one(A)),one_one(A))
         => ( ! [X1: A,Y1: A,X2: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R2,X1),X2)
                  & aa(A,$o,aa(A,fun(A,$o),R2,Y1),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R2,aa(A,A,aa(A,fun(A,A),times_times(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),times_times(A),X2),Y23)) )
           => ( finite_finite2(B,S)
             => ( ! [X: B] :
                    ( member(B,X,S)
                   => aa(A,$o,aa(A,fun(A,$o),R2,aa(B,A,H,X)),aa(B,A,G,X)) )
               => aa(A,$o,aa(A,fun(A,$o),R2,groups7121269368397514597t_prod(B,A,H,S)),groups7121269368397514597t_prod(B,A,G,S)) ) ) ) ) ) ).

% prod.related
tff(fact_4029_prod_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),Xa: A] :
          ( finite_finite2(A,A3)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) = $ite(member(A,Xa,A3),groups7121269368397514597t_prod(A,B,G,A3),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xa)),groups7121269368397514597t_prod(A,B,G,A3))) ) ) ) ).

% prod.insert_if
tff(fact_4030_prod_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),B3: set(A)] :
          ( finite_finite2(A,A3)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = groups7121269368397514597t_prod(A,B,aa(set(A),fun(A,B),aTP_Lamp_hu(fun(A,B),fun(set(A),fun(A,B)),G),B3),A3) ) ) ) ).

% prod.inter_restrict
tff(fact_4031_prod_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),collect(A,aTP_Lamp_hv(fun(A,B),fun(A,$o),G)))) = groups7121269368397514597t_prod(A,B,G,A3) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_4032_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_hw(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_4033_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_hx(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M),set_or1337092689740270186AtMost(nat,Na,M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_4034_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I: A,F2: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I))
             => ( ! [I3: A] :
                    ( member(A,I3,I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,I3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),groups7121269368397514597t_prod(A,B,F2,I5)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_4035_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( member(A,I3,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),groups7121269368397514597t_prod(A,B,F2,I5)) ) ) ) ) ).

% less_1_prod
tff(fact_4036_prod_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B3: set(A),A3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
         => ( finite_finite2(A,A3)
           => ( groups7121269368397514597t_prod(A,B,G,A3) = aa(B,B,aa(B,fun(B,B),times_times(B),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),groups7121269368397514597t_prod(A,B,G,B3)) ) ) ) ) ).

% prod.subset_diff
tff(fact_4037_prod_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),B3: set(A)] :
          ( finite_finite2(A,A3)
         => ( groups7121269368397514597t_prod(A,B,G,A3) = aa(B,B,aa(B,fun(B,B),times_times(B),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))) ) ) ) ).

% prod.Int_Diff
tff(fact_4038_prod_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B3)
           => ( aa(B,B,aa(B,fun(B,B),times_times(B),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) = aa(B,B,aa(B,fun(B,B),times_times(B),groups7121269368397514597t_prod(A,B,G,A3)),groups7121269368397514597t_prod(A,B,G,B3)) ) ) ) ) ).

% prod.union_inter
tff(fact_4039_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_4040_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_4041_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_4042_prod_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),P: fun(A,$o),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_hy(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H),G),A3) = aa(B,B,aa(B,fun(B,B),times_times(B),groups7121269368397514597t_prod(A,B,H,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,P)))),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P))))) ) ) ) ).

% prod.If_cases
tff(fact_4043_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_hr(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,Na))) ) ).

% prod.lessThan_Suc_shift
tff(fact_4044_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_hr(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_4045_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_hr(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,Na)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_4046_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ! [I3: A] :
                ( member(A,I3,A3)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I3))
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I3)),aa(A,B,G,I3)) ) )
           => ( ( A3 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),groups7121269368397514597t_prod(A,B,F2,A3)),groups7121269368397514597t_prod(A,B,G,A3)) ) ) ) ) ).

% prod_mono_strict
tff(fact_4047_even__prod__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),groups7121269368397514597t_prod(A,B,F2,A3))
          <=> ? [X4: A] :
                ( member(A,X4,A3)
                & aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),aa(A,B,F2,X4)) ) ) ) ) ).

% even_prod_iff
tff(fact_4048_prod_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B3)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
                 => ( aa(A,B,G,X) = one_one(B) ) )
             => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),times_times(B),groups7121269368397514597t_prod(A,B,G,A3)),groups7121269368397514597t_prod(A,B,G,B3)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_4049_prod_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),Xa: A,G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( member(A,Xa,A3)
           => ( groups7121269368397514597t_prod(A,B,G,A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xa)),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) ) ) ) ) ).

% prod.remove
tff(fact_4050_prod_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),Xa: A] :
          ( finite_finite2(A,A3)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xa)),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) ) ) ) ).

% prod.insert_remove
tff(fact_4051_prod_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B3)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
             => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),times_times(B),groups7121269368397514597t_prod(A,B,G,A3)),groups7121269368397514597t_prod(A,B,G,B3)) ) ) ) ) ) ).

% prod.union_disjoint
tff(fact_4052_prod_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B3)
           => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)))),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ) ).

% prod.union_diff2
tff(fact_4053_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_4054_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc: A] :
      set_fo6178422350223883121st_nat(A,F2,A2,B2,Acc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A2),Acc,set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F2,A2),Acc))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_4055_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,Xa: fun(nat,fun(A,A)),Xaa: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xa,Xaa,Xb,Xc) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Xaa),Xc,set_fo6178422350223883121st_nat(A,Xa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xaa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),Xa,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_4056_prod_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A2: A,B2: fun(A,B),C2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_hz(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C2),S) = $ite(member(A,A2,S),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A2)),groups7121269368397514597t_prod(A,B,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))),groups7121269368397514597t_prod(A,B,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))) ) ) ) ).

% prod.delta_remove
tff(fact_4057_norm__prod__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),Z: fun(A,B),W: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Z,I3))),one_one(real)) )
         => ( ! [I3: A] :
                ( member(A,I3,I5)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,W,I3))),one_one(real)) )
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),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_ia(fun(A,B),fun(fun(A,B),fun(A,real)),Z),W)),I5)) ) ) ) ).

% norm_prod_diff
tff(fact_4058_prod__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom_divide(B)
     => ! [A3: set(A),F2: fun(A,B),A2: A] :
          ( finite_finite2(A,A3)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => ( groups7121269368397514597t_prod(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = $ite(member(A,A2,A3),aa(B,B,aa(B,fun(B,B),divide_divide(B),groups7121269368397514597t_prod(A,B,F2,A3)),aa(A,B,F2,A2)),groups7121269368397514597t_prod(A,B,F2,A3)) ) ) ) ) ).

% prod_diff1
tff(fact_4059_prod__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( field(B)
     => ! [A3: set(A),B3: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B3)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
                 => ( aa(A,B,F2,X) != zero_zero(B) ) )
             => ( groups7121269368397514597t_prod(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),groups7121269368397514597t_prod(A,B,F2,A3)),groups7121269368397514597t_prod(A,B,F2,B3))),groups7121269368397514597t_prod(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ) ) ).

% prod_Un
tff(fact_4060_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Na)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_ib(A,fun(nat,A),A2),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ) ).

% pochhammer_Suc_prod
tff(fact_4061_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] : comm_s3205402744901411588hammer(A,A2,Na) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ic(A,fun(nat,fun(nat,A)),A2),Na),set_or1337092689740270186AtMost(nat,one_one(nat),Na)) ) ).

% pochhammer_prod_rev
tff(fact_4062_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_id(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.in_pairs
tff(fact_4063_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ie(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_4064_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Na)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ic(A,fun(nat,fun(nat,A)),A2),Na),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_4065_Maclaurin__cos__expansion2,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),Xa)
            & ( aa(real,real,cos(real),Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_if(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_4066_Maclaurin__minus__cos__expansion,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),T4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),zero_zero(real))
            & ( aa(real,real,cos(real),Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_if(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_4067_Maclaurin__cos__expansion,axiom,
    ! [Xa: real,Na: nat] :
    ? [T4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xa))
      & ( aa(real,real,cos(real),Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_if(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_4068_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C2: fun(nat,A),Xa: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ig(fun(nat,A),fun(A,fun(nat,A)),C2),Xa))
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ih(fun(nat,A),fun(A,fun(nat,A)),C2),Xa)),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ig(fun(nat,A),fun(A,fun(nat,A)),C2),Xa))) ) ) ).

% diffs_equiv
tff(fact_4069_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_4070_norm__fact,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Na: nat] : real_V7770717601297561774m_norm(A,semiring_char_0_fact(A,Na)) = semiring_char_0_fact(real,Na) ) ).

% norm_fact
tff(fact_4071_of__real__fact,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Na: nat] : aa(real,A,real_Vector_of_real(A),semiring_char_0_fact(real,Na)) = semiring_char_0_fact(A,Na) ) ).

% of_real_fact
tff(fact_4072_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_4073_prod__pos__nat__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( finite_finite2(A,A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),groups7121269368397514597t_prod(A,nat,F2,A3))
      <=> ! [X4: A] :
            ( member(A,X4,A3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_4074_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_4075_int__prod,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A)] : aa(nat,int,semiring_1_of_nat(int),groups7121269368397514597t_prod(A,nat,F2,A3)) = groups7121269368397514597t_prod(A,int,aTP_Lamp_fr(fun(A,nat),fun(A,int),F2),A3) ).

% int_prod
tff(fact_4076_diffs__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [F2: fun(nat,real),X5: nat] : aa(nat,A,diffs(A,aTP_Lamp_ii(fun(nat,real),fun(nat,A),F2)),X5) = aa(real,A,real_Vector_of_real(A),aa(nat,real,diffs(real,F2),X5)) ) ).

% diffs_of_real
tff(fact_4077_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,Na)) ) ).

% fact_gt_zero
tff(fact_4078_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,Na)),zero_zero(A)) ) ).

% fact_not_neg
tff(fact_4079_diffs__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X5: nat] : aa(nat,A,diffs(A,aTP_Lamp_ij(fun(nat,A),fun(nat,A),C2)),X5) = aa(A,A,uminus_uminus(A),aa(nat,A,diffs(A,C2),X5)) ) ).

% diffs_minus
tff(fact_4080_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,Na)) ) ) ) ).

% fact_less_mono
tff(fact_4081_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_4082_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_4083_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,Na)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Na),Na))) ) ).

% fact_le_power
tff(fact_4084_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_gt(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_4085_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_fj(nat,nat),set_or1337092689740270186AtMost(nat,one_one(nat),Na))) ) ).

% fact_prod
tff(fact_4086_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)))),semiring_char_0_fact(A,Na)) ) ) ).

% choose_dvd
tff(fact_4087_diffs__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X5: nat] : aa(nat,A,diffs(A,C2),X5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,X5))),aa(nat,A,C2,aa(nat,nat,suc,X5))) ) ).

% diffs_def
tff(fact_4088_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_4089_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_gt(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_4090_termdiff__converges__all,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xa: A] :
          ( ! [X: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),C2),X))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_il(fun(nat,A),fun(A,fun(nat,A)),C2),Xa)) ) ) ).

% termdiff_converges_all
tff(fact_4091_ln__prod,axiom,
    ! [A: $tType,I5: set(A),F2: fun(A,real)] :
      ( finite_finite2(A,I5)
     => ( ! [I3: A] :
            ( member(A,I3,I5)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,I3)) )
       => ( aa(real,real,ln_ln(real),groups7121269368397514597t_prod(A,real,F2,I5)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_im(fun(A,real),fun(A,real),F2)),I5) ) ) ) ).

% ln_prod
tff(fact_4092_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_4093_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_4094_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          semiring_char_0_fact(A,M) = $ite(M = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))) ) ).

% fact_num_eq_if
tff(fact_4095_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( semiring_char_0_fact(A,Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_4096_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,aa(A,fun(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_4097_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,K6: real,C2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),K6)
         => ( ! [X: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),K6)
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),C2),X)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_in(A,fun(fun(nat,A),fun(nat,A)),Xa),C2)) ) ) ) ).

% termdiff_converges
tff(fact_4098_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xa: real,Na: nat,Diff: fun(nat,fun(A,real))] :
          ( ( Xa = zero_zero(real) )
         => ( ( Na != zero_zero(nat) )
           => ( aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_io(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Xa),Diff)),set_ord_lessThan(nat,Na)) = aa(A,real,aa(nat,fun(A,real),Diff,zero_zero(nat)),zero_zero(A)) ) ) ) ) ).

% Maclaurin_zero
tff(fact_4099_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)
     => ? [B4: 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_ip(real,fun(fun(nat,real),fun(nat,real)),H),J)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),B4),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Na)),semiring_char_0_fact(real,Na)))) ) ).

% Maclaurin_lemma
tff(fact_4100_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,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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_4101_cos__coeff__def,axiom,
    ! [X5: nat] :
      aa(nat,real,cos_coeff,X5) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X5),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X5),aa(num,nat,numeral_numeral(nat),bit0(one2))))),semiring_char_0_fact(real,X5)),zero_zero(real)) ).

% cos_coeff_def
tff(fact_4102_cos__paired,axiom,
    ! [Xa: real] : aa(real,$o,sums(real,aTP_Lamp_iq(real,fun(nat,real),Xa)),aa(real,real,cos(real),Xa)) ).

% cos_paired
tff(fact_4103_sin__paired,axiom,
    ! [Xa: real] : aa(real,$o,sums(real,aTP_Lamp_ir(real,fun(nat,real),Xa)),aa(real,real,sin(real),Xa)) ).

% sin_paired
tff(fact_4104_Maclaurin__sin__expansion3,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),Xa)
            & ( aa(real,real,sin(real),Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_is(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_4105_Maclaurin__sin__expansion4,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ? [T4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T4)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),Xa)
          & ( aa(real,real,sin(real),Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_is(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_4106_Maclaurin__sin__expansion2,axiom,
    ! [Xa: real,Na: nat] :
    ? [T4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xa))
      & ( aa(real,real,sin(real),Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_is(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_4107_Maclaurin__sin__expansion,axiom,
    ! [Xa: real,Na: nat] :
    ? [T4: real] : aa(real,real,sin(real),Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_is(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ).

% Maclaurin_sin_expansion
tff(fact_4108_sin__coeff__0,axiom,
    aa(nat,real,sin_coeff,zero_zero(nat)) = zero_zero(real) ).

% sin_coeff_0
tff(fact_4109_fact__less__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Na)) ) ) ).

% fact_less_mono_nat
tff(fact_4110_diffs__sin__coeff,axiom,
    diffs(real,sin_coeff) = cos_coeff ).

% diffs_sin_coeff
tff(fact_4111_fact__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
     => ( semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),Na)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na))) ) ) ).

% fact_diff_Suc
tff(fact_4112_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),R)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Na),R)) ) ).

% fact_div_fact_le_pow
tff(fact_4113_diffs__cos__coeff,axiom,
    ! [X5: nat] : aa(nat,real,diffs(real,cos_coeff),X5) = aa(real,real,uminus_uminus(real),aa(nat,real,sin_coeff,X5)) ).

% diffs_cos_coeff
tff(fact_4114_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_fj(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Na),M))) ) ) ).

% fact_eq_fact_times
tff(fact_4115_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_fj(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_4116_sin__coeff__Suc,axiom,
    ! [Na: nat] : aa(nat,real,sin_coeff,aa(nat,nat,suc,Na)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,cos_coeff,Na)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na))) ).

% sin_coeff_Suc
tff(fact_4117_cos__coeff__Suc,axiom,
    ! [Na: nat] : aa(nat,real,cos_coeff,aa(nat,nat,suc,Na)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),aa(nat,real,sin_coeff,Na))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na))) ).

% cos_coeff_Suc
tff(fact_4118_sin__coeff__def,axiom,
    ! [X5: nat] :
      aa(nat,real,sin_coeff,X5) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X5),zero_zero(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X5),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),semiring_char_0_fact(real,X5))) ).

% sin_coeff_def
tff(fact_4119_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)),one_one(nat)),Na,one_one(nat))),semiring_char_0_fact(nat,K))) ) ).

% binomial_code
tff(fact_4120_Maclaurin__exp__lt,axiom,
    ! [Xa: real,Na: nat] :
      ( ( Xa != zero_zero(real) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T4))
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xa))
            & ( aa(real,real,exp(real),Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_it(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,exp(real),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_4121_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),Na) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Na)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_4122_maxt__sound,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T),Xa)
       => ( vEBT_vebt_maxt(T) = some(nat,Xa) ) ) ) ).

% maxt_sound
tff(fact_4123_maxbmo,axiom,
    ! [T: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_maxt(T) = some(nat,Xa) )
     => aa(nat,$o,vEBT_V8194947554948674370ptions(T),Xa) ) ).

% maxbmo
tff(fact_4124_maxt__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Maxi: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_maxt(T) = some(nat,Maxi) )
       => aa(nat,$o,vEBT_vebt_member(T),Maxi) ) ) ).

% maxt_member
tff(fact_4125_exp__inj__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa(real,real,exp(real),Xa) = aa(real,real,exp(real),Y) )
    <=> ( Xa = Y ) ) ).

% exp_inj_iff
tff(fact_4126_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,Na: nat,Maxi: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_maxt(T) = some(nat,Maxi) )
       => ( aa(nat,$o,vEBT_vebt_member(T),Xa)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_4127_set__vebt__maxt_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_maxt(T) = some(nat,Xa) )
      <=> ( member(nat,Xa,vEBT_set_vebt(T))
          & ! [X4: nat] :
              ( member(nat,X4,vEBT_set_vebt(T))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Xa) ) ) ) ) ).

% set_vebt_maxt'
tff(fact_4128_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_maxt(T) = none(nat) )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot(set(nat)) ) ) ) ).

% maxt_corr_help_empty
tff(fact_4129_set__vebt__maxt,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_maxt(T) = some(nat,Xa) )
      <=> vEBT_VEBT_max_in_set(vEBT_set_vebt(T),Xa) ) ) ).

% set_vebt_maxt
tff(fact_4130_maxt__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( vEBT_vebt_maxt(T) = some(nat,Xa) )
       => vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T),Xa) ) ) ).

% maxt_corr
tff(fact_4131_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_4132_of__int__floor__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( ( aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xa)) = Xa )
        <=> ? [N2: int] : Xa = aa(int,A,ring_1_of_int(A),N2) ) ) ).

% of_int_floor_cancel
tff(fact_4133_exp__less__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,exp(real),Xa)),aa(real,real,exp(real),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ).

% exp_less_cancel_iff
tff(fact_4134_exp__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,exp(real),Xa)),aa(real,real,exp(real),Y)) ) ).

% exp_less_mono
tff(fact_4135_exp__le__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),Xa)),aa(real,real,exp(real),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ).

% exp_le_cancel_iff
tff(fact_4136_abs__exp__cancel,axiom,
    ! [Xa: real] : aa(real,real,abs_abs(real),aa(real,real,exp(real),Xa)) = aa(real,real,exp(real),Xa) ).

% abs_exp_cancel
tff(fact_4137_ln__exp,axiom,
    ! [Xa: real] : aa(real,real,ln_ln(real),aa(real,real,exp(real),Xa)) = Xa ).

% ln_exp
tff(fact_4138_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_4139_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_4140_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_4141_binomial__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa(nat,nat,binomial(Na),K) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K) ) ).

% binomial_eq_0_iff
tff(fact_4142_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_4143_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_4144_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_4145_exp__eq__one__iff,axiom,
    ! [Xa: real] :
      ( ( aa(real,real,exp(real),Xa) = one_one(real) )
    <=> ( Xa = zero_zero(real) ) ) ).

% exp_eq_one_iff
tff(fact_4146_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_4147_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_4148_floor__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,Xa)),Z) ) ).

% floor_diff_of_int
tff(fact_4149_one__less__exp__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,exp(real),Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% one_less_exp_iff
tff(fact_4150_exp__less__one__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,exp(real),Xa)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% exp_less_one_iff
tff(fact_4151_exp__le__one__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),Xa)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% exp_le_one_iff
tff(fact_4152_one__le__exp__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,exp(real),Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% one_le_exp_iff
tff(fact_4153_exp__ln,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,exp(real),aa(real,real,ln_ln(real),Xa)) = Xa ) ) ).

% exp_ln
tff(fact_4154_exp__ln__iff,axiom,
    ! [Xa: real] :
      ( ( aa(real,real,exp(real),aa(real,real,ln_ln(real),Xa)) = Xa )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% exp_ln_iff
tff(fact_4155_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa) ) ) ).

% zero_le_floor
tff(fact_4156_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),V)),Xa) ) ) ).

% numeral_le_floor
tff(fact_4157_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A)) ) ) ).

% floor_less_zero
tff(fact_4158_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(num,A,numeral_numeral(A),V)) ) ) ).

% floor_less_numeral
tff(fact_4159_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa) ) ) ).

% zero_less_floor
tff(fact_4160_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ).

% floor_le_zero
tff(fact_4161_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa) ) ) ).

% one_le_floor
tff(fact_4162_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ).

% floor_less_one
tff(fact_4163_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_4164_floor__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V: num] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,Xa)),aa(num,int,numeral_numeral(int),V)) ) ).

% floor_diff_numeral
tff(fact_4165_floor__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: num,Na: nat] : archim6421214686448440834_floor(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Na)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na) ) ).

% floor_numeral_power
tff(fact_4166_floor__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,Xa)),one_one(int)) ) ).

% floor_diff_one
tff(fact_4167_floor__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A2)),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),A2)),aa(num,int,numeral_numeral(int),B2)) ).

% floor_divide_eq_div_numeral
tff(fact_4168_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),Xa) ) ) ).

% numeral_less_floor
tff(fact_4169_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))) ) ) ).

% floor_le_numeral
tff(fact_4170_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa) ) ) ).

% one_less_floor
tff(fact_4171_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% floor_le_one
tff(fact_4172_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),Xa) ) ) ).

% neg_numeral_le_floor
tff(fact_4173_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ) ).

% floor_less_neg_numeral
tff(fact_4174_floor__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(num,int,numeral_numeral(int),B2)) ).

% floor_one_divide_eq_div_numeral
tff(fact_4175_floor__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: 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),A2)),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(num,int,numeral_numeral(int),B2)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_4176_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),Xa) ) ) ).

% neg_numeral_less_floor
tff(fact_4177_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))) ) ) ).

% floor_le_neg_numeral
tff(fact_4178_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B2: 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),B2)))) = 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),B2)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_4179_exp__not__eq__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,A,exp(A),Xa) != zero_zero(A) ) ).

% exp_not_eq_zero
tff(fact_4180_ln__unique,axiom,
    ! [Y: real,Xa: real] :
      ( ( aa(real,real,exp(real),Y) = Xa )
     => ( aa(real,real,ln_ln(real),Xa) = Y ) ) ).

% ln_unique
tff(fact_4181_exp__less__cancel,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,exp(real),Xa)),aa(real,real,exp(real),Y))
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ).

% exp_less_cancel
tff(fact_4182_exp__times__arg__commute,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),A3)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,exp(A),A3)) ) ).

% exp_times_arg_commute
tff(fact_4183_norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Xa))),aa(real,real,exp(real),real_V7770717601297561774m_norm(A,Xa))) ) ).

% norm_exp
tff(fact_4184_of__real__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: real] : aa(real,A,real_Vector_of_real(A),aa(real,real,exp(real),Xa)) = aa(A,A,exp(A),aa(real,A,real_Vector_of_real(A),Xa)) ) ).

% of_real_exp
tff(fact_4185_binomial__eq__0,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K)
     => ( aa(nat,nat,binomial(Na),K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_4186_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xa))),Xa) ) ).

% of_int_floor_le
tff(fact_4187_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Y)) ) ) ).

% floor_mono
tff(fact_4188_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% floor_less_cancel
tff(fact_4189_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_4190_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_4191_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_4192_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,aa(nat,fun(nat,nat),power_power(nat),Na),R)) ) ).

% binomial_le_pow
tff(fact_4193_exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ? [X: real] : aa(real,real,exp(real),X) = Y ) ).

% exp_total
tff(fact_4194_exp__gt__zero,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,exp(real),Xa)) ).

% exp_gt_zero
tff(fact_4195_not__exp__less__zero,axiom,
    ! [Xa: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,exp(real),Xa)),zero_zero(real)) ).

% not_exp_less_zero
tff(fact_4196_exp__ge__zero,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,exp(real),Xa)) ).

% exp_ge_zero
tff(fact_4197_not__exp__le__zero,axiom,
    ! [Xa: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),Xa)),zero_zero(real)) ).

% not_exp_le_zero
tff(fact_4198_floor__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),archimedean_ceiling(A,Xa)) ) ).

% floor_le_ceiling
tff(fact_4199_mult__exp__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),Xa)),aa(A,A,exp(A),Y)) = aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) ) ).

% mult_exp_exp
tff(fact_4200_exp__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa) )
         => ( aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),Xa)),aa(A,A,exp(A),Y)) ) ) ) ).

% exp_add_commuting
tff(fact_4201_exp__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,exp(A),Xa)),aa(A,A,exp(A),Y)) ) ).

% exp_diff
tff(fact_4202_floor__le__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),archimedean_round(A,Xa)) ) ).

% floor_le_round
tff(fact_4203_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_4204_Suc__times__binomial__add,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,A2)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),aa(nat,nat,suc,A2))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,B2)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),A2)) ).

% Suc_times_binomial_add
tff(fact_4205_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),Xa) ) ) ).

% le_floor_iff
tff(fact_4206_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% floor_less_iff
tff(fact_4207_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_4208_int__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),archim6421214686448440834_floor(A,Xa)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),Xa)) ) ).

% int_add_floor
tff(fact_4209_floor__add__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),Z) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(int,A,ring_1_of_int(A),Z))) ) ).

% floor_add_int
tff(fact_4210_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Y))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))) ) ).

% le_floor_add
tff(fact_4211_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K))) ) ) ) ).

% choose_mult
tff(fact_4212_binomial__absorb__comp,axiom,
    ! [Na: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),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,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),K)) ).

% binomial_absorb_comp
tff(fact_4213_floor__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Na: nat] :
          ( ( Xa = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xa)) )
         => ( archim6421214686448440834_floor(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),archim6421214686448440834_floor(A,Xa)),Na) ) ) ) ).

% floor_power
tff(fact_4214_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_4215_exp__gt__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,exp(real),Xa)) ) ).

% exp_gt_one
tff(fact_4216_ceiling__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archimedean_ceiling(A,Xa) = aa(int,int,uminus_uminus(int),archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),Xa))) ) ).

% ceiling_def
tff(fact_4217_floor__minus,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),Xa)) = aa(int,int,uminus_uminus(int),archimedean_ceiling(A,Xa)) ) ).

% floor_minus
tff(fact_4218_ceiling__minus,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archimedean_ceiling(A,aa(A,A,uminus_uminus(A),Xa)) = aa(int,int,uminus_uminus(int),archim6421214686448440834_floor(A,Xa)) ) ).

% ceiling_minus
tff(fact_4219_exp__ge__add__one__self,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),aa(real,real,exp(real),Xa)) ).

% exp_ge_add_one_self
tff(fact_4220_exp__minus__inverse,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),Xa)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xa))) = one_one(A) ) ).

% exp_minus_inverse
tff(fact_4221_exp__of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Na: nat,Xa: 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)),Xa)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Xa)),Na) ) ).

% exp_of_nat_mult
tff(fact_4222_exp__of__nat2__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Na: nat] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),aa(nat,A,semiring_1_of_nat(A),Na))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Xa)),Na) ) ).

% exp_of_nat2_mult
tff(fact_4223_log__ln,axiom,
    ! [Xa: real] : aa(real,real,ln_ln(real),Xa) = aa(real,real,log(aa(real,real,exp(real),one_one(real))),Xa) ).

% log_ln
tff(fact_4224_exp__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( aa(B,B,exp(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),I5)) = groups7121269368397514597t_prod(A,B,aTP_Lamp_iu(fun(A,B),fun(A,B),F2),I5) ) ) ) ).

% exp_sum
tff(fact_4225_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_4226_one__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),one_one(A))) ) ).

% one_add_floor
tff(fact_4227_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),K)) ).

% binomial_absorption
tff(fact_4228_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_4229_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,archim6421214686448440834_floor(A,A2))),aa(int,nat,nat2,archim6421214686448440834_floor(A,B2)))),aa(int,nat,nat2,archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))) ) ).

% le_mult_nat_floor
tff(fact_4230_nat__floor__neg,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,Xa)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_4231_floor__eq3,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na)))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,Xa)) = Na ) ) ) ).

% floor_eq3
tff(fact_4232_exp__ge__add__one__self__aux,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),aa(real,real,exp(real),Xa)) ) ).

% exp_ge_add_one_self_aux
tff(fact_4233_le__nat__floor,axiom,
    ! [Xa: nat,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Xa)),A2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),aa(int,nat,nat2,archim6421214686448440834_floor(real,A2))) ) ).

% le_nat_floor
tff(fact_4234_ceiling__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          archimedean_ceiling(A,Xa) = $ite(Xa = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Xa),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),one_one(int))) ) ).

% ceiling_altdef
tff(fact_4235_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y)
     => ? [X: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),minus_minus(real),Y),one_one(real)))
          & ( aa(real,real,exp(real),X) = Y ) ) ) ).

% lemma_exp_total
tff(fact_4236_ln__ge__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,ln_ln(real),Xa))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),Y)),Xa) ) ) ).

% ln_ge_iff
tff(fact_4237_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,Xa)),archim6421214686448440834_floor(A,Xa))),one_one(int)) ) ).

% ceiling_diff_floor_le_1
tff(fact_4238_ln__x__over__x__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Y)),Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Xa)),Xa)) ) ) ).

% ln_x_over_x_mono
tff(fact_4239_binomial__fact__lemma,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)))),aa(nat,nat,binomial(Na),K)) = semiring_char_0_fact(nat,Na) ) ) ).

% binomial_fact_lemma
tff(fact_4240_floor__eq,axiom,
    ! [Na: int,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(int,real,ring_1_of_int(real),Na)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Na)),one_one(real)))
       => ( archim6421214686448440834_floor(real,Xa) = Na ) ) ) ).

% floor_eq
tff(fact_4241_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_4242_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_4243_real__of__int__floor__gt__diff__one,axiom,
    ! [R: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(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_4244_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,aa(real,fun(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_4245_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xa: A,A2: A] :
          powr(A,Xa,A2) = $ite(Xa = zero_zero(A),zero_zero(A),aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,ln_ln(A),Xa)))) ) ).

% powr_def
tff(fact_4246_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))
           => ( archim6421214686448440834_floor(A,Xa) = Z ) ) ) ) ).

% floor_unique
tff(fact_4247_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A2: int] :
          ( ( archim6421214686448440834_floor(A,Xa) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A2)),one_one(A))) ) ) ) ).

% floor_eq_iff
tff(fact_4248_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),T: A] :
          ( aa(int,$o,P,archim6421214686448440834_floor(A,T))
        <=> ! [I2: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I2)),T)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),T),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I2)),one_one(A))) )
             => aa(int,$o,P,I2) ) ) ) ).

% floor_split
tff(fact_4249_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_4250_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_4251_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Na),K))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_4252_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_4253_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_4254_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xa) ) ) ).

% less_floor_iff
tff(fact_4255_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))) ) ) ).

% floor_le_iff
tff(fact_4256_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A2)),archim6421214686448440834_floor(A,B2))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ) ).

% le_mult_floor
tff(fact_4257_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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% binomial_le_pow2
tff(fact_4258_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_4259_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xa))),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),one_one(int)))) ) ) ).

% floor_correct
tff(fact_4260_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_4261_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),aa(num,num,bit1,one2))) ).

% exp_le
tff(fact_4262_floor__eq4,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Na)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na)))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,Xa)) = Na ) ) ) ).

% floor_eq4
tff(fact_4263_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_4264_floor__eq2,axiom,
    ! [Na: int,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Na)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Na)),one_one(real)))
       => ( archim6421214686448440834_floor(real,Xa) = Na ) ) ) ).

% floor_eq2
tff(fact_4265_floor__divide__real__eq__div,axiom,
    ! [B2: int,A2: real] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),aa(int,real,ring_1_of_int(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),archim6421214686448440834_floor(real,A2)),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_4266_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Na: nat,Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),aa(nat,A,semiring_1_of_nat(A),Na)))),Na) = aa(A,A,exp(A),Xa) ) ) ) ).

% exp_divide_power_eq
tff(fact_4267_tanh__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,tanh(A),Xa) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,exp(A),Xa)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xa)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),Xa)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xa)))) ) ).

% tanh_altdef
tff(fact_4268_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$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),Q2)))),Q2)),P2) ) ) ).

% floor_divide_lower
tff(fact_4269_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_4270_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_4271_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_4272_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_4273_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),K)) ) ) ).

% binomial_addition_formula
tff(fact_4274_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)))) ) ) ) ).

% binomial_fact
tff(fact_4275_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),K))) ) ) ) ).

% fact_binomial
tff(fact_4276_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_4277_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,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% exp_double
tff(fact_4278_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$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),Q2)))),one_one(A))),Q2)) ) ) ).

% floor_divide_upper
tff(fact_4279_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% choose_two
tff(fact_4280_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archimedean_round(A,Xa) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% round_def
tff(fact_4281_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_4282_exp__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),Xa)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% exp_bound
tff(fact_4283_real__exp__bound__lemma,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),Xa)),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),Xa))) ) ) ).

% real_exp_bound_lemma
tff(fact_4284_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),Na))),Xa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),aa(nat,real,semiring_1_of_nat(real),Na)))),Na)),aa(real,real,exp(real),Xa)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_4285_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,semiring_1_of_nat(real),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),aa(nat,real,semiring_1_of_nat(real),Na)))),Na)),aa(real,real,exp(real),aa(real,real,uminus_uminus(real),Xa))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_4286_floor__log__eq__powr__iff,axiom,
    ! [Xa: real,B2: real,K: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(B2),Xa)) = K )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,aa(int,real,ring_1_of_int(real),K))),Xa)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),powr(real,B2,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_4287_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_4288_Maclaurin__exp__le,axiom,
    ! [Xa: real,Na: nat] :
    ? [T4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xa))
      & ( aa(real,real,exp(real),Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_it(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,exp(real),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ).

% Maclaurin_exp_le
tff(fact_4289_exp__lower__Taylor__quadratic,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,exp(real),Xa)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_4290_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_4291_log__base__10__eq2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),Xa) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),aa(real,real,exp(real),one_one(real)))),aa(real,real,ln_ln(real),Xa)) ) ) ).

% log_base_10_eq2
tff(fact_4292_tanh__real__altdef,axiom,
    ! [Xa: real] : aa(real,real,tanh(real),Xa) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(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)))),Xa)))),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)))),Xa)))) ).

% tanh_real_altdef
tff(fact_4293_floor__log__nat__eq__if,axiom,
    ! [B2: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Na)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
         => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),Na) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_4294_log__base__10__eq1,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),Xa) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),aa(real,real,exp(real),one_one(real)))),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))))),aa(real,real,ln_ln(real),Xa)) ) ) ).

% log_base_10_eq1
tff(fact_4295_central__binomial__lower__bound,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),Na)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na)))),aa(nat,real,semiring_1_of_nat(real),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_4296_vebt__maxti__hT,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : time_htt(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_maxti(Tia),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_iv(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),T),Tia),one_one(nat)) ).

% vebt_maxti_hT
tff(fact_4297_vebt__maxti__h,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_maxti(Tia),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_iv(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),T),Tia)) ).

% vebt_maxti_h
tff(fact_4298_set__vebt__maxt_H_H,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( vEBT_vebt_maxt(T) = $ite(vEBT_set_vebt(T) = bot_bot(set(nat)),none(nat),some(nat,lattic643756798349783984er_Max(nat,vEBT_set_vebt(T)))) ) ) ).

% set_vebt_maxt''
tff(fact_4299_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_iw(nat,fun(nat,A),Na)),set_ord_atMost(nat,Na))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ) ) ).

% choose_odd_sum
tff(fact_4300_TBOUND__vebt__maxti,axiom,
    ! [T: vEBT_VEBTi] : time_TBOUND(option(nat),vEBT_vebt_maxti(T),one_one(nat)) ).

% TBOUND_vebt_maxti
tff(fact_4301_Max__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A] : lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = Xa ) ).

% Max_singleton
tff(fact_4302_Max__divisors__self__nat,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => ( lattic643756798349783984er_Max(nat,collect(nat,aTP_Lamp_cu(nat,fun(nat,$o),Na))) = Na ) ) ).

% Max_divisors_self_nat
tff(fact_4303_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,A3)),Xa)
            <=> ! [X4: A] :
                  ( member(A,X4,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_4304_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),lattic643756798349783984er_Max(A,A3)),Xa)
            <=> ! [X4: A] :
                  ( member(A,X4,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Xa) ) ) ) ) ) ).

% Max_less_iff
tff(fact_4305_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_4306_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_4307_atMost__0,axiom,
    set_ord_atMost(nat,zero_zero(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_4308_not__empty__eq__Iic__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [H: A] : bot_bot(set(A)) != set_ord_atMost(A,H) ) ).

% not_empty_eq_Iic_eq_empty
tff(fact_4309_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_atMost(A,U) = collect(A,aTP_Lamp_ix(A,fun(A,$o),U)) ) ).

% atMost_def
tff(fact_4310_atMost__Suc,axiom,
    ! [K: nat] : set_ord_atMost(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,K)),set_ord_atMost(nat,K)) ).

% atMost_Suc
tff(fact_4311_Max__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => member(A,lattic643756798349783984er_Max(A,A3),A3) ) ) ) ).

% Max_in
tff(fact_4312_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( lattic643756798349783984er_Max(A,A3) = M )
            <=> ( member(A,M,A3)
                & ! [X4: A] :
                    ( member(A,X4,A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),M) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_4313_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),lattic643756798349783984er_Max(A,A3))
            <=> ? [X4: A] :
                  ( member(A,X4,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X4) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_4314_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( M = lattic643756798349783984er_Max(A,A3) )
            <=> ( member(A,M,A3)
                & ! [X4: A] :
                    ( member(A,X4,A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),M) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_4315_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,A3)),Xa)
             => ! [A8: A] :
                  ( member(A,A8,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A8),Xa) ) ) ) ) ) ).

% Max.boundedE
tff(fact_4316_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( member(A,A5,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),Xa) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,A3)),Xa) ) ) ) ) ).

% Max.boundedI
tff(fact_4317_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),lattic643756798349783984er_Max(A,A3))
            <=> ? [X4: A] :
                  ( member(A,X4,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X4) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_4318_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( finite_finite2(A,A3)
         => ( ! [B6: A] :
                ( member(A,B6,A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B6),A2) )
           => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)) = A2 ) ) ) ) ).

% Max_insert2
tff(fact_4319_atMost__nat__numeral,axiom,
    ! [K: num] : set_ord_atMost(nat,aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),aa(nat,fun(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_4320_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atMost(A,A2)),set_ord_lessThan(A,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% Iic_subset_Iio_iff
tff(fact_4321_sum__choose__upper,axiom,
    ! [M: nat,Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_iy(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_4322_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( finite_finite2(A,B3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,A3)),lattic643756798349783984er_Max(A,B3)) ) ) ) ) ).

% Max.subset_imp
tff(fact_4323_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M7: set(A),N3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M7),N3)
         => ( ( M7 != bot_bot(set(A)) )
           => ( finite_finite2(A,N3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,M7)),lattic643756798349783984er_Max(A,N3)) ) ) ) ) ).

% Max_mono
tff(fact_4324_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_es(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Na))) ) ).

% sum.atMost_Suc_shift
tff(fact_4325_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_gg(fun(nat,A),fun(nat,A),F2)),set_ord_atMost(nat,I)) = aa(A,A,aa(A,fun(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_4326_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Na: nat,D2: fun(nat,A)] :
          ( ! [X4: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_iz(fun(nat,A),fun(A,fun(nat,A)),C2),X4)),set_ord_atMost(nat,Na)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_iz(fun(nat,A),fun(A,fun(nat,A)),D2),X4)),set_ord_atMost(nat,Na))
        <=> ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Na)
             => ( aa(nat,A,C2,I2) = aa(nat,A,D2,I2) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_4327_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_hr(fun(nat,A),fun(nat,A),G),set_ord_atMost(nat,Na))) ) ).

% prod.atMost_Suc_shift
tff(fact_4328_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: fun(nat,A),B3: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,A2,N))
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,A2),set_ord_atMost(nat,N))),B3)
           => summable(A,A2) ) ) ) ).

% bounded_imp_summable
tff(fact_4329_divide__nat__def,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = $ite(Na = zero_zero(nat),zero_zero(nat),lattic643756798349783984er_Max(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ja(nat,fun(nat,fun(nat,$o)),M),Na)))) ).

% divide_nat_def
tff(fact_4330_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_jb(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_ord_atMost(nat,Na)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jd(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Na)),set_ord_lessThan(nat,Na)) ) ).

% sum.nested_swap'
tff(fact_4331_ivl__disj__un__singleton_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_lessThan(A,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = set_ord_atMost(A,U) ) ).

% ivl_disj_un_singleton(2)
tff(fact_4332_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_4333_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Na: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_je(fun(nat,fun(nat,A)),fun(nat,A),A2),set_ord_atMost(nat,Na)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jg(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Na),set_ord_lessThan(nat,Na)) ) ).

% prod.nested_swap'
tff(fact_4334_sum__choose__lower,axiom,
    ! [R: nat,Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_jh(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_4335_choose__rising__sum_I1_J,axiom,
    ! [Na: nat,M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ji(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_4336_choose__rising__sum_I2_J,axiom,
    ! [Na: nat,M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ji(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_4337_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C2: fun(nat,A),Na: nat,K: nat] :
          ( ! [W2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jj(fun(nat,A),fun(A,fun(nat,A)),C2),W2)),set_ord_atMost(nat,Na)) = zero_zero(A)
         => ( aa(nat,$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_4338_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Na: nat] :
          ( ! [X4: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_iz(fun(nat,A),fun(A,fun(nat,A)),C2),X4)),set_ord_atMost(nat,Na)) = zero_zero(A)
        <=> ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Na)
             => ( aa(nat,A,C2,I2) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_4339_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_es(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Na))) ) ).

% sum.atMost_shift
tff(fact_4340_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_4341_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_hr(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,Na))) ) ).

% prod.atMost_shift
tff(fact_4342_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_jk(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_4343_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_jl(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_4344_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xa)),set_ord_atMost(nat,Na))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,Na))) ) ).

% sum_gp_basic
tff(fact_4345_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Na: nat] :
          ( finite_finite2(A,collect(A,aa(nat,fun(A,$o),aTP_Lamp_jm(fun(nat,A),fun(nat,fun(A,$o)),C2),Na)))
        <=> ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Na)
              & ( aa(nat,A,C2,I2) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_4346_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,$o),ord_less_eq(nat),K),Na)
           => finite_finite2(A,collect(A,aa(nat,fun(A,$o),aTP_Lamp_jm(fun(nat,A),fun(nat,fun(A,$o)),C2),Na))) ) ) ) ).

% polyfun_roots_finite
tff(fact_4347_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),A2: A,Na: nat] :
          ( ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),set_ord_atMost(nat,Na)) = zero_zero(A) )
         => ~ ! [B6: fun(nat,A)] :
                ~ ! [Z5: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(A,fun(nat,A)),C2),Z5)),set_ord_atMost(nat,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z5),A2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(A,fun(nat,A)),B6),Z5)),set_ord_lessThan(nat,Na))) ) ) ).

% polyfun_linear_factor_root
tff(fact_4348_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),Na: nat,A2: A] :
        ? [B6: fun(nat,A)] :
        ! [Z5: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(A,fun(nat,A)),C2),Z5)),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,aa(A,fun(A,A),minus_minus(A),Z5),A2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(A,fun(nat,A)),B6),Z5)),set_ord_lessThan(nat,Na)))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),set_ord_atMost(nat,Na))) ) ).

% polyfun_linear_factor
tff(fact_4349_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Na: nat,Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xa)),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xa)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)))) ) ) ) ).

% sum_power_shift
tff(fact_4350_atLeast1__atMost__eq__remove0,axiom,
    ! [Na: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Na) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_atMost(nat,Na)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_4351_choose__row__sum,axiom,
    ! [Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,binomial(Na)),set_ord_atMost(nat,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ).

% choose_row_sum
tff(fact_4352_summable__Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_jo(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_jo(fun(nat,A),fun(nat,real),B2))
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_jq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ).

% summable_Cauchy_product
tff(fact_4353_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_jo(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_jo(fun(nat,A),fun(nat,real),B2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A2)),suminf(A,B2)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_jq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ) ).

% Cauchy_product
tff(fact_4354_binomial,axiom,
    ! [A2: nat,B2: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),Na) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_jr(nat,fun(nat,fun(nat,fun(nat,nat))),A2),B2),Na)),set_ord_atMost(nat,Na)) ).

% binomial
tff(fact_4355_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_fi(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Na)) ) ).

% sum.in_pairs_0
tff(fact_4356_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [M: nat,A2: fun(nat,A),Na: nat,B2: fun(nat,A),Xa: A] :
          ( ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),I3)
             => ( aa(nat,A,A2,I3) = zero_zero(A) ) )
         => ( ! [J3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),J3)
               => ( aa(nat,A,B2,J3) = 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_jn(fun(nat,A),fun(A,fun(nat,A)),A2),Xa)),set_ord_atMost(nat,M))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(A,fun(nat,A)),B2),Xa)),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_jt(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A2),B2),Xa)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) ) ) ) ) ).

% polynomial_product
tff(fact_4357_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_id(fun(nat,A),fun(nat,A),G),set_ord_atMost(nat,Na)) ) ).

% prod.in_pairs_0
tff(fact_4358_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Na: nat,K: A] :
          ( ! [X4: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_iz(fun(nat,A),fun(A,fun(nat,A)),C2),X4)),set_ord_atMost(nat,Na)) = K
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X4: nat] :
                ( member(nat,X4,set_or1337092689740270186AtMost(nat,one_one(nat),Na))
               => ( aa(nat,A,C2,X4) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_4359_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),Na) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ju(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Na)),set_ord_atMost(nat,Na)) ) ).

% binomial_ring
tff(fact_4360_polynomial__product__nat,axiom,
    ! [M: nat,A2: fun(nat,nat),Na: nat,B2: fun(nat,nat),Xa: nat] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),I3)
         => ( aa(nat,nat,A2,I3) = zero_zero(nat) ) )
     => ( ! [J3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),J3)
           => ( aa(nat,nat,B2,J3) = 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_jv(fun(nat,nat),fun(nat,fun(nat,nat)),A2),Xa)),set_ord_atMost(nat,M))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_jv(fun(nat,nat),fun(nat,fun(nat,nat)),B2),Xa)),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_jx(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A2),B2),Xa)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) ) ) ) ).

% polynomial_product_nat
tff(fact_4361_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,B2: A,Na: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),Na) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_jy(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Na)),set_ord_atMost(nat,Na)) ) ).

% pochhammer_binomial_sum
tff(fact_4362_choose__square__sum,axiom,
    ! [Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_jz(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_4363_Cauchy__product__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_jo(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_jo(fun(nat,A),fun(nat,real),B2))
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_jq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A2)),suminf(A,B2))) ) ) ) ).

% Cauchy_product_sums
tff(fact_4364_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_ka(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_kb(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_4365_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_kc(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_kd(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_4366_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,Z: A,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Na) = A2 )
          <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_ke(nat,fun(A,fun(A,fun(nat,A))),Na),Z),A2)),set_ord_atMost(nat,Na)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_4367_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xa: A,Na: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xa)),set_ord_atMost(nat,Na)) = $ite(Xa = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,Na)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa))) ) ).

% sum_gp0
tff(fact_4368_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_kf(nat,fun(nat,A),Na)),set_ord_atMost(nat,Na)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_4369_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,A2: fun(nat,A),Xa: A,Y: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(A,fun(nat,A)),A2),Xa)),set_ord_atMost(nat,Na))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),set_ord_atMost(nat,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)),aa(set(nat),A,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_kh(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Na),A2),Xa),Y)),set_ord_lessThan(nat,Na))) ) ) ) ).

% polyfun_diff_alt
tff(fact_4370_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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ).

% binomial_r_part_sum
tff(fact_4371_choose__linear__sum,axiom,
    ! [Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ki(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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))) ).

% choose_linear_sum
tff(fact_4372_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_kj(nat,fun(nat,A),Na)),set_ord_atMost(nat,Na)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_4373_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E: real,C2: fun(nat,A),Na: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
         => ? [M8: real] :
            ! [Z5: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),M8),real_V7770717601297561774m_norm(A,Z5))
             => aa(real,$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_bz(fun(nat,A),fun(A,fun(nat,A)),C2),Z5)),set_ord_atMost(nat,Na)))),aa(real,real,aa(real,fun(real,real),times_times(real),E),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Z5)),aa(nat,nat,suc,Na)))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_4374_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,A2: fun(nat,A),Xa: A,Y: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(A,fun(nat,A)),A2),Xa)),set_ord_atMost(nat,Na))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),set_ord_atMost(nat,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)),aa(set(nat),A,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_kl(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Na),A2),Xa),Y)),set_ord_lessThan(nat,Na))) ) ) ) ).

% polyfun_diff
tff(fact_4375_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_km(nat,fun(nat,A),Na)),set_ord_atMost(nat,Na))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ) ) ).

% choose_even_sum
tff(fact_4376_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ko(A,fun(A,fun(nat,A)),Xa),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sin(A),Xa)),aa(A,A,sin(A),Y))) ) ).

% sin_x_sin_y
tff(fact_4377_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_kq(A,fun(A,fun(nat,A)),Xa),Y)),aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))) ) ).

% sums_cos_x_plus_y
tff(fact_4378_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ks(A,fun(A,fun(nat,A)),Xa),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),Xa)),aa(A,A,cos(A),Y))) ) ).

% cos_x_cos_y
tff(fact_4379_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_kt(A,fun(nat,A),A2)),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,A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_4380_mult__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [Xa: A,A2: real,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Xa),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) = aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y)) ) ).

% mult_scaleR_right
tff(fact_4381_mult__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [A2: real,Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),Y) = aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y)) ) ).

% mult_scaleR_left
tff(fact_4382_scaleR__scaleR,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,Xa: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xa)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),times_times(real),A2),B2)),Xa) ) ).

% scaleR_scaleR
tff(fact_4383_scaleR__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: A,U: real,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,U),B2)) )
        <=> ( ( A2 = B2 )
            | ( U = one_one(real) ) ) ) ) ).

% scaleR_eq_iff
tff(fact_4384_scaleR__power,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xa: real,Y: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,real_V8093663219630862766scaleR(A,Xa),Y)),Na) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Na)) ) ).

% scaleR_power
tff(fact_4385_Max__divisors__self__int,axiom,
    ! [Na: int] :
      ( ( Na != zero_zero(int) )
     => ( lattic643756798349783984er_Max(int,collect(int,aTP_Lamp_df(int,fun(int,$o),Na))) = aa(int,int,abs_abs(int),Na) ) ) ).

% Max_divisors_self_int
tff(fact_4386_scaleR__collapse,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),U)),A2)),aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) = A2 ) ).

% scaleR_collapse
tff(fact_4387_norm__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: real,Xa: A] : real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,abs_abs(real),A2)),real_V7770717601297561774m_norm(A,Xa)) ) ).

% norm_scaleR
tff(fact_4388_scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,W: num,A2: 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)),A2)) = 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))),A2) ) ).

% scaleR_times
tff(fact_4389_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [V: num,W: num,A2: 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)),A2)) = 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))),A2) ) ).

% inverse_scaleR_times
tff(fact_4390_fraction__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,V: num,W: num,A2: 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)),A2)) = 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))),A2) ) ).

% fraction_scaleR_times
tff(fact_4391_scaleR__half__double,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: 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),A2),A2)) = A2 ) ).

% scaleR_half_double
tff(fact_4392_summable__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A),R: real] :
          ( summable(A,X7)
         => summable(A,aa(real,fun(nat,A),aTP_Lamp_ku(fun(nat,A),fun(real,fun(nat,A)),X7),R)) ) ) ).

% summable_scaleR_right
tff(fact_4393_scaleR__sum__right,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,F2: fun(B,A),A3: set(B)] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kv(real,fun(fun(B,A),fun(B,A)),A2),F2)),A3) ) ).

% scaleR_sum_right
tff(fact_4394_scaleR__right_Osum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,G: fun(B,A),A3: set(B)] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kv(real,fun(fun(B,A),fun(B,A)),A2),G)),A3) ) ).

% scaleR_right.sum
tff(fact_4395_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_4396_real__scaleR__def,axiom,
    ! [A2: real,Xa: real] : aa(real,real,real_V8093663219630862766scaleR(real,A2),Xa) = aa(real,real,aa(real,fun(real,real),times_times(real),A2),Xa) ).

% real_scaleR_def
tff(fact_4397_scaleR__right__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xa: A,Y: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ).

% scaleR_right_distrib
tff(fact_4398_sums__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A),A2: A,R: real] :
          ( aa(A,$o,sums(A,X7),A2)
         => aa(A,$o,sums(A,aa(real,fun(nat,A),aTP_Lamp_ku(fun(nat,A),fun(real,fun(nat,A)),X7),R)),aa(A,A,real_V8093663219630862766scaleR(A,R),A2)) ) ) ).

% sums_scaleR_right
tff(fact_4399_scaleR__left_Oadd,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: real,Y: real,Xaa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)),Xaa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Xa),Xaa)),aa(A,A,real_V8093663219630862766scaleR(A,Y),Xaa)) ) ).

% scaleR_left.add
tff(fact_4400_scaleR__left__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,Xa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),Xa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xa)) ) ).

% scaleR_left_distrib
tff(fact_4401_scaleR__conv__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R: real,Xa: A] : aa(A,A,real_V8093663219630862766scaleR(A,R),Xa) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(real,A,real_Vector_of_real(A),R)),Xa) ) ).

% scaleR_conv_of_real
tff(fact_4402_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_4403_scaleR__left_Osum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [G: fun(B,real),A3: set(B),Xa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(set(B),real,groups7311177749621191930dd_sum(B,real,G),A3)),Xa) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_kw(fun(B,real),fun(A,fun(B,A)),G),Xa)),A3) ) ).

% scaleR_left.sum
tff(fact_4404_scaleR__sum__left,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [F2: fun(B,real),A3: set(B),Xa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(set(B),real,groups7311177749621191930dd_sum(B,real,F2),A3)),Xa) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_kw(fun(B,real),fun(A,fun(B,A)),F2),Xa)),A3) ) ).

% scaleR_sum_left
tff(fact_4405_suminf__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A),R: real] :
          ( summable(A,X7)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,R),suminf(A,X7)) = suminf(A,aa(real,fun(nat,A),aTP_Lamp_ku(fun(nat,A),fun(real,fun(nat,A)),X7),R)) ) ) ) ).

% suminf_scaleR_right
tff(fact_4406_summable__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,real),Xa: A] :
          ( summable(real,X7)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_kx(fun(nat,real),fun(A,fun(nat,A)),X7),Xa)) ) ) ).

% summable_scaleR_left
tff(fact_4407_sums__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,real),A2: real,Xa: A] :
          ( aa(real,$o,sums(real,X7),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_kx(fun(nat,real),fun(A,fun(nat,A)),X7),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)) ) ) ).

% sums_scaleR_left
tff(fact_4408_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,A2),K)),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc_Suc
tff(fact_4409_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_4410_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_4411_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(real,$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),B2),A2) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_4412_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A,U: real,V: real,A2: A] :
          ( ( Xa = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V)),A2) )
        <=> $ite(V = zero_zero(real),Xa = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,V),Xa) = aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) ) ) ).

% eq_vector_fraction_iff
tff(fact_4413_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,V: real,A2: A,Xa: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V)),A2) = Xa )
        <=> $ite(V = zero_zero(real),Xa = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,U),A2) = aa(A,A,real_V8093663219630862766scaleR(A,V),Xa)) ) ) ).

% vector_fraction_eq_iff
tff(fact_4414_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E: A,C2: A,B2: real,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2)),E)),C2)),D2) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_4415_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E: A,C2: A,B2: real,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E)),D2))
        <=> aa(A,$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,aa(real,fun(real,real),minus_minus(real),B2),A2)),E)),D2)) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_4416_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_4417_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_4418_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)),zero_zero(A))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_4419_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_4420_suminf__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,real),Xa: A] :
          ( summable(real,X7)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,suminf(real,X7)),Xa) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_kx(fun(nat,real),fun(A,fun(nat,A)),X7),Xa)) ) ) ) ).

% suminf_scaleR_left
tff(fact_4421_scaleR__2,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),bit0(one2))),Xa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Xa) ) ).

% scaleR_2
tff(fact_4422_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K)) ) ).

% gbinomial_addition_formula
tff(fact_4423_gbinomial__absorb__comp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,A2),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K)) ) ).

% gbinomial_absorb_comp
tff(fact_4424_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,A2),K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A2),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1
tff(fact_4425_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A2),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1'
tff(fact_4426_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,gbinomial(A,A2),K)) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_4427_sin__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,$o,sums(A,aTP_Lamp_ky(A,fun(nat,A),Xa)),aa(A,A,sin(A),Xa)) ) ).

% sin_converges
tff(fact_4428_sin__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X5: A] : aa(A,A,sin(A),X5) = suminf(A,aTP_Lamp_ky(A,fun(nat,A),X5)) ) ).

% sin_def
tff(fact_4429_cos__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,$o,sums(A,aTP_Lamp_kz(A,fun(nat,A),Xa)),aa(A,A,cos(A),Xa)) ) ).

% cos_converges
tff(fact_4430_cos__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X5: A] : aa(A,A,cos(A),X5) = suminf(A,aTP_Lamp_kz(A,fun(nat,A),X5)) ) ).

% cos_def
tff(fact_4431_summable__norm__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : summable(real,aTP_Lamp_la(A,fun(nat,real),Xa)) ) ).

% summable_norm_sin
tff(fact_4432_summable__norm__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : summable(real,aTP_Lamp_lb(A,fun(nat,real),Xa)) ) ).

% summable_norm_cos
tff(fact_4433_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,gbinomial(A,A2),K)) ) ).

% Suc_times_gbinomial
tff(fact_4434_gbinomial__absorption,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K)) ) ).

% gbinomial_absorption
tff(fact_4435_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),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,A2),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_4436_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_lc(A,fun(nat,A),A2)),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),A2),aa(nat,A,semiring_1_of_nat(A),Na))),one_one(A))),Na) ) ).

% gbinomial_parallel_sum
tff(fact_4437_sin__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,$o,sums(A,aTP_Lamp_ld(A,fun(nat,A),Xa)),aa(A,A,sin(A),Xa)) ) ).

% sin_minus_converges
tff(fact_4438_cos__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,$o,sums(A,aTP_Lamp_le(A,fun(nat,A),Xa)),aa(A,A,cos(A),Xa)) ) ).

% cos_minus_converges
tff(fact_4439_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))) ) ).

% gbinomial_rec
tff(fact_4440_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),aa(nat,A,gbinomial(A,A2),K)) ) ).

% gbinomial_factors
tff(fact_4441_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,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na))),one_one(A))),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Na)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),Na)) ) ).

% gbinomial_index_swap
tff(fact_4442_gbinomial__negated__upper,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),K)),A2)),one_one(A))),K)) ) ).

% gbinomial_negated_upper
tff(fact_4443_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A2)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),K)) ) ).

% gbinomial_minus
tff(fact_4444_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_4445_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),A2),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer
tff(fact_4446_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer'
tff(fact_4447_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_lf(A,fun(nat,A),A2)),set_ord_atMost(nat,M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),M)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),M)) ) ).

% gbinomial_sum_lower_neg
tff(fact_4448_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_lg(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_4449_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A2: A,Xa: 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_lh(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),Xa),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_li(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),Xa),Y)),set_ord_atMost(nat,M)) ) ).

% gbinomial_partial_sum_poly
tff(fact_4450_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_lj(A,fun(nat,A),A2),set_or1337092689740270186AtMost(nat,zero_zero(nat),K))),semiring_char_0_fact(A,aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc
tff(fact_4451_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_4452_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_lk(nat,fun(nat,A),M)),set_ord_atMost(nat,M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M) ) ).

% gbinomial_sum_nat_pow2
tff(fact_4453_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A2: A,Xa: 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_lh(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),Xa),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_ll(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),Xa),Y)),set_ord_atMost(nat,M)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_4454_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] :
          aa(nat,A,gbinomial(A,A2),K) = $ite(K = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_lm(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)),one_one(A))),semiring_char_0_fact(A,K))) ) ).

% gbinomial_code
tff(fact_4455_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_kt(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_4456_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,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ).

% gbinomial_r_part_sum
tff(fact_4457_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          archimedean_round(A,Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,Xa)),archimedean_ceiling(A,Xa),archim6421214686448440834_floor(A,Xa)) ) ).

% round_altdef
tff(fact_4458_exp__first__two__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,A,exp(A),Xa) = 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)),Xa)),suminf(A,aTP_Lamp_ln(A,fun(nat,A),Xa))) ) ).

% exp_first_two_terms
tff(fact_4459_height__compose__summary,axiom,
    ! [Summary2: vEBT_VEBT,Info2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Summary2))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info2,Deg,TreeList,Summary2))) ).

% height_compose_summary
tff(fact_4460_Maclaurin__sin__bound,axiom,
    ! [Xa: real,Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,sin(real),Xa)),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_is(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Xa)),Na))) ).

% Maclaurin_sin_bound
tff(fact_4461_deg__deg__n,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList,Summary2),Na)
     => ( Deg = Na ) ) ).

% deg_deg_n
tff(fact_4462_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(Tree,aa(nat,nat,suc,aa(nat,nat,suc,Na)))
     => ? [Info: option(product_prod(nat,nat)),TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] : Tree = vEBT_Node(Info,aa(nat,nat,suc,aa(nat,nat,suc,Na)),TreeList2,S2) ) ).

% deg_SUcn_Node
tff(fact_4463_inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A2)) = A2 ) ).

% inverse_inverse_eq
tff(fact_4464_inverse__eq__iff__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,inverse_inverse(A),B2) )
        <=> ( A2 = B2 ) ) ) ).

% inverse_eq_iff_eq
tff(fact_4465_frac__frac,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archimedean_frac(A,archimedean_frac(A,Xa)) = archimedean_frac(A,Xa) ) ).

% frac_frac
tff(fact_4466_inverse__nonzero__iff__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_4467_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_4468_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_4469_inverse__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xa: A] :
          ( ( aa(A,A,inverse_inverse(A),Xa) = one_one(A) )
        <=> ( Xa = one_one(A) ) ) ) ).

% inverse_eq_1_iff
tff(fact_4470_inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) ) ).

% inverse_mult_distrib
tff(fact_4471_inverse__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) ) ).

% inverse_divide
tff(fact_4472_inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A2)) ) ).

% inverse_minus_eq
tff(fact_4473_abs__inverse,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A2)) ) ).

% abs_inverse
tff(fact_4474_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_4475_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_4476_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_4477_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_4478_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% inverse_negative_iff_negative
tff(fact_4479_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% inverse_positive_iff_positive
tff(fact_4480_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_4481_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_4482_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_4483_left__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),A2) = one_one(A) ) ) ) ).

% left_inverse
tff(fact_4484_right__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),A2)) = one_one(A) ) ) ) ).

% right_inverse
tff(fact_4485_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_4486_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_4487_power__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),A2)),Na) = aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ).

% power_inverse
tff(fact_4488_mult__commute__imp__mult__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Y: A,Xa: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Y)),Xa) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),aa(A,A,inverse_inverse(A),Y)) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_4489_inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,inverse_inverse(A),B2) )
         => ( A2 = B2 ) ) ) ).

% inverse_eq_imp_eq
tff(fact_4490_real__sqrt__inverse,axiom,
    ! [Xa: real] : aa(real,real,sqrt,aa(real,real,inverse_inverse(real),Xa)) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xa)) ).

% real_sqrt_inverse
tff(fact_4491_nonzero__imp__inverse__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) != zero_zero(A) ) ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_4492_nonzero__inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A2)) = A2 ) ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_4493_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,inverse_inverse(A),B2) )
         => ( ( A2 != zero_zero(A) )
           => ( ( B2 != zero_zero(A) )
             => ( A2 = B2 ) ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_4494_inverse__zero__imp__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = zero_zero(A) )
         => ( A2 = zero_zero(A) ) ) ) ).

% inverse_zero_imp_zero
tff(fact_4495_field__class_Ofield__inverse__zero,axiom,
    ! [A: $tType] :
      ( field(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% field_class.field_inverse_zero
tff(fact_4496_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R: real,Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R),real_V7770717601297561774m_norm(A,Xa))
         => ( aa(real,$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),Xa))),aa(real,real,inverse_inverse(real),R)) ) ) ) ).

% norm_inverse_le_norm
tff(fact_4497_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% inverse_less_imp_less
tff(fact_4498_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% less_imp_inverse_less
tff(fact_4499_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_4500_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_4501_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
         => ( ( A2 != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_4502_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
         => ( ( A2 != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_4503_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A)) ) ) ).

% negative_imp_inverse_negative
tff(fact_4504_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ).

% positive_imp_inverse_positive
tff(fact_4505_nonzero__inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_4506_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_4507_inverse__unique,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = one_one(A) )
         => ( aa(A,A,inverse_inverse(A),A2) = B2 ) ) ) ).

% inverse_unique
tff(fact_4508_nonzero__inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_4509_inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ).

% inverse_eq_divide
tff(fact_4510_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),B2)) ) ).

% field_class.field_divide_inverse
tff(fact_4511_divide__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),B2)) ) ).

% divide_inverse
tff(fact_4512_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B2)),A2) ) ).

% divide_inverse_commute
tff(fact_4513_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xa)),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xa)),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)) ) ).

% power_mult_power_inverse_commute
tff(fact_4514_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)),aa(A,A,inverse_inverse(A),Xa)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)) ) ).

% power_mult_inverse_distrib
tff(fact_4515_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: nat,Xb: 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),Xa))),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa))) ) ).

% mult_inverse_of_nat_commute
tff(fact_4516_nonzero__abs__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A2)) ) ) ) ).

% nonzero_abs_inverse
tff(fact_4517_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: int,Xb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa))),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa))) ) ).

% mult_inverse_of_int_commute
tff(fact_4518_exp__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xa)) = aa(A,A,inverse_inverse(A),aa(A,A,exp(A),Xa)) ) ).

% exp_minus
tff(fact_4519_powr__minus,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xa: A,A2: A] : powr(A,Xa,aa(A,A,uminus_uminus(A),A2)) = aa(A,A,inverse_inverse(A),powr(A,Xa,A2)) ) ).

% powr_minus
tff(fact_4520_divide__real__def,axiom,
    ! [Xa: real,Y: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),Y) = aa(real,real,aa(real,fun(real,real),times_times(real),Xa),aa(real,real,inverse_inverse(real),Y)) ).

% divide_real_def
tff(fact_4521_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,Xa)) ) ).

% frac_ge_0
tff(fact_4522_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),archimedean_frac(A,Xa)),one_one(A)) ) ).

% frac_lt_1
tff(fact_4523_frac__1__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),one_one(A))) = archimedean_frac(A,Xa) ) ).

% frac_1_eq
tff(fact_4524_exp__fdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X5: nat] : aa(nat,A,diffs(A,aTP_Lamp_lo(nat,A)),X5) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,X5)) ) ).

% exp_fdiffs
tff(fact_4525_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_4526_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_4527_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% le_imp_inverse_le
tff(fact_4528_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% inverse_le_imp_le
tff(fact_4529_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),Xa)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa) ) ) ) ).

% inverse_le_1_iff
tff(fact_4530_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),Xa))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ) ).

% one_less_inverse_iff
tff(fact_4531_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% one_less_inverse
tff(fact_4532_division__ring__inverse__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_add
tff(fact_4533_inverse__add,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,inverse_inverse(A),A2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% inverse_add
tff(fact_4534_field__class_Ofield__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),A2) = one_one(A) ) ) ) ).

% field_class.field_inverse
tff(fact_4535_division__ring__inverse__diff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_4536_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_4537_inverse__powr,axiom,
    ! [Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( powr(real,aa(real,real,inverse_inverse(real),Y),A2) = aa(real,real,inverse_inverse(real),powr(real,Y,A2)) ) ) ).

% inverse_powr
tff(fact_4538_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),Xa))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),one_one(A)) ) ) ) ).

% one_le_inverse_iff
tff(fact_4539_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),Xa)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa) ) ) ) ).

% inverse_less_1_iff
tff(fact_4540_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% one_le_inverse
tff(fact_4541_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ) ).

% inverse_less_iff
tff(fact_4542_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ).

% inverse_le_iff
tff(fact_4543_inverse__diff__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))),aa(A,A,inverse_inverse(A),B2))) ) ) ) ) ).

% inverse_diff_inverse
tff(fact_4544_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)))),Xa) ) ) ).

% reals_Archimedean
tff(fact_4545_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,Xa: 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),Xa)),C2) = Y )
          <=> ( Xa = aa(A,A,aa(A,fun(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_4546_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,Y: A,Xa: 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),Xa)),C2) )
          <=> ( aa(A,A,aa(A,fun(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)) = Xa ) ) ) ) ).

% real_vector_eq_affinity
tff(fact_4547_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_divideR_le_eq
tff(fact_4548_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% pos_le_divideR_eq
tff(fact_4549_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% neg_divideR_le_eq
tff(fact_4550_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_le_divideR_eq
tff(fact_4551_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_less_divideR_eq
tff(fact_4552_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% neg_divideR_less_eq
tff(fact_4553_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% pos_less_divideR_eq
tff(fact_4554_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_divideR_less_eq
tff(fact_4555_forall__pos__mono__1,axiom,
    ! [P: fun(real,$o),E: real] :
      ( ! [D3: real,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D3),E2)
         => ( aa(real,$o,P,D3)
           => aa(real,$o,P,E2) ) )
     => ( ! [N: nat] : aa(real,$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)),E)
         => aa(real,$o,P,E) ) ) ) ).

% forall_pos_mono_1
tff(fact_4556_real__arch__inverse,axiom,
    ! [E: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
    <=> ? [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))),E) ) ) ).

% real_arch_inverse
tff(fact_4557_forall__pos__mono,axiom,
    ! [P: fun(real,$o),E: real] :
      ( ! [D3: real,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D3),E2)
         => ( aa(real,$o,P,D3)
           => aa(real,$o,P,E2) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero(nat) )
           => aa(real,$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)),E)
         => aa(real,$o,P,E) ) ) ) ).

% forall_pos_mono
tff(fact_4558_sqrt__divide__self__eq,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,Xa)),Xa) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xa)) ) ) ).

% sqrt_divide_self_eq
tff(fact_4559_ln__inverse,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,ln_ln(real),aa(real,real,inverse_inverse(real),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,ln_ln(real),Xa)) ) ) ).

% ln_inverse
tff(fact_4560_frac__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : archimedean_frac(A,Xa) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xa))) ) ).

% frac_def
tff(fact_4561_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : summable(A,aTP_Lamp_lp(A,fun(nat,A),Xa)) ) ).

% summable_exp
tff(fact_4562_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Xa: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw),Xa) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
tff(fact_4563_summable__exp__generic,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : summable(A,aTP_Lamp_lq(A,fun(nat,A),Xa)) ) ).

% summable_exp_generic
tff(fact_4564_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ? [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N))),Xa) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_4565_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M: nat,Na: nat] :
          ( ( Xa != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xa)),M)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_4566_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_4567_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_4568_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_4569_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_4570_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_4571_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_4572_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_4573_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_4574_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( ( archimedean_frac(A,Xa) = Xa )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ) ).

% frac_eq
tff(fact_4575_log__inverse,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,log(A2),aa(real,real,inverse_inverse(real),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,log(A2),Xa)) ) ) ) ) ).

% log_inverse
tff(fact_4576_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),archimedean_frac(A,Y))),one_one(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),archimedean_frac(A,Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),archimedean_frac(A,Y))),one_one(A))) ) ).

% frac_add
tff(fact_4577_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList,Summary2),Xa) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
tff(fact_4578_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Xa: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(some(product_prod(nat,nat),V),zero_zero(nat),Uy,Uz),Xa) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
tff(fact_4579_exp__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,$o,sums(A,aTP_Lamp_lq(A,fun(nat,A),Xa)),aa(A,A,exp(A),Xa)) ) ).

% exp_converges
tff(fact_4580_exp__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X5: A] : aa(A,A,exp(A),X5) = suminf(A,aTP_Lamp_lq(A,fun(nat,A),X5)) ) ).

% exp_def
tff(fact_4581_summable__norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : summable(real,aTP_Lamp_lr(A,fun(nat,real),Xa)) ) ).

% summable_norm_exp
tff(fact_4582_exp__plus__inverse__exp,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,exp(real),Xa)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),Xa)))) ).

% exp_plus_inverse_exp
tff(fact_4583_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Xa: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(some(product_prod(nat,nat),V),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc),Xa) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
tff(fact_4584_plus__inverse__ge__2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,inverse_inverse(real),Xa))) ) ).

% plus_inverse_ge_2
tff(fact_4585_real__inv__sqrt__pow2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xa))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,inverse_inverse(real),Xa) ) ) ).

% real_inv_sqrt_pow2
tff(fact_4586_tan__cot,axiom,
    ! [Xa: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(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)))),Xa)) = aa(real,real,inverse_inverse(real),aa(real,real,tan(real),Xa)) ).

% tan_cot
tff(fact_4587_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),archimedean_frac(A,Y))),one_one(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Y)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Y))),one_one(int))) ) ).

% floor_add
tff(fact_4588_real__le__x__sinh,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,exp(real),Xa)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),Xa)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% real_le_x_sinh
tff(fact_4589_real__le__abs__sinh,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,exp(real),Xa)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),Xa)))),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% real_le_abs_sinh
tff(fact_4590_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A,Y: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa) )
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Na))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),Na)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ls(A,fun(A,fun(nat,fun(nat,A))),Xa),Y),Na)),set_ord_atMost(nat,Na)) ) ) ) ).

% exp_series_add_commuting
tff(fact_4591_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,A,exp(A),Xa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),suminf(A,aTP_Lamp_lt(A,fun(nat,A),Xa))) ) ).

% exp_first_term
tff(fact_4592_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( aa(A,A,cos(A),Xa) != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),aa(A,A,cos(A),Xa))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ) ).

% tan_sec
tff(fact_4593_powr__real__of__int,axiom,
    ! [Xa: real,Na: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(int,real,ring_1_of_int(real),Na)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(int,nat,nat2,Na)),aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Na))))) ) ) ).

% powr_real_of_int
tff(fact_4594_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A,K: nat] : aa(A,A,exp(A),Xa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_lq(A,fun(nat,A),Xa)),set_ord_lessThan(nat,K))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_lu(A,fun(nat,fun(nat,A)),Xa),K))) ) ).

% exp_first_terms
tff(fact_4595_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_VEBT_insert(vEBT_Node(Info2,Deg,TreeList,Summary2),Xa) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)),Xa),vEBT_Node(Info2,Deg,TreeList,Summary2),vEBT_vebt_insert(vEBT_Node(Info2,Deg,TreeList,Summary2),Xa)) ).

% VEBT_internal.insert'.simps(2)
tff(fact_4596_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Na)
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Ma)
          & ? [M4: nat] :
              ( ( some(nat,M4) = vEBT_vebt_mint(Summary2) )
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ).

% mintlistlength
tff(fact_4597_both__member__options__ding,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList,Summary2),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(Info2,Deg,TreeList,Summary2)),Xa) ) ) ) ).

% both_member_options_ding
tff(fact_4598_pred__max,axiom,
    ! [Deg: nat,Ma: nat,Xa: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa)
       => ( vEBT_vebt_pred(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = some(nat,Ma) ) ) ) ).

% pred_max
tff(fact_4599_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Xa)
       => ( vEBT_vebt_succ(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = none(nat) ) ) ) ).

% geqmaxNone
tff(fact_4600_insert__simp__mima,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        | ( Xa = Ma ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
       => ( vEBT_vebt_insert(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2) ) ) ) ).

% insert_simp_mima
tff(fact_4601_delt__out__of__range,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
       => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2) ) ) ) ).

% delt_out_of_range
tff(fact_4602_del__single__cont,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & ( Xa = Ma ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
       => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary2) ) ) ) ).

% del_single_cont
tff(fact_4603_tdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_d_e_l_e_t_e(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Mi)),Deg,TreeList,Summary2),Xa)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))) ) ).

% tdeletemimi
tff(fact_4604_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V1232361888498592333_e_t_e(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Mi)),Deg,TreeList,Summary2),Xa)),one_one(nat)) ) ).

% tdeletemimi'
tff(fact_4605_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)) ) ) ).

% mi_ma_2_deg
tff(fact_4606_succ__min,axiom,
    ! [Deg: nat,Xa: nat,Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
       => ( vEBT_vebt_succ(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = some(nat,Mi) ) ) ) ).

% succ_min
tff(fact_4607_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Deg)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2)),Xa)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
          | ( Xa = Mi )
          | ( Xa = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_4608_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B)),X5: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),aTP_Lamp_lv(set(product_prod(A,B)),fun(A,fun(B,$o)),R2)),aTP_Lamp_lv(set(product_prod(A,B)),fun(A,fun(B,$o)),S)),X5),Xa2)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa2),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),R2),S)) ) ).

% sup_Un_eq2
tff(fact_4609_inf__Int__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B)),X5: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),aTP_Lamp_lv(set(product_prod(A,B)),fun(A,fun(B,$o)),R2)),aTP_Lamp_lv(set(product_prod(A,B)),fun(A,fun(B,$o)),S)),X5),Xa2)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa2),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),R2),S)) ) ).

% inf_Int_eq2
tff(fact_4610_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( ! [X4: A,Xa3: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3),R2)
        <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3),S) )
    <=> ( R2 = S ) ) ).

% pred_equals_eq2
tff(fact_4611_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),aTP_Lamp_lv(set(product_prod(A,B)),fun(A,fun(B,$o)),R2)),aTP_Lamp_lv(set(product_prod(A,B)),fun(A,fun(B,$o)),S))
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S) ) ).

% pred_subset_eq2
tff(fact_4612_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X5: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X5),Xa2)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa2),bot_bot(set(product_prod(A,B)))) ) ).

% bot_empty_eq2
tff(fact_4613_in__children__def,axiom,
    ! [Na: nat,TreeList: list(vEBT_VEBT),Xa: nat] :
      ( vEBT_V5917875025757280293ildren(Na,TreeList,Xa)
    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,Na))),vEBT_VEBT_low(Xa,Na)) ) ).

% in_children_def
tff(fact_4614_div__half__nat,axiom,
    ! [Y: nat,Xa: nat] :
      ( ( Y != zero_zero(nat) )
     => ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),Y)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Y)) = $let(
            q: nat,
            q:= aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)),
            $let(
              r: nat,
              r:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),q),Y)),
              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),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,aa(nat,fun(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_4615_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xa: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary2: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Deg)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2)),Xa) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_4616_predicate2I,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
      ( ! [X: A,Y3: B] :
          ( aa(B,$o,aa(A,fun(B,$o),P,X),Y3)
         => aa(B,$o,aa(A,fun(B,$o),Q,X),Y3) )
     => aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q) ) ).

% predicate2I
tff(fact_4617_sup2CI,axiom,
    ! [A: $tType,B: $tType,B3: fun(A,fun(B,$o)),Xa: A,Y: B,A3: fun(A,fun(B,$o))] :
      ( ( ~ aa(B,$o,aa(A,fun(B,$o),B3,Xa),Y)
       => aa(B,$o,aa(A,fun(B,$o),A3,Xa),Y) )
     => aa(B,$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))),A3),B3),Xa),Y) ) ).

% sup2CI
tff(fact_4618_inf2I,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,$o)),Xa: A,Y: B,B3: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),A3,Xa),Y)
     => ( aa(B,$o,aa(A,fun(B,$o),B3,Xa),Y)
       => aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),A3),B3),Xa),Y) ) ) ).

% inf2I
tff(fact_4619_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,Xa: nat,TreeList: list(vEBT_VEBT),Ma: nat,Summary2: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Xa)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => ( vEBT_vebt_succ(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = none(nat) ) ) ) ) ).

% succ_list_to_short
tff(fact_4620_pred__list__to__short,axiom,
    ! [Deg: nat,Xa: nat,Ma: nat,TreeList: list(vEBT_VEBT),Mi: nat,Summary2: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => ( vEBT_vebt_pred(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = none(nat) ) ) ) ) ).

% pred_list_to_short
tff(fact_4621_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2)),Xa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
        & ( ( Xa = Mi )
          | ( Xa = Ma )
          | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
            & aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% member_inv
tff(fact_4622_predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o)),Xa: A,Y: B] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
     => ( aa(B,$o,aa(A,fun(B,$o),P,Xa),Y)
       => aa(B,$o,aa(A,fun(B,$o),Q,Xa),Y) ) ) ).

% predicate2D
tff(fact_4623_rev__predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xa: A,Y: B,Q: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),P,Xa),Y)
     => ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
       => aa(B,$o,aa(A,fun(B,$o),Q,Xa),Y) ) ) ).

% rev_predicate2D
tff(fact_4624_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,nat,size_size(A),Xa) != aa(A,nat,size_size(A),Y) )
         => ( Xa != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_4625_inf2E,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,$o)),B3: fun(A,fun(B,$o)),Xa: A,Y: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),A3),B3),Xa),Y)
     => ~ ( aa(B,$o,aa(A,fun(B,$o),A3,Xa),Y)
         => ~ aa(B,$o,aa(A,fun(B,$o),B3,Xa),Y) ) ) ).

% inf2E
tff(fact_4626_inf2D1,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,$o)),B3: fun(A,fun(B,$o)),Xa: A,Y: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),A3),B3),Xa),Y)
     => aa(B,$o,aa(A,fun(B,$o),A3,Xa),Y) ) ).

% inf2D1
tff(fact_4627_inf2D2,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,$o)),B3: fun(A,fun(B,$o)),Xa: A,Y: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),A3),B3),Xa),Y)
     => aa(B,$o,aa(A,fun(B,$o),B3,Xa),Y) ) ).

% inf2D2
tff(fact_4628_sup2I2,axiom,
    ! [A: $tType,B: $tType,B3: fun(A,fun(B,$o)),Xa: A,Y: B,A3: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),B3,Xa),Y)
     => aa(B,$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))),A3),B3),Xa),Y) ) ).

% sup2I2
tff(fact_4629_sup2I1,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,$o)),Xa: A,Y: B,B3: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),A3,Xa),Y)
     => aa(B,$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))),A3),B3),Xa),Y) ) ).

% sup2I1
tff(fact_4630_sup2E,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,$o)),B3: fun(A,fun(B,$o)),Xa: A,Y: B] :
      ( aa(B,$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,$o))),A3),B3),Xa),Y)
     => ( ~ aa(B,$o,aa(A,fun(B,$o),A3,Xa),Y)
       => aa(B,$o,aa(A,fun(B,$o),B3,Xa),Y) ) ) ).

% sup2E
tff(fact_4631_bot2E,axiom,
    ! [A: $tType,B: $tType,Xa: A,Y: B] : ~ aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),Xa),Y) ).

% bot2E
tff(fact_4632_obtain__list__from__elements,axiom,
    ! [A: $tType,Na: nat,P: fun(A,fun(nat,$o))] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Na)
         => ? [Li: A] : aa(nat,$o,aa(A,fun(nat,$o),P,Li),I3) )
     => ~ ! [L2: list(A)] :
            ( ( aa(list(A),nat,size_size(list(A)),L2) = Na )
           => ~ ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Na)
                 => aa(nat,$o,aa(A,fun(nat,$o),P,aa(nat,A,nth(A,L2),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_4633_xor__num_Ocases,axiom,
    ! [Xa: product_prod(num,num)] :
      ( ( Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2) )
     => ( ! [N: num] : Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),bit0(N))
       => ( ! [N: num] : Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N))
         => ( ! [M4: num] : Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M4)),one2)
           => ( ! [M4: num,N: num] : Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M4)),bit0(N))
             => ( ! [M4: num,N: num] : Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M4)),aa(num,num,bit1,N))
               => ( ! [M4: num] : Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),one2)
                 => ( ! [M4: num,N: num] : Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),bit0(N))
                   => ~ ! [M4: num,N: num] : Xa != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),aa(num,num,bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_4634_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option(product_prod(nat,nat)),V: nat,TreeList: list(vEBT_VEBT),Sa: vEBT_VEBT,Xa: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy,aa(nat,nat,suc,V),TreeList,Sa),Xa)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_4635_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeList,Vd),Xa)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_4636_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list(vEBT_VEBT),Vc: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V),TreeList,Vc),Xa)
    <=> ( ( Xa = Mi )
        | ( Xa = Ma )
        | $let(
            pos: nat,
            pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_4637_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xa,Xaa)
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
           => ~ ( ( Xaa = Mi2 )
                | ( Xaa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Vc2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)
             => ~ ( ( Xaa = Mi2 )
                  | ( Xaa = Ma2 )
                  | $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
         => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [Vd2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)
               => ~ $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_4638_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_T_m_e_m_b_e_r(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
        $ite(
          Xa = Mi,
          one_one(nat),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
            $ite(
              Xa = Ma,
              one_one(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
                  one_one(nat),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                    $ite(
                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
                      one_one(nat),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),
                        $let(
                          h: nat,
                          h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat)) )) )) )) )) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_4639_vebt__member__code_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2)),Xa)
    <=> $ite(
          ( ( Deg = zero_zero(nat) )
          | ( Deg = aa(nat,nat,suc,zero_zero(nat)) ) ),
          $false,
          $ite(
            Xa = Mi,
            $true,
            $ite(
              Xa = Ma,
              $true,
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
                $false,
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
                  $false,
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% vebt_member_code(3)
tff(fact_4640_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)),Xa)
    <=> $ite(
          Xa = Mi,
          $true,
          $ite(
            Xa = Ma,
            $true,
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
              $false,
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
                $false,
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_4641_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Na: nat,Va: nat] :
      ( vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Na)
     => ( ( Na = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
       => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Mi)
         => ( ( Ma != Mi )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary2)))))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)) ) ) ) ) ).

% nested_mint
tff(fact_4642_size__union,axiom,
    ! [A: $tType,M7: 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)),M7),N3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),M7)),aa(multiset(A),nat,size_size(multiset(A)),N3)) ).

% size_union
tff(fact_4643_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_4644_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Deg)
     => ( ( Mi != Ma )
       => ( the2(nat,vEBT_vebt_maxt(Summary2)) = vEBT_VEBT_high(Ma,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% summaxma
tff(fact_4645_nonempty__has__size,axiom,
    ! [A: $tType,S: multiset(A)] :
      ( ( S != zero_zero(multiset(A)) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(multiset(A),nat,size_size(multiset(A)),S)) ) ).

% nonempty_has_size
tff(fact_4646_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_4647_num_Osize_I5_J,axiom,
    ! [X22: num] : aa(num,nat,size_size(num),bit0(X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_4648_num_Osize_I6_J,axiom,
    ! [X32: num] : aa(num,nat,size_size(num),aa(num,num,bit1,X32)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X32)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(6)
tff(fact_4649_del__x__mia,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = $let(
                xn: nat,
                xn:= aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary2)))))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $ite(
                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                    $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                      $let(
                        newlist: list(vEBT_VEBT),
                        newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                        $ite(
                          vEBT_VEBT_minNull(newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary2,h),
                            vEBT_Node(some(product_prod(nat,nat),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),xn),
                                  $ite(
                                    xn = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) ),
                                    Ma ))),Deg,newlist,sn) ),
                          vEBT_Node(some(product_prod(nat,nat),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),xn),
                                $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma))),Deg,newlist,Summary2) ) ) ),
                    vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2) ) ) ) ) ) ) ) ).

% del_x_mia
tff(fact_4650_del__x__mi__lets__in__minNull,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary2: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT),Sn: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary2)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( vEBT_VEBT_minNull(Newnode)
                       => ( ( Sn = vEBT_vebt_delete(Summary2,H) )
                         => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(some(product_prod(nat,nat),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(Sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),the2(nat,maxs)))))) ),
                                      Ma ))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_4651_del__x__mi__lets__in,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary2: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( Xa = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary2)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = $ite(
                            vEBT_VEBT_minNull(Newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary2,H),
                              vEBT_Node(some(product_prod(nat,nat),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),the2(nat,maxs)))))) ),
                                      Ma ))),Deg,Newlist,sn) ),
                            vEBT_Node(some(product_prod(nat,nat),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg,Newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_4652_nth__update__invalid,axiom,
    ! [A: $tType,I: nat,L: list(A),J: nat,Xa: A] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(nat,A,nth(A,list_update(A,L,J,Xa)),I) = aa(nat,A,nth(A,L),I) ) ) ).

% nth_update_invalid
tff(fact_4653_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( ~ vEBT_VEBT_minNull(Newnode)
                 => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                     => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(some(product_prod(nat,nat),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                $ite(Xa = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg,Newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_4654_del__x__mi__lets__in__not__minNull,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary2: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( Xa = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary2)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( ~ vEBT_VEBT_minNull(Newnode)
                       => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(some(product_prod(nat,nat),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg,Newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_4655_del__x__not__mia,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
               => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L),
                      $let(
                        newlist: list(vEBT_VEBT),
                        newlist:= list_update(vEBT_VEBT,TreeList,H,newnode),
                        $ite(
                          vEBT_VEBT_minNull(newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary2,H),
                            vEBT_Node(some(product_prod(nat,nat),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                  $ite(
                                    Xa = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) ),
                                    Ma ))),Deg,newlist,sn) ),
                          vEBT_Node(some(product_prod(nat,nat),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                $ite(Xa = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),H)))),Ma))),Deg,newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_4656_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Sn: vEBT_VEBT,Summary2: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( vEBT_VEBT_minNull(Newnode)
                 => ( ( Sn = vEBT_vebt_delete(Summary2,H) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                       => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(some(product_prod(nat,nat),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                  $ite(
                                    Xa = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(Sn),
                                      $ite(maxs = none(nat),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),the2(nat,maxs)))))) ),
                                    Ma ))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_4657_del__x__not__mi,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                   => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = $ite(
                          vEBT_VEBT_minNull(Newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary2,H),
                            vEBT_Node(some(product_prod(nat,nat),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                  $ite(
                                    Xa = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),the2(nat,maxs)))))) ),
                                    Ma ))),Deg,Newlist,sn) ),
                          vEBT_Node(some(product_prod(nat,nat),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                $ite(Xa = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg,Newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_4658_del__in__range,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Xa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = $let(
                xn: nat,
                xn:= 
                  $ite(Xa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa),
                $let(
                  minn: nat,
                  minn:= 
                    $ite(Xa = Mi,xn,Mi),
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(
                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                      $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                        $let(
                          newlist: list(vEBT_VEBT),
                          newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                          $ite(
                            vEBT_VEBT_minNull(newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary2,h),
                              vEBT_Node(some(product_prod(nat,nat),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                    $ite(
                                      xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) ),
                                      Ma ))),Deg,newlist,sn) ),
                            vEBT_Node(some(product_prod(nat,nat),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                  $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma))),Deg,newlist,Summary2) ) ) ),
                      vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2) ) ) ) ) ) ) ) ) ).

% del_in_range
tff(fact_4659_del__x__mi,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary2: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat] :
      ( ( ( Xa = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary2)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L),
                        $let(
                          newlist: list(vEBT_VEBT),
                          newlist:= list_update(vEBT_VEBT,TreeList,H,newnode),
                          $ite(
                            vEBT_VEBT_minNull(newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary2,H),
                              vEBT_Node(some(product_prod(nat,nat),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) ),
                                      Ma ))),Deg,newlist,sn) ),
                            vEBT_Node(some(product_prod(nat,nat),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),H)))),Ma))),Deg,newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_4660_union__less__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: multiset(A),C3: multiset(A),B3: multiset(A),D4: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),A3),C3)
         => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),B3),D4)
           => aa(multiset(A),$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)),A3),B3)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),D4)) ) ) ) ).

% union_less_mono
tff(fact_4661_union__le__mono2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: multiset(A),D4: multiset(A),C3: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),B3),D4)
         => aa(multiset(A),$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)),C3),B3)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),D4)) ) ) ).

% union_le_mono2
tff(fact_4662_union__le__mono1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: multiset(A),D4: multiset(A),C3: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),B3),D4)
         => aa(multiset(A),$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)),B3),C3)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),D4),C3)) ) ) ).

% union_le_mono1
tff(fact_4663_nth__list__update_H,axiom,
    ! [A: $tType,L: list(A),I: nat,Xa: A,J: nat] :
      aa(nat,A,nth(A,list_update(A,L,I,Xa)),J) = $ite(
        ( ( I = J )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)) ),
        Xa,
        aa(nat,A,nth(A,L),J) ) ).

% nth_list_update'
tff(fact_4664_vebt__delete_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_delete(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xa) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa) ),
        vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),
        $ite(
          ( ( Xa = Mi )
          & ( Xa = Ma ) ),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),
          $let(
            xn: nat,
            xn:= 
              $ite(Xa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa),
            $let(
              minn: nat,
              minn:= 
                $ite(Xa = Mi,xn,Mi),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  $let(
                    newnode: vEBT_VEBT,
                    newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                    $let(
                      newlist: list(vEBT_VEBT),
                      newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                      $ite(
                        vEBT_VEBT_minNull(newnode),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= vEBT_vebt_delete(Summary2,h),
                          vEBT_Node(some(product_prod(nat,nat),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                $ite(
                                  xn = Ma,
                                  $let(
                                    maxs: option(nat),
                                    maxs:= vEBT_vebt_maxt(sn),
                                    $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) ),
                                  Ma ))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,sn) ),
                        vEBT_Node(some(product_prod(nat,nat),
                            aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                              $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,Summary2) ) ) ),
                  vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_4665_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_V1232361888498592333_e_t_e(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xa) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa) ),
        one_one(nat),
        $ite(
          ( ( Xa = Mi )
          & ( Xa = Ma ) ),
          one_one(nat),
          $let(
            xn: nat,
            xn:= 
              $ite(Xa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa),
            $let(
              l: nat,
              l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                    $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),vEBT_V1232361888498592333_e_t_e(Summary2,h),one_one(nat))),
                  one_one(nat) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
tff(fact_4666_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list(vEBT_VEBT),Xa: nat,Ma: nat,Summary2: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( Xa != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mi),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary2,vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary2)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_4667_max_Oidem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A2),A2) = A2 ) ).

% max.idem
tff(fact_4668_max_Oleft__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ).

% max.left_idem
tff(fact_4669_max_Oright__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ).

% max.right_idem
tff(fact_4670_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb1
tff(fact_4671_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_4672_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% max.bounded_iff
tff(fact_4673_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb3
tff(fact_4674_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb4
tff(fact_4675_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z) ) ) ) ).

% max_less_iff_conj
tff(fact_4676_max__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),bot_bot(A)) = Xa ) ).

% max_bot2
tff(fact_4677_max__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),bot_bot(A)),Xa) = Xa ) ).

% max_bot
tff(fact_4678_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_4679_max__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),B2) = zero_zero(nat) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_4680_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),A2) = A2 ).

% max_nat.left_neutral
tff(fact_4681_max__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),B2) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_4682_max__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),zero_zero(nat)) = A2 ).

% max_nat.right_neutral
tff(fact_4683_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_4684_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_4685_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),Xa)),zero_zero(A)) = aa(num,A,numeral_numeral(A),Xa) ) ).

% max_0_1(4)
tff(fact_4686_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Xa)) = aa(num,A,numeral_numeral(A),Xa) ) ).

% max_0_1(3)
tff(fact_4687_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_4688_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_4689_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_4690_max__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),Xa)),one_one(A)) = aa(num,A,numeral_numeral(A),Xa) ) ).

% max_0_1(6)
tff(fact_4691_max__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),aa(num,A,numeral_numeral(A),Xa)) = aa(num,A,numeral_numeral(A),Xa) ) ).

% max_0_1(5)
tff(fact_4692_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_4693_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_4694_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_4695_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_4696_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_4697_insert__simp__norm,axiom,
    ! [Xa: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary2: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( Xa != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Xa),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary2,vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary2)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_4698_Max__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),lattic643756798349783984er_Max(A,A3)) ) ) ) ) ).

% Max_insert
tff(fact_4699_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% max_add_distrib_left
tff(fact_4700_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),ord_max(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z)) ) ).

% max_add_distrib_right
tff(fact_4701_nat__add__max__left,axiom,
    ! [M: nat,Na: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Na)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),Q2)) ).

% nat_add_max_left
tff(fact_4702_nat__add__max__right,axiom,
    ! [M: nat,Na: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),Q2)) = 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),Q2)) ).

% nat_add_max_right
tff(fact_4703_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X5: A,Xa2: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),X5),Xa2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Xa2),Xa2,X5) ) ).

% max_def_raw
tff(fact_4704_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2),B2,A2) ) ).

% max_def
tff(fact_4705_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y) = Xa ) ) ) ).

% max_absorb1
tff(fact_4706_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_4707_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,D2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),D2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ) ).

% max.mono
tff(fact_4708_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ) ) ).

% max.orderE
tff(fact_4709_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% max.orderI
tff(fact_4710_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% max.boundedE
tff(fact_4711_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2) ) ) ) ).

% max.boundedI
tff(fact_4712_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ) ) ).

% max.order_iff
tff(fact_4713_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ).

% max.cobounded1
tff(fact_4714_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ).

% max.cobounded2
tff(fact_4715_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xa)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y) ) ) ) ).

% le_max_iff_disj
tff(fact_4716_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb_iff1
tff(fact_4717_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb_iff2
tff(fact_4718_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.coboundedI1
tff(fact_4719_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.coboundedI2
tff(fact_4720_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_4721_sup__max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( sup_sup(A) = ord_max(A) ) ) ).

% sup_max
tff(fact_4722_nat__mult__max__right,axiom,
    ! [M: nat,Na: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),Q2)) = 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),Q2)) ).

% nat_mult_max_right
tff(fact_4723_nat__mult__max__left,axiom,
    ! [M: nat,Na: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Na)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q2)) ).

% nat_mult_max_left
tff(fact_4724_max_Oassoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) ) ).

% max.assoc
tff(fact_4725_max_Ocommute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = aa(A,A,aa(A,fun(A,A),ord_max(A),B2),A2) ) ).

% max.commute
tff(fact_4726_max_Oleft__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) ) ).

% max.left_commute
tff(fact_4727_of__int__max,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: int,Y: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_max(int),Xa),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(int,A,ring_1_of_int(A),Xa)),aa(int,A,ring_1_of_int(A),Y)) ) ).

% of_int_max
tff(fact_4728_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Xa),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_max
tff(fact_4729_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z)) ) ).

% max_diff_distrib_left
tff(fact_4730_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.strict_coboundedI2
tff(fact_4731_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.strict_coboundedI1
tff(fact_4732_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
tff(fact_4733_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% max.strict_boundedE
tff(fact_4734_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xa)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Y) ) ) ) ).

% less_max_iff_disj
tff(fact_4735_mset__le__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A),N3: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M7),N3)
         => ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),N3),M7) ) ) ).

% mset_le_asym
tff(fact_4736_mset__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K6: multiset(A),M7: multiset(A),N3: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),K6),M7)
         => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M7),N3)
           => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),K6),N3) ) ) ) ).

% mset_le_trans
tff(fact_4737_mset__le__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A)] : ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M7),M7) ) ).

% mset_le_irrefl
tff(fact_4738_mset__le__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A),N3: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M7),N3)
         => ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),N3),M7) ) ) ).

% mset_le_not_sym
tff(fact_4739_mset__le__not__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A)] : ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M7),M7) ) ).

% mset_le_not_refl
tff(fact_4740_less__eq__multiset__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A),N3: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less_eq(multiset(A)),M7),N3)
        <=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M7),N3)
            | ( M7 = N3 ) ) ) ) ).

% less_eq_multiset_def
tff(fact_4741_nat__minus__add__max,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)),M) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),M) ).

% nat_minus_add_max
tff(fact_4742_Max_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( B3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
             => ( aa(A,A,aa(A,fun(A,A),ord_max(A),lattic643756798349783984er_Max(A,B3)),lattic643756798349783984er_Max(A,A3)) = lattic643756798349783984er_Max(A,A3) ) ) ) ) ) ).

% Max.subset
tff(fact_4743_Max_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X: A,Y3: A] : member(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A)))))
             => member(A,lattic643756798349783984er_Max(A,A3),A3) ) ) ) ) ).

% Max.closed
tff(fact_4744_Max_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( ~ member(A,Xa,A3)
           => ( ( A3 != bot_bot(set(A)) )
             => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),lattic643756798349783984er_Max(A,A3)) ) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_4745_Max_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( finite_finite2(A,B3)
             => ( ( B3 != bot_bot(set(A)) )
               => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),lattic643756798349783984er_Max(A,A3)),lattic643756798349783984er_Max(A,B3)) ) ) ) ) ) ) ).

% Max.union
tff(fact_4746_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( member(A,Xa,A3)
           => ( lattic643756798349783984er_Max(A,A3) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))))) ) ) ) ) ).

% Max.remove
tff(fact_4747_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))))) ) ) ) ).

% Max.insert_remove
tff(fact_4748_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_insert(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xa) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),Mi,Xa),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
            & ~ ( ( Xa = Mi )
                | ( Xa = Ma ) ) ),
            vEBT_Node(some(product_prod(nat,nat),
                aa(nat,product_prod(nat,nat),
                  aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),Xa,Mi)),
                  aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_vebt_insert(Summary2,h),Summary2)),
            vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_4749_nth__list__update__eq,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I,Xa)),I) = Xa ) ) ).

% nth_list_update_eq
tff(fact_4750_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q2: 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),Q2),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))),Q2)),one_one(A))),aa(A,A,aa(A,fun(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))),Q2)),R)) ) ).

% divmod_step_eq
tff(fact_4751_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),Q2),zero_zero(extended_enat)) = Q2 ).

% max_enat_simps(2)
tff(fact_4752_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),zero_zero(extended_enat)),Q2) = Q2 ).

% max_enat_simps(3)
tff(fact_4753_sup__int__def,axiom,
    sup_sup(int) = ord_max(int) ).

% sup_int_def
tff(fact_4754_finite__maxlen,axiom,
    ! [A: $tType,M7: set(list(A))] :
      ( finite_finite2(list(A),M7)
     => ? [N: nat] :
        ! [X5: list(A)] :
          ( member(list(A),X5,M7)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),X5)),N) ) ) ).

% finite_maxlen
tff(fact_4755_length__induct,axiom,
    ! [A: $tType,P: fun(list(A),$o),Xs: list(A)] :
      ( ! [Xs2: list(A)] :
          ( ! [Ys: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),nat,size_size(list(A)),Xs2))
             => aa(list(A),$o,P,Ys) )
         => aa(list(A),$o,P,Xs2) )
     => aa(list(A),$o,P,Xs) ) ).

% length_induct
tff(fact_4756_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys2),I3) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_4757_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: fun(nat,fun(A,$o))] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),K)
         => ? [X_12: A] : aa(A,$o,aa(nat,fun(A,$o),P,I2),X_12) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
          & ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),K)
             => aa(A,$o,aa(nat,fun(A,$o),P,I2),aa(nat,A,nth(A,Xs3),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_4758_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs = Ys2 )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
        & ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys2),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_4759_nth__list__update,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xa: A,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I,Xa)),J) = $ite(I = J,Xa,aa(nat,A,nth(A,Xs),J)) ) ) ).

% nth_list_update
tff(fact_4760_list__update__same__conv,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ( list_update(A,Xs,I,Xa) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),I) = Xa ) ) ) ).

% list_update_same_conv
tff(fact_4761_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_T_i_n_s_e_r_t(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2)))))),
        $let(
          xn: nat,
          xn:= 
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),Mi,Xa),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
              & ~ ( ( Xa = Mi )
                  | ( Xa = Ma ) ) ),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),
                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t(Summary2,h),one_one(nat))),
              one_one(nat) ) ) )) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_4762_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_T_m_e_m_b_e_r2(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          Xa = Mi,
          zero_zero(nat),
          $ite(
            Xa = Ma,
            zero_zero(nat),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
              zero_zero(nat),
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
                zero_zero(nat),
                $ite(
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
                  & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) ),
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)) ),
                  zero_zero(nat) ) ) ) ) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
tff(fact_4763_vebt__maxtilist,axiom,
    ! [I: nat,Ts: list(vEBT_VEBT),Tsi: list(vEBT_VEBTi)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Ts))
     => hoare_hoare_triple(option(nat),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Ts),Tsi),vEBT_vebt_maxti(aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tsi),I)),aa(list(vEBT_VEBTi),fun(option(nat),assn),aa(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn)),aTP_Lamp_lw(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),I),Ts),Tsi)) ) ).

% vebt_maxtilist
tff(fact_4764_vebt__mintilist,axiom,
    ! [I: nat,Ts: list(vEBT_VEBT),Tsi: list(vEBT_VEBTi)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Ts))
     => hoare_hoare_triple(option(nat),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Ts),Tsi),vEBT_vebt_minti(aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tsi),I)),aa(list(vEBT_VEBTi),fun(option(nat),assn),aa(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn)),aTP_Lamp_lx(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),I),Ts),Tsi)) ) ).

% vebt_mintilist
tff(fact_4765_sup__enat__def,axiom,
    sup_sup(extended_enat) = ord_max(extended_enat) ).

% sup_enat_def
tff(fact_4766_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_e_m_b_e_r2(T,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))) ) ).

% member_bound_height'
tff(fact_4767_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e(Xa,Xaa) = Y )
     => ( ( ? [A5: $o,B6: $o] : Xa = vEBT_Leaf((A5),(B6))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A5: $o,B6: $o] : Xa = vEBT_Leaf((A5),(B6))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A5: $o,B6: $o] : Xa = vEBT_Leaf((A5),(B6))
             => ( ? [N: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary)
               => ( Y != one_one(nat) ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( Y != $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                              one_one(nat),
                              $ite(
                                ( ( Xaa = Mi2 )
                                & ( Xaa = Ma2 ) ),
                                one_one(nat),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary)))))),Xaa),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                          $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),vEBT_V1232361888498592333_e_t_e(Summary,h),one_one(nat))),
                                        one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
tff(fact_4768_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_T_p_r_e_d2(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xa) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
        one_one(nat),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
              $let(
                minlow: option(nat),
                minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( minlow != none(nat) )
                  & vEBT_VEBT_greater(some(nat,l),minlow) ),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary2,h)),one_one(nat)) ) ),
              one_one(nat) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_4769_vebt__delete_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xa,Xaa) = Y )
     => ( ! [A5: $o,B6: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B6)) )
           => ( ( Xaa = zero_zero(nat) )
             => ( Y != vEBT_Leaf($false,(B6)) ) ) )
       => ( ! [A5: $o] :
              ( ? [B6: $o] : Xa = vEBT_Leaf((A5),(B6))
             => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != vEBT_Leaf((A5),$false) ) ) )
         => ( ! [A5: $o,B6: $o] :
                ( ( Xa = vEBT_Leaf((A5),(B6)) )
               => ( ? [N: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N))
                 => ( Y != vEBT_Leaf((A5),(B6)) ) ) )
           => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary) )
                 => ( Y != vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst: list(vEBT_VEBT),Smry: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst,Smry) )
                   => ( Y != vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst,Smry) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr: list(vEBT_VEBT),Sm: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) )
                     => ( Y != vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( Y != $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                              vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary),
                              $ite(
                                ( ( Xaa = Mi2 )
                                & ( Xaa = Ma2 ) ),
                                vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary)))))),Xaa),
                                  $let(
                                    minn: nat,
                                    minn:= 
                                      $ite(Xaa = Mi2,xn,Mi2),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          newnode: vEBT_VEBT,
                                          newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                                          $let(
                                            newlist: list(vEBT_VEBT),
                                            newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                            $ite(
                                              vEBT_VEBT_minNull(newnode),
                                              $let(
                                                sn: vEBT_VEBT,
                                                sn:= vEBT_vebt_delete(Summary,h),
                                                vEBT_Node(some(product_prod(nat,nat),
                                                    aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                      $ite(
                                                        xn = Ma2,
                                                        $let(
                                                          maxs: option(nat),
                                                          maxs:= vEBT_vebt_maxt(sn),
                                                          $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) ),
                                                        Ma2 ))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),newlist,sn) ),
                                              vEBT_Node(some(product_prod(nat,nat),
                                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                    $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),newlist,Summary) ) ) ),
                                        vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_4770_Leaf__0__not,axiom,
    ! [A2: $o,B2: $o] : ~ vEBT_invar_vebt(vEBT_Leaf((A2),(B2)),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_4771_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( vEBT_invar_vebt(T,one_one(nat))
    <=> ? [A10: $o,B10: $o] : T = vEBT_Leaf((A10),(B10)) ) ).

% deg1Leaf
tff(fact_4772_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( vEBT_invar_vebt(T,one_one(nat))
     => ? [A5: $o,B6: $o] : T = vEBT_Leaf((A5),(B6)) ) ).

% deg_1_Leaf
tff(fact_4773_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T,Na)
     => ( ( Na = one_one(nat) )
       => ? [A5: $o,B6: $o] : T = vEBT_Leaf((A5),(B6)) ) ) ).

% deg_1_Leafy
tff(fact_4774_greater__shift,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa)
    <=> vEBT_VEBT_greater(some(nat,Xa),some(nat,Y)) ) ).

% greater_shift
tff(fact_4775_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_space2,vEBT_Leaf((A2),(B2))) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% VEBT_internal.space'.simps(1)
tff(fact_4776_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Va: nat] :
      vEBT_T_p_r_e_d(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Va))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite((B2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
tff(fact_4777_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] : vEBT_T_s_u_c_c(vEBT_Leaf((Uu),(B2)),zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
tff(fact_4778_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xa: nat] :
      vEBT_T_i_n_s_e_r_t(vEBT_Leaf((A2),(B2)),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(Xa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
tff(fact_4779_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_space,vEBT_Leaf((A2),(B2))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.space.simps(1)
tff(fact_4780_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A2: $o,Uw: $o] : vEBT_T_p_r_e_d(vEBT_Leaf((A2),(Uw)),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
tff(fact_4781_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xa: nat] :
      vEBT_T_m_e_m_b_e_r(vEBT_Leaf((A2),(B2)),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
        $ite(Xa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
tff(fact_4782_VEBT__internal_Oinsert_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_insert(Xa,Xaa) = Y )
     => ( ! [A5: $o,B6: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B6)) )
           => ( Y != vEBT_vebt_insert(vEBT_Leaf((A5),(B6)),Xaa) ) )
       => ~ ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info,Deg2,TreeList2,Summary) )
             => ( Y != $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)),Xaa),vEBT_Node(Info,Deg2,TreeList2,Summary),vEBT_vebt_insert(vEBT_Node(Info,Deg2,TreeList2,Summary),Xaa)) ) ) ) ) ).

% VEBT_internal.insert'.elims
tff(fact_4783_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d2(Xa,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A5: $o,Uw2: $o] : Xa = vEBT_Leaf((A5),(Uw2))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A5: $o,B6: $o] : Xa = vEBT_Leaf((A5),(B6))
             => ( ? [Va2: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va2))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [V4: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( Y != $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                              one_one(nat),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                    $let(
                                      minlow: option(nat),
                                      minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                      $ite(
                                        ( ( minlow != none(nat) )
                                        & vEBT_VEBT_greater(some(nat,l),minlow) ),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary,h)),one_one(nat)) ) ),
                                    one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_4784_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xa,Xaa)
      <=> (Y) )
     => ( ! [A5: $o,B6: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B6)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A5),
                    $ite(Xaa = one_one(nat),(B6),$false) ) ) )
       => ( ( ? [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
           => (Y) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S2: vEBT_VEBT] : Xa = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2)
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_4785_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xa,Xaa)
     => ( ! [A5: $o,B6: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B6)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A5),
                  $ite(Xaa = one_one(nat),(B6),$false) ) )
       => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [S2: vEBT_VEBT] : Xa = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2)
             => ~ $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_4786_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xa,Xaa)
     => ( ! [A5: $o,B6: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B6)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A5),
                $ite(Xaa = one_one(nat),(B6),$false) ) )
       => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa != vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
         => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S2: vEBT_VEBT] : Xa = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2)
               => $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_4787_vebt__member_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
     => ( ! [A5: $o,B6: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B6)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A5),
                  $ite(Xaa = one_one(nat),(B6),$false) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Summary: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)
             => ~ $ite(
                    Xaa = Mi2,
                    $true,
                    $ite(
                      Xaa = Ma2,
                      $true,
                      $ite(
                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                        $false,
                        $ite(
                          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_4788_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xa,Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xa != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xa != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( ( Xaa = Mi2 )
                  | ( Xaa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)
                 => ( ( Xaa = Mi2 )
                    | ( Xaa = Ma2 )
                    | $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
             => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_4789_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xa,Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => (Y) )
       => ( ( ? [Ux: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy2)
           => (Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
             => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_4790_vebt__member_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
     => ( ! [A5: $o,B6: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B6)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A5),
                $ite(Xaa = one_one(nat),(B6),$false) ) )
       => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
         => ( ! [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa != vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy2,Uz2)
           => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa != vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)
                   => $ite(
                        Xaa = Mi2,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_4791_vebt__member_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
      <=> (Y) )
     => ( ! [A5: $o,B6: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B6)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A5),
                    $ite(Xaa = one_one(nat),(B6),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => (Y) )
         => ( ( ? [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy2,Uz2)
             => (Y) )
           => ( ( ? [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => (Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi2,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                $false,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_4792_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(Xa,Xaa) = Y )
     => ( ( ? [A5: $o,B6: $o] : Xa = vEBT_Leaf((A5),(B6))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                          $ite(
                            Xaa = Mi2,
                            zero_zero(nat),
                            $ite(
                              Xaa = Ma2,
                              zero_zero(nat),
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                zero_zero(nat),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  zero_zero(nat),
                                  $ite(
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xaa)
                                    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Ma2) ),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)) ),
                                    zero_zero(nat) ) ) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
tff(fact_4793_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(Xa,Xaa) = Y )
     => ( ( ? [A5: $o,B6: $o] : Xa = vEBT_Leaf((A5),(B6))
         => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
         => ( ( ? [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy2,Uz2)
             => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
           => ( ( ? [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $ite(
                            Xaa = Mi2,
                            one_one(nat),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xaa = Ma2,
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                        one_one(nat),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),
                                          $let(
                                            h: nat,
                                            h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat)) )) )) )) )) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_4794_vebt__insert_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xa,Xaa) = Y )
     => ( ! [A5: $o,B6: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B6)) )
           => ( Y != $ite(
                  Xaa = zero_zero(nat),
                  vEBT_Leaf($true,(B6)),
                  $ite(Xaa = one_one(nat),vEBT_Leaf((A5),$true),vEBT_Leaf((A5),(B6))) ) ) )
       => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info,zero_zero(nat),Ts2,S2) )
             => ( Y != vEBT_Node(Info,zero_zero(nat),Ts2,S2) ) )
         => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
               => ( Y != vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) ) )
           => ( ! [V4: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary) )
                 => ( Y != vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                              & ~ ( ( Xaa = Mi2 )
                                  | ( Xaa = Ma2 ) ) ),
                              vEBT_Node(some(product_prod(nat,nat),
                                  aa(nat,product_prod(nat,nat),
                                    aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Xaa,Mi2)),
                                    aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),list_update(vEBT_VEBT,TreeList2,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_vebt_insert(Summary,h),Summary)),
                              vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_4795_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(Xa,Xaa) = Y )
     => ( ( ? [A5: $o,B6: $o] : Xa = vEBT_Leaf((A5),(B6))
         => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : Xa = vEBT_Node(Info,zero_zero(nat),Ts2,S2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : Xa = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V4: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary)
               => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2)))))),
                          $let(
                            xn: nat,
                            xn:= 
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                            $let(
                              h: nat,
                              h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(
                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                & ~ ( ( Xaa = Mi2 )
                                    | ( Xaa = Ma2 ) ) ),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t(Summary,h),one_one(nat))),
                                one_one(nat) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_4796_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(Xa,Xaa) = Y )
     => ( ( ? [A5: $o,B6: $o] : Xa = vEBT_Leaf((A5),(B6))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : Xa = vEBT_Node(Info,zero_zero(nat),Ts2,S2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : Xa = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V4: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                              & ~ ( ( Xaa = Mi2 )
                                  | ( Xaa = Ma2 ) ) ),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t2(Summary,h),one_one(nat))),
                              one_one(nat) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_4797_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(Xa,Xaa) = Y )
     => ( ( ? [Uu2: $o,B6: $o] : Xa = vEBT_Leaf((Uu2),(B6))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xa = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa(nat,nat,suc,N)
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [Ux: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V4: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)
                 => ( Y != one_one(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( Y != $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                            one_one(nat),
                            $let(
                              l: nat,
                              l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                  $let(
                                    maxlow: option(nat),
                                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                    $ite(
                                      ( ( maxlow != none(nat) )
                                      & vEBT_VEBT_less(some(nat,l),maxlow) ),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary,h)),one_one(nat)) ) ),
                                  one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_4798_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_T_s_u_c_c2(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xa) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
        one_one(nat),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( maxlow != none(nat) )
                  & vEBT_VEBT_less(some(nat,l),maxlow) ),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary2,h)),one_one(nat)) ) ),
              one_one(nat) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_4799_less__shift,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
    <=> vEBT_VEBT_less(some(nat,Xa),some(nat,Y)) ) ).

% less_shift
tff(fact_4800_insert_H__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(T,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))) ) ).

% insert'_bound_height
tff(fact_4801_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_T_i_n_s_e_r_t2(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xa) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),Mi,Xa),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
            & ~ ( ( Xa = Mi )
                | ( Xa = Ma ) ) ),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t2(Summary2,h),one_one(nat))),
            one_one(nat) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_4802_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_d_e_l_e_t_e(Xa,Xaa) = Y )
     => ( ( ? [A5: $o,B6: $o] : Xa = vEBT_Leaf((A5),(B6))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A5: $o,B6: $o] : Xa = vEBT_Leaf((A5),(B6))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A5: $o,B6: $o] : Xa = vEBT_Leaf((A5),(B6))
             => ( ? [N: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary)
               => ( Y != one_one(nat) ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                              $ite(
                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                                | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)),
                                    aa(nat,nat,
                                      aa(nat,fun(nat,nat),plus_plus(nat),
                                        aa(nat,nat,
                                          aa(nat,fun(nat,nat),plus_plus(nat),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),
                                              $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(Summary)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary)))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),one_one(nat)))),
                                          one_one(nat))),
                                      $let(
                                        xn: nat,
                                        xn:= 
                                          $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary)))))),Xaa),
                                        $let(
                                          l: nat,
                                          l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $let(
                                            h: nat,
                                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                            $ite(
                                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_d_e_l_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l))),
                                                $let(
                                                  newnode: vEBT_VEBT,
                                                  newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l),
                                                  $let(
                                                    newlist: list(vEBT_VEBT),
                                                    newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_i_n_N_u_l_l(newnode))),
                                                      $ite(
                                                        vEBT_VEBT_minNull(newnode),
                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_d_e_l_e_t_e(Summary,h))),
                                                          $let(
                                                            sn: vEBT_VEBT,
                                                            sn:= vEBT_vebt_delete(Summary,h),
                                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                              $ite(
                                                                xn = Ma2,
                                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(sn))),
                                                                  $let(
                                                                    maxs: option(nat),
                                                                    maxs:= vEBT_vebt_maxt(sn),
                                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                                                      $ite(maxs = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) )),
                                                                one_one(nat) )) )),
                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                          $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h))),one_one(nat))) )) ) )),
                                              one_one(nat) ) ) ) )) )) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
tff(fact_4803_set__n__deg__not__0,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,M: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList))
         => vEBT_invar_vebt(X,Na) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na) ) ) ).

% set_n_deg_not_0
tff(fact_4804_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
     => ( ( ? [A5: $o,B6: $o] : A1 = vEBT_Leaf((A5),(B6))
         => ( A22 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList2: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M4: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary) )
             => ( ( A22 = Deg2 )
               => ( ! [X5: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X5,set2(vEBT_VEBT,TreeList2))
                     => vEBT_invar_vebt(X5,N) )
                 => ( vEBT_invar_vebt(Summary,M4)
                   => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M4) )
                     => ( ( M4 = N )
                       => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M4) )
                         => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_13)
                           => ~ ! [X5: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X5,set2(vEBT_VEBT,TreeList2))
                                 => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X_13) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M4: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary) )
               => ( ( A22 = Deg2 )
                 => ( ! [X5: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X5,set2(vEBT_VEBT,TreeList2))
                       => vEBT_invar_vebt(X5,N) )
                   => ( vEBT_invar_vebt(Summary,M4)
                     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M4) )
                       => ( ( M4 = aa(nat,nat,suc,N) )
                         => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M4) )
                           => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_13)
                             => ~ ! [X5: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X5,set2(vEBT_VEBT,TreeList2))
                                   => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X_13) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Deg2,TreeList2,Summary) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X5: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X5,set2(vEBT_VEBT,TreeList2))
                         => vEBT_invar_vebt(X5,N) )
                     => ( vEBT_invar_vebt(Summary,M4)
                       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M4) )
                         => ( ( M4 = N )
                           => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M4) )
                             => ( ! [I4: nat] :
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M4))
                                   => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_12)
                                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I4) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X5: vEBT_VEBT] :
                                        ( member(vEBT_VEBT,X5,set2(vEBT_VEBT,TreeList2))
                                       => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X_13) ) )
                                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M4))
                                             => ( ( ( vEBT_VEBT_high(Ma2,N) = I4 )
                                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N)) )
                                                & ! [X5: nat] :
                                                    ( ( ( vEBT_VEBT_high(X5,N) = I4 )
                                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X5,N)) )
                                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X5)
                                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X5),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Deg2,TreeList2,Summary) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X5: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X5,set2(vEBT_VEBT,TreeList2))
                           => vEBT_invar_vebt(X5,N) )
                       => ( vEBT_invar_vebt(Summary,M4)
                         => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M4) )
                           => ( ( M4 = aa(nat,nat,suc,N) )
                             => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M4) )
                               => ( ! [I4: nat] :
                                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M4))
                                     => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_12)
                                      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I4) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X5: vEBT_VEBT] :
                                          ( member(vEBT_VEBT,X5,set2(vEBT_VEBT,TreeList2))
                                         => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X_13) ) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M4))
                                               => ( ( ( vEBT_VEBT_high(Ma2,N) = I4 )
                                                   => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N)) )
                                                  & ! [X5: nat] :
                                                      ( ( ( vEBT_VEBT_high(X5,N) = I4 )
                                                        & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X5,N)) )
                                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X5)
                                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X5),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_4805_inthall,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Na: nat] :
      ( ! [X: A] :
          ( member(A,X,set2(A,Xs))
         => aa(A,$o,P,X) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),Na)) ) ) ).

% inthall
tff(fact_4806_height__compose__child,axiom,
    ! [T: vEBT_VEBT,TreeList: list(vEBT_VEBT),Info2: option(product_prod(nat,nat)),Deg: nat,Summary2: vEBT_VEBT] :
      ( member(vEBT_VEBT,T,set2(vEBT_VEBT,TreeList))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info2,Deg,TreeList,Summary2))) ) ).

% height_compose_child
tff(fact_4807_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Deg)
     => ( ( Mi = Ma )
       => ( ! [X5: vEBT_VEBT] :
              ( member(vEBT_VEBT,X5,set2(vEBT_VEBT,TreeList))
             => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X_13) )
          & ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_13) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_4808_set__swap,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => ( set2(A,list_update(A,list_update(A,Xs,I,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I))) = set2(A,Xs) ) ) ) ).

% set_swap
tff(fact_4809_length__pos__if__in__set,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      ( member(A,Xa,set2(A,Xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_pos_if_in_set
tff(fact_4810_nth__mem,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => member(A,aa(nat,A,nth(A,Xs),Na),set2(A,Xs)) ) ).

% nth_mem
tff(fact_4811_list__ball__nth,axiom,
    ! [A: $tType,Na: nat,Xs: list(A),P: fun(A,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ! [X: A] :
            ( member(A,X,set2(A,Xs))
           => aa(A,$o,P,X) )
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_4812_in__set__conv__nth,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      ( member(A,Xa,set2(A,Xs))
    <=> ? [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
          & ( aa(nat,A,nth(A,Xs),I2) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_4813_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Xa: A] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I3)) )
     => ( member(A,Xa,set2(A,Xs))
       => aa(A,$o,P,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_4814_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X4: A] :
          ( member(A,X4,set2(A,Xs))
         => aa(A,$o,P,X4) )
    <=> ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_4815_all__set__conv__nth,axiom,
    ! [A: $tType,L: list(A),P: fun(A,$o)] :
      ( ! [X4: A] :
          ( member(A,X4,set2(A,L))
         => aa(A,$o,P,X4) )
    <=> ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
         => aa(A,$o,P,aa(nat,A,nth(A,L),I2)) ) ) ).

% all_set_conv_nth
tff(fact_4816_set__update__memI,axiom,
    ! [A: $tType,Na: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => member(A,Xa,set2(A,list_update(A,Xs,Na,Xa))) ) ).

% set_update_memI
tff(fact_4817_in__set__upd__eq,axiom,
    ! [A: $tType,I: nat,L: list(A),Xa: A,Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( member(A,Xa,set2(A,list_update(A,L,I,Y)))
      <=> ( ( Xa = Y )
          | ( member(A,Xa,set2(A,L))
            & ! [Y5: A] : member(A,Xa,set2(A,list_update(A,L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_4818_in__set__upd__cases,axiom,
    ! [A: $tType,Xa: A,L: list(A),I: nat,Y: A] :
      ( member(A,Xa,set2(A,list_update(A,L,I,Y)))
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
         => ( Xa != Y ) )
       => member(A,Xa,set2(A,L)) ) ) ).

% in_set_upd_cases
tff(fact_4819_in__set__upd__eq__aux,axiom,
    ! [A: $tType,I: nat,L: list(A),Xa: A,Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( member(A,Xa,set2(A,list_update(A,L,I,Y)))
      <=> ( ( Xa = Y )
          | ! [Y5: A] : member(A,Xa,set2(A,list_update(A,L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_4820_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xa: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set2(A,list_update(A,Xs,I,Xa))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),set2(A,Xs))) ).

% set_update_subset_insert
tff(fact_4821_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_T_m_a_x_t(vEBT_Leaf((A2),(B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite((B2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
tff(fact_4822_finite__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),Na: nat] :
      ( finite_finite2(A,A3)
     => finite_finite2(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_ly(set(A),fun(nat,fun(list(A),$o)),A3),Na))) ) ).

% finite_lists_length_eq
tff(fact_4823_insert__swap__set__eq,axiom,
    ! [A: $tType,I: nat,L: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(nat,A,nth(A,L),I)),set2(A,list_update(A,L,I,Xa))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),set2(A,L)) ) ) ).

% insert_swap_set_eq
tff(fact_4824_maxt__bound,axiom,
    ! [T: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_a_x_t(T)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ).

% maxt_bound
tff(fact_4825_finite__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),Na: nat] :
      ( finite_finite2(A,A3)
     => finite_finite2(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_lz(set(A),fun(nat,fun(list(A),$o)),A3),Na))) ) ).

% finite_lists_length_le
tff(fact_4826_mint__bound,axiom,
    ! [T: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_i_n_t(T)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ).

% mint_bound
tff(fact_4827_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_T_m_i_n_t(vEBT_Leaf((A2),(B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite((A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
tff(fact_4828_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(Xa) = Y )
     => ( ! [A5: $o,B6: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B6)) )
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                  $ite((B6),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy2,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_4829_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,Summary2: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList))
         => vEBT_invar_vebt(X,Na) )
     => ( vEBT_invar_vebt(Summary2,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = Na )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M) )
             => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_1)
               => ( ! [X: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList))
                     => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary2),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_4830_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,Summary2: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList))
         => vEBT_invar_vebt(X,Na) )
     => ( vEBT_invar_vebt(Summary2,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = aa(nat,nat,suc,Na) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M) )
             => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_1)
               => ( ! [X: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList))
                     => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary2),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_4831_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(Xa) = Y )
     => ( ! [A5: $o] :
            ( ? [B6: $o] : Xa = vEBT_Leaf((A5),(B6))
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                  $ite((A5),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy2,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_4832_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,Summary2: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList))
         => vEBT_invar_vebt(X,Na) )
     => ( vEBT_invar_vebt(Summary2,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = Na )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M) )
             => ( ! [I3: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                   => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),X_12)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I3) ) )
               => ( ( ( Mi = Ma )
                   => ! [X: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList))
                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                             => ( ( ( vEBT_VEBT_high(Ma,Na) = I3 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),vEBT_VEBT_low(Ma,Na)) )
                                & ! [X: nat] :
                                    ( ( ( vEBT_VEBT_high(X,Na) = I3 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),vEBT_VEBT_low(X,Na)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_4833_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_T_p_r_e_d(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
          one_one(nat),
          $let(
            l: nat,
            l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $let(
              h: nat,
              h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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(aa(num,num,bit1,bit0(one2))))),one_one(nat))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),
                      $ite(
                        ( ( minlow != none(nat) )
                        & vEBT_VEBT_greater(some(nat,l),minlow) ),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                        $let(
                          pr: option(nat),
                          pr:= vEBT_vebt_pred(Summary2,h),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary2,h))),one_one(nat))),
                            $ite(pr = none(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,pr)))))) ) )) ),
                  one_one(nat) )) ) ) )) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_4834_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_T_s_u_c_c(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
          one_one(nat),
          $let(
            l: nat,
            l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $let(
              h: nat,
              h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),
                    $let(
                      maxlow: option(nat),
                      maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                        $ite(
                          ( ( maxlow != none(nat) )
                          & vEBT_VEBT_less(some(nat,l),maxlow) ),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                          $let(
                            sc: option(nat),
                            sc:= vEBT_vebt_succ(Summary2,h),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary2,h))),one_one(nat))),
                              $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,sc)))))) ) )) )),
                  one_one(nat) )) ) ) )) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_4835_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,Summary2: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList))
         => vEBT_invar_vebt(X,Na) )
     => ( vEBT_invar_vebt(Summary2,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = aa(nat,nat,suc,Na) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M) )
             => ( ! [I3: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                   => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),X_12)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I3) ) )
               => ( ( ( Mi = Ma )
                   => ! [X: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList))
                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                             => ( ( ( vEBT_VEBT_high(Ma,Na) = I3 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),vEBT_VEBT_low(Ma,Na)) )
                                & ! [X: nat] :
                                    ( ( ( vEBT_VEBT_high(X,Na) = I3 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),vEBT_VEBT_low(X,Na)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_4836_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d(Xa,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A5: $o,Uw2: $o] : Xa = vEBT_Leaf((A5),(Uw2))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ) ) )
         => ( ! [A5: $o,B6: $o] :
                ( ( Xa = vEBT_Leaf((A5),(B6)) )
               => ( ? [Va2: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va2))
                 => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                        $ite((B6),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [V4: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                one_one(nat),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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(aa(num,num,bit1,bit0(one2))))),one_one(nat))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),
                                            $ite(
                                              ( ( minlow != none(nat) )
                                              & vEBT_VEBT_greater(some(nat,l),minlow) ),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                              $let(
                                                pr: option(nat),
                                                pr:= vEBT_vebt_pred(Summary,h),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary,h))),one_one(nat))),
                                                  $ite(pr = none(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,pr)))))) ) )) ),
                                        one_one(nat) )) ) ) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_4837_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(Xa,Xaa) = Y )
     => ( ( ? [Uu2: $o,B6: $o] : Xa = vEBT_Leaf((Uu2),(B6))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xa = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa(nat,nat,suc,N)
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [Ux: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V4: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)
                 => ( Y != one_one(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                              one_one(nat),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                        $let(
                                          maxlow: option(nat),
                                          maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                            $ite(
                                              ( ( maxlow != none(nat) )
                                              & vEBT_VEBT_less(some(nat,l),maxlow) ),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                              $let(
                                                sc: option(nat),
                                                sc:= vEBT_vebt_succ(Summary,h),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary,h))),one_one(nat))),
                                                  $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,sc)))))) ) )) )),
                                      one_one(nat) )) ) ) )) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_4838_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_T_d_e_l_e_t_e(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
        $ite(
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
          | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa) ),
          one_one(nat),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
            $ite(
              ( ( Xa = Mi )
              & ( Xa = Ma ) ),
              aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)),
              aa(nat,nat,
                aa(nat,fun(nat,nat),plus_plus(nat),
                  aa(nat,nat,
                    aa(nat,fun(nat,nat),plus_plus(nat),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),
                        $ite(Xa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(Summary2)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary2)))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),one_one(nat)))),
                    one_one(nat))),
                $let(
                  xn: nat,
                  xn:= 
                    $ite(Xa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa),
                  $let(
                    l: nat,
                    l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $let(
                      h: nat,
                      h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(
                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_d_e_l_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l))),
                          $let(
                            newnode: vEBT_VEBT,
                            newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l),
                            $let(
                              newlist: list(vEBT_VEBT),
                              newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_i_n_N_u_l_l(newnode))),
                                $ite(
                                  vEBT_VEBT_minNull(newnode),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_d_e_l_e_t_e(Summary2,h))),
                                    $let(
                                      sn: vEBT_VEBT,
                                      sn:= vEBT_vebt_delete(Summary2,h),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                        $ite(
                                          xn = Ma,
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(sn))),
                                            $let(
                                              maxs: option(nat),
                                              maxs:= vEBT_vebt_maxt(sn),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                                $ite(maxs = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) )),
                                          one_one(nat) )) )),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                    $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h))),one_one(nat))) )) ) )),
                        one_one(nat) ) ) ) )) )) )) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
tff(fact_4839_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
    <=> ( ( ? [A10: $o,B10: $o] : A1 = vEBT_Leaf((A10),(B10))
          & ( A22 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,set2(vEBT_VEBT,TreeList3))
               => vEBT_invar_vebt(X4,N2) )
            & vEBT_invar_vebt(Summary3,N2)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),N2) )
            & ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X_12)
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,set2(vEBT_VEBT,TreeList3))
               => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_12) ) )
        | ? [TreeList3: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,set2(vEBT_VEBT,TreeList3))
               => vEBT_invar_vebt(X4,N2) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N2))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N2)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,suc,N2)) )
            & ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X_12)
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,set2(vEBT_VEBT,TreeList3))
               => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_12) ) )
        | ? [TreeList3: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,set2(vEBT_VEBT,TreeList3))
               => vEBT_invar_vebt(X4,N2) )
            & vEBT_invar_vebt(Summary3,N2)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),N2) )
            & ! [I2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
               => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),X_12)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),I2) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X4,set2(vEBT_VEBT,TreeList3))
                 => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_12) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi3),Ma3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A22))
            & ( ( Mi3 != Ma3 )
             => ! [I2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                 => ( ( ( vEBT_VEBT_high(Ma3,N2) = I2 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),vEBT_VEBT_low(Ma3,N2)) )
                    & ! [X4: nat] :
                        ( ( ( vEBT_VEBT_high(X4,N2) = I2 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),vEBT_VEBT_low(X4,N2)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi3),X4)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Ma3) ) ) ) ) ) )
        | ? [TreeList3: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,set2(vEBT_VEBT,TreeList3))
               => vEBT_invar_vebt(X4,N2) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N2))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N2)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,suc,N2)) )
            & ! [I2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N2)))
               => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),X_12)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),I2) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X4,set2(vEBT_VEBT,TreeList3))
                 => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_12) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi3),Ma3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A22))
            & ( ( Mi3 != Ma3 )
             => ! [I2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N2)))
                 => ( ( ( vEBT_VEBT_high(Ma3,N2) = I2 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),vEBT_VEBT_low(Ma3,N2)) )
                    & ! [X4: nat] :
                        ( ( ( vEBT_VEBT_high(X4,N2) = I2 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I2)),vEBT_VEBT_low(X4,N2)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi3),X4)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Ma3) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_4840_setceilmax,axiom,
    ! [Sa: vEBT_VEBT,M: nat,Listy: list(vEBT_VEBT),Na: nat] :
      ( vEBT_invar_vebt(Sa,M)
     => ( ! [X: vEBT_VEBT] :
            ( member(vEBT_VEBT,X,set2(vEBT_VEBT,Listy))
           => vEBT_invar_vebt(X,Na) )
       => ( ( M = aa(nat,nat,suc,Na) )
         => ( ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,set2(vEBT_VEBT,Listy))
               => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))) ) )
           => ( ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Sa)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) )
             => ( aa(nat,int,semiring_1_of_nat(int),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),aa(vEBT_VEBT,fun(set(vEBT_VEBT),set(vEBT_VEBT)),insert(vEBT_VEBT),Sa),set2(vEBT_VEBT,Listy))))) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_4841_max__idx__list,axiom,
    ! [I: nat,X13: list(vEBT_VEBT),Na: nat,X14: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I)))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14)),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),set2(vEBT_VEBT,X13)))))))) ) ).

% max_idx_list
tff(fact_4842_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_d_e_l_e_t_e(Xa,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),zero_zero(nat))) ) ) )
         => ( ! [A5: $o,B6: $o] :
                ( ( Xa = vEBT_Leaf((A5),(B6)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A5: $o,B6: $o] :
                  ( ( Xa = vEBT_Leaf((A5),(B6)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),aa(nat,nat,suc,aa(nat,nat,suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary)),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary)),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary) )
                       => ( ( Y = one_one(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                  $ite(
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                                    | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                      $ite(
                                        ( ( Xaa = Mi2 )
                                        & ( Xaa = Ma2 ) ),
                                        aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)),
                                        aa(nat,nat,
                                          aa(nat,fun(nat,nat),plus_plus(nat),
                                            aa(nat,nat,
                                              aa(nat,fun(nat,nat),plus_plus(nat),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),
                                                  $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(Summary)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary)))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),one_one(nat)))),
                                              one_one(nat))),
                                          $let(
                                            xn: nat,
                                            xn:= 
                                              $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary)))))),Xaa),
                                            $let(
                                              l: nat,
                                              l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                              $let(
                                                h: nat,
                                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                                $ite(
                                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_d_e_l_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l))),
                                                    $let(
                                                      newnode: vEBT_VEBT,
                                                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l),
                                                      $let(
                                                        newlist: list(vEBT_VEBT),
                                                        newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_i_n_N_u_l_l(newnode))),
                                                          $ite(
                                                            vEBT_VEBT_minNull(newnode),
                                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_d_e_l_e_t_e(Summary,h))),
                                                              $let(
                                                                sn: vEBT_VEBT,
                                                                sn:= vEBT_vebt_delete(Summary,h),
                                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                                  $ite(
                                                                    xn = Ma2,
                                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(sn))),
                                                                      $let(
                                                                        maxs: option(nat),
                                                                        maxs:= vEBT_vebt_maxt(sn),
                                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                                                          $ite(maxs = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) )),
                                                                    one_one(nat) )) )),
                                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                              $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h))),one_one(nat))) )) ) )),
                                                  one_one(nat) ) ) ) )) )) )) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_4843_image__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),Xa: B,A3: set(B)] :
      ( ( B2 = aa(B,A,F2,Xa) )
     => ( member(B,Xa,A3)
       => member(A,B2,aa(set(B),set(A),image(B,A,F2),A3)) ) ) ).

% image_eqI
tff(fact_4844_image__ident,axiom,
    ! [A: $tType,Y6: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_ma(A,A)),Y6) = Y6 ).

% image_ident
tff(fact_4845_height__compose__list,axiom,
    ! [T: vEBT_VEBT,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( member(vEBT_VEBT,T,set2(vEBT_VEBT,TreeList))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T)),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),aa(vEBT_VEBT,fun(set(vEBT_VEBT),set(vEBT_VEBT)),insert(vEBT_VEBT),Summary2),set2(vEBT_VEBT,TreeList))))) ) ).

% height_compose_list
tff(fact_4846_image__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] : aa(set(B),set(A),image(B,A,F2),bot_bot(set(B))) = bot_bot(set(A)) ).

% image_empty
tff(fact_4847_empty__is__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(B),set(A),image(B,A,F2),A3) )
    <=> ( A3 = bot_bot(set(B)) ) ) ).

% empty_is_image
tff(fact_4848_image__is__empty,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( ( aa(set(B),set(A),image(B,A,F2),A3) = bot_bot(set(A)) )
    <=> ( A3 = bot_bot(set(B)) ) ) ).

% image_is_empty
tff(fact_4849_insert__image,axiom,
    ! [B: $tType,A: $tType,Xa: A,A3: set(A),F2: fun(A,B)] :
      ( member(A,Xa,A3)
     => ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),aa(A,B,F2,Xa)),aa(set(A),set(B),image(A,B,F2),A3)) = aa(set(A),set(B),image(A,B,F2),A3) ) ) ).

% insert_image
tff(fact_4850_image__insert,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A2: B,B3: set(B)] : aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(B,A,F2,A2)),aa(set(B),set(A),image(B,A,F2),B3)) ).

% image_insert
tff(fact_4851_max__ins__scaled,axiom,
    ! [Na: nat,X14: vEBT_VEBT,M: nat,X13: list(vEBT_VEBT)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),lattic643756798349783984er_Max(nat,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14)),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),set2(vEBT_VEBT,X13))))))) ).

% max_ins_scaled
tff(fact_4852_height__i__max,axiom,
    ! [I: nat,X13: list(vEBT_VEBT),Foo: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I))),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Foo),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),set2(vEBT_VEBT,X13))))) ) ).

% height_i_max
tff(fact_4853_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S) = S ) ).

% image_add_0
tff(fact_4854_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_4855_image__add__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [C2: A,A2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),C2)),set_ord_atMost(A,A2)) = set_ord_atMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)) ) ).

% image_add_atMost
tff(fact_4856_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_mb(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_4857_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D2: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_mc(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_4858_if__image__distrib,axiom,
    ! [A: $tType,B: $tType,P: fun(B,$o),F2: fun(B,A),G: fun(B,A),S: set(B)] : aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_md(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F2),G)),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),collect(B,P)))),aa(set(B),set(A),image(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),collect(B,aTP_Lamp_me(fun(B,$o),fun(B,$o),P))))) ).

% if_image_distrib
tff(fact_4859_Max__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [A3: set(A),C2: B] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( lattic643756798349783984er_Max(B,aa(set(A),set(B),image(A,B,aTP_Lamp_mf(B,fun(A,B),C2)),A3)) = C2 ) ) ) ) ).

% Max_const
tff(fact_4860_Min__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [A3: set(A),C2: B] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( lattic643756798350308766er_Min(B,aa(set(A),set(B),image(A,B,aTP_Lamp_mf(B,fun(A,B),C2)),A3)) = C2 ) ) ) ) ).

% Min_const
tff(fact_4861_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D2)
         => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),D2),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_4862_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D2)
         => ( aa(set(A),set(A),image(A,A,aTP_Lamp_mg(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),D2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D2)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_4863_minus__Max__eq__Min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S: set(A)] :
          ( finite_finite2(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),lattic643756798349783984er_Max(A,S)) = lattic643756798350308766er_Min(A,aa(set(A),set(A),image(A,A,uminus_uminus(A)),S)) ) ) ) ) ).

% minus_Max_eq_Min
tff(fact_4864_minus__Min__eq__Max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S: set(A)] :
          ( finite_finite2(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),lattic643756798350308766er_Min(A,S)) = lattic643756798349783984er_Max(A,aa(set(A),set(A),image(A,A,uminus_uminus(A)),S)) ) ) ) ) ).

% minus_Min_eq_Max
tff(fact_4865_image__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B3: set(B)] : aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image(B,A,F2),A3)),aa(set(B),set(A),image(B,A,F2),B3)) ).

% image_Un
tff(fact_4866_image__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),aa(set(A),set(B),image(A,B,F2),B3)) ) ).

% image_mono
tff(fact_4867_image__subsetI,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),B3: set(B)] :
      ( ! [X: A] :
          ( member(A,X,A3)
         => member(B,aa(A,B,F2,X),B3) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),B3) ) ).

% image_subsetI
tff(fact_4868_subset__imageE,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F2: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(B),set(A),image(B,A,F2),A3))
     => ~ ! [C9: set(B)] :
            ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C9),A3)
           => ( B3 != aa(set(B),set(A),image(B,A,F2),C9) ) ) ) ).

% subset_imageE
tff(fact_4869_image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F2),A3)),B3)
    <=> ! [X4: B] :
          ( member(B,X4,A3)
         => member(A,aa(B,A,F2,X4),B3) ) ) ).

% image_subset_iff
tff(fact_4870_subset__image__iff,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F2: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(B),set(A),image(B,A,F2),A3))
    <=> ? [AA: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),AA),A3)
          & ( B3 = aa(set(B),set(A),image(B,A,F2),AA) ) ) ) ).

% subset_image_iff
tff(fact_4871_imageE,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),A3: set(B)] :
      ( member(A,B2,aa(set(B),set(A),image(B,A,F2),A3))
     => ~ ! [X: B] :
            ( ( B2 = aa(B,A,F2,X) )
           => ~ member(B,X,A3) ) ) ).

% imageE
tff(fact_4872_image__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),A3: set(C)] : aa(set(B),set(A),image(B,A,F2),aa(set(C),set(B),image(C,B,G),A3)) = aa(set(C),set(A),image(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_mh(fun(B,A),fun(fun(C,B),fun(C,A)),F2),G)),A3) ).

% image_image
tff(fact_4873_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),P: fun(A,$o)] : collect(A,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_mi(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),F2),A3),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_mj(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),F2),A3),P))) ).

% Compr_image_eq
tff(fact_4874_imageI,axiom,
    ! [B: $tType,A: $tType,Xa: A,A3: set(A),F2: fun(A,B)] :
      ( member(A,Xa,A3)
     => member(B,aa(A,B,F2,Xa),aa(set(A),set(B),image(A,B,F2),A3)) ) ).

% imageI
tff(fact_4875_image__iff,axiom,
    ! [A: $tType,B: $tType,Z: A,F2: fun(B,A),A3: set(B)] :
      ( member(A,Z,aa(set(B),set(A),image(B,A,F2),A3))
    <=> ? [X4: B] :
          ( member(B,X4,A3)
          & ( Z = aa(B,A,F2,X4) ) ) ) ).

% image_iff
tff(fact_4876_bex__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(A,$o)] :
      ( ? [X5: A] :
          ( member(A,X5,aa(set(B),set(A),image(B,A,F2),A3))
          & aa(A,$o,P,X5) )
     => ? [X: B] :
          ( member(B,X,A3)
          & aa(A,$o,P,aa(B,A,F2,X)) ) ) ).

% bex_imageD
tff(fact_4877_image__cong,axiom,
    ! [B: $tType,A: $tType,M7: set(A),N3: set(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( M7 = N3 )
     => ( ! [X: A] :
            ( member(A,X,N3)
           => ( aa(A,B,F2,X) = aa(A,B,G,X) ) )
       => ( aa(set(A),set(B),image(A,B,F2),M7) = aa(set(A),set(B),image(A,B,G),N3) ) ) ) ).

% image_cong
tff(fact_4878_ball__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,aa(set(B),set(A),image(B,A,F2),A3))
         => aa(A,$o,P,X) )
     => ! [X5: B] :
          ( member(B,X5,A3)
         => aa(A,$o,P,aa(B,A,F2,X5)) ) ) ).

% ball_imageD
tff(fact_4879_rev__image__eqI,axiom,
    ! [B: $tType,A: $tType,Xa: A,A3: set(A),B2: B,F2: fun(A,B)] :
      ( member(A,Xa,A3)
     => ( ( B2 = aa(A,B,F2,Xa) )
       => member(B,B2,aa(set(A),set(B),image(A,B,F2),A3)) ) ) ).

% rev_image_eqI
tff(fact_4880_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( ~ finite_finite2(A,A3)
     => ( finite_finite2(B,aa(set(A),set(B),image(A,B,F2),A3))
       => ? [X: A] :
            ( member(A,X,A3)
            & ~ finite_finite2(A,collect(A,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_mk(set(A),fun(fun(A,B),fun(A,fun(A,$o))),A3),F2),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_4881_image__Int__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B3: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),image(B,A,F2),A3)),aa(set(B),set(A),image(B,A,F2),B3))) ).

% image_Int_subset
tff(fact_4882_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B3: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(B),set(A),image(B,A,F2),A3)),aa(set(B),set(A),image(B,A,F2),B3))),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B3))) ).

% image_diff_subset
tff(fact_4883_image__constant,axiom,
    ! [A: $tType,B: $tType,Xa: A,A3: set(A),C2: B] :
      ( member(A,Xa,A3)
     => ( aa(set(A),set(B),image(A,B,aTP_Lamp_ml(B,fun(A,B),C2)),A3) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),C2),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_4884_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,C2: A,A3: set(B)] :
      aa(set(B),set(A),image(B,A,aTP_Lamp_mm(A,fun(B,A),C2)),A3) = $ite(A3 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),C2),bot_bot(set(A)))) ).

% image_constant_conv
tff(fact_4885_sum_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),H: fun(A,B),G: fun(A,C)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),S) = aa(set(C),B,groups7311177749621191930dd_sum(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_mo(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S),H),G)),aa(set(A),set(C),image(A,C,G),S)) ) ) ) ).

% sum.image_gen
tff(fact_4886_prod_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),H: fun(A,B),G: fun(A,C)] :
          ( finite_finite2(A,S)
         => ( groups7121269368397514597t_prod(A,B,H,S) = groups7121269368397514597t_prod(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_mp(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S),H),G),aa(set(A),set(C),image(A,C,G),S)) ) ) ) ).

% prod.image_gen
tff(fact_4887_the__elem__image__unique,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B),Xa: A] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ! [Y3: A] :
            ( member(A,Y3,A3)
           => ( aa(A,B,F2,Y3) = aa(A,B,F2,Xa) ) )
       => ( the_elem(B,aa(set(A),set(B),image(A,B,F2),A3)) = aa(A,B,F2,Xa) ) ) ) ).

% the_elem_image_unique
tff(fact_4888_sum_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [S: set(A),T3: set(B),G: fun(A,B),H: fun(A,C)] :
          ( finite_finite2(A,S)
         => ( finite_finite2(B,T3)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),S)),T3)
             => ( aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_mr(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S),G),H)),T3) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,H),S) ) ) ) ) ) ).

% sum.group
tff(fact_4889_prod_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [S: set(A),T3: set(B),G: fun(A,B),H: fun(A,C)] :
          ( finite_finite2(A,S)
         => ( finite_finite2(B,T3)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),S)),T3)
             => ( groups7121269368397514597t_prod(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_ms(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S),G),H),T3) = groups7121269368397514597t_prod(A,C,H,S) ) ) ) ) ) ).

% prod.group
tff(fact_4890_set__image__eq__pointwiseI,axiom,
    ! [B: $tType,A: $tType,L: list(A),L3: list(A),F2: fun(A,B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(list(A),nat,size_size(list(A)),L3) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),L))
           => ( aa(A,B,F2,aa(nat,A,nth(A,L),I3)) = aa(A,B,F2,aa(nat,A,nth(A,L3),I3)) ) )
       => ( aa(set(A),set(B),image(A,B,F2),set2(A,L)) = aa(set(A),set(B),image(A,B,F2),set2(A,L3)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_4891_in__set__image__conv__nth,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xa: B,L: list(B)] :
      ( member(A,aa(B,A,F2,Xa),aa(set(B),set(A),image(B,A,F2),set2(B,L)))
    <=> ? [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),L))
          & ( aa(B,A,F2,aa(nat,B,nth(B,L),I2)) = aa(B,A,F2,Xa) ) ) ) ).

% in_set_image_conv_nth
tff(fact_4892_scaleR__image__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Xa: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(set(A),set(A),image(A,A,real_V8093663219630862766scaleR(A,C2)),set_or1337092689740270186AtMost(A,Xa,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),Xa),aa(A,A,real_V8093663219630862766scaleR(A,C2),Y)) ) ) ) ).

% scaleR_image_atLeastAtMost
tff(fact_4893_hom__Max__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,H,X)),aa(A,A,H,Y3))
         => ( finite_finite2(A,N3)
           => ( ( N3 != bot_bot(set(A)) )
             => ( aa(A,A,H,lattic643756798349783984er_Max(A,N3)) = lattic643756798349783984er_Max(A,aa(set(A),set(A),image(A,A,H),N3)) ) ) ) ) ) ).

% hom_Max_commute
tff(fact_4894_Max__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S: set(A),F2: fun(A,B),K: B] :
          ( finite_finite2(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( lattic643756798349783984er_Max(B,aa(set(A),set(B),image(A,B,aa(B,fun(A,B),aTP_Lamp_mt(fun(A,B),fun(B,fun(A,B)),F2),K)),S)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),lattic643756798349783984er_Max(B,aa(set(A),set(B),image(A,B,F2),S))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_4895_Min__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S: set(A),F2: fun(A,B),K: B] :
          ( finite_finite2(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( lattic643756798350308766er_Min(B,aa(set(A),set(B),image(A,B,aa(B,fun(A,B),aTP_Lamp_mt(fun(A,B),fun(B,fun(A,B)),F2),K)),S)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),lattic643756798350308766er_Min(B,aa(set(A),set(B),image(A,B,F2),S))),K) ) ) ) ) ).

% Min_add_commute
tff(fact_4896_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xa: A,Y: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,Xa,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),Xa),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),Xa),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),Xa)),bot_bot(set(A))) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_4897_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xa: A,Y: A] :
          aa(set(A),set(A),image(A,A,aTP_Lamp_mu(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,Xa,Y)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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),Xa),C2))),
            bot_bot(set(A)) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_4898_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_mv(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2))) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_4899_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_mw(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2))) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_4900_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_mx(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_4901_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_my(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_4902_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(Xa,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu2: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(B6)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(B6))),zero_zero(nat))) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv2),(Uw2))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux,Uy2,Uz2)),Xaa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2)),Xaa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                  one_one(nat),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),
                                        $ite(
                                          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                            $let(
                                              maxlow: option(nat),
                                              maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                                $ite(
                                                  ( ( maxlow != none(nat) )
                                                  & vEBT_VEBT_less(some(nat,l),maxlow) ),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                                  $let(
                                                    sc: option(nat),
                                                    sc:= vEBT_vebt_succ(Summary,h),
                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary,h))),one_one(nat))),
                                                      $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,sc)))))) ) )) )),
                                          one_one(nat) )) ) ) )) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_4903_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d(Xa,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),zero_zero(nat))) ) ) )
         => ( ! [A5: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((A5),(Uw2)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A5: $o,B6: $o] :
                  ( ( Xa = vEBT_Leaf((A5),(B6)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite((B6),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Xaa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve)),Xaa)) ) )
                 => ( ! [V4: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi) )
                       => ( ( Y = one_one(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                    one_one(nat),
                                    $let(
                                      l: nat,
                                      l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $let(
                                        h: nat,
                                        h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,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(aa(num,num,bit1,bit0(one2))))),one_one(nat))),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                            $let(
                                              minlow: option(nat),
                                              minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),
                                                $ite(
                                                  ( ( minlow != none(nat) )
                                                  & vEBT_VEBT_greater(some(nat,l),minlow) ),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                                  $let(
                                                    pr: option(nat),
                                                    pr:= vEBT_vebt_pred(Summary,h),
                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary,h))),one_one(nat))),
                                                      $ite(pr = none(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,pr)))))) ) )) ),
                                            one_one(nat) )) ) ) )) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_4904_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e(Xa,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),zero_zero(nat))) ) ) )
         => ( ! [A5: $o,B6: $o] :
                ( ( Xa = vEBT_Leaf((A5),(B6)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A5: $o,B6: $o] :
                  ( ( Xa = vEBT_Leaf((A5),(B6)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),aa(nat,nat,suc,aa(nat,nat,suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary)),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary)),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary) )
                       => ( ( Y = one_one(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                                  | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                                  one_one(nat),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    one_one(nat),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary)))))),Xaa),
                                      $let(
                                        l: nat,
                                        l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                              $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),vEBT_V1232361888498592333_e_t_e(Summary,h),one_one(nat))),
                                            one_one(nat) ) ) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
tff(fact_4905_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A3: set(A),F2: fun(nat,A),Na: nat] :
      ( ( A3 = aa(set(nat),set(A),image(nat,A,F2),collect(nat,aTP_Lamp_cy(nat,fun(nat,$o),Na))) )
     => finite_finite2(A,A3) ) ).

% nat_seg_image_imp_finite
tff(fact_4906_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
    <=> ? [N2: nat,F6: fun(nat,A)] : A3 = aa(set(nat),set(A),image(nat,A,F6),collect(nat,aTP_Lamp_cy(nat,fun(nat,$o),N2))) ) ).

% finite_conv_nat_seg_image
tff(fact_4907_in__image__insert__iff,axiom,
    ! [A: $tType,B3: set(set(A)),Xa: A,A3: set(A)] :
      ( ! [C9: set(A)] :
          ( member(set(A),C9,B3)
         => ~ member(A,Xa,C9) )
     => ( member(set(A),A3,aa(set(set(A)),set(set(A)),image(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa)),B3))
      <=> ( member(A,Xa,A3)
          & member(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))),B3) ) ) ) ).

% in_image_insert_iff
tff(fact_4908_image__int__atLeastAtMost,axiom,
    ! [A2: nat,B2: nat] : aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,A2,B2)) = set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastAtMost
tff(fact_4909_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_4910_lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_ord_lessThan(nat,Na))) ).

% lessThan_Suc_eq_insert_0
tff(fact_4911_atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_ord_atMost(nat,aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_ord_atMost(nat,Na))) ).

% atMost_Suc_eq_insert_0
tff(fact_4912_vebt__delete_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xa,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = vEBT_Leaf($false,(B6)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),zero_zero(nat))) ) ) )
         => ( ! [A5: $o,B6: $o] :
                ( ( Xa = vEBT_Leaf((A5),(B6)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = vEBT_Leaf((A5),$false) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A5: $o,B6: $o] :
                  ( ( Xa = vEBT_Leaf((A5),(B6)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                     => ( ( Y = vEBT_Leaf((A5),(B6)) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),aa(nat,nat,suc,aa(nat,nat,suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary) )
                   => ( ( Y = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary)),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst: list(vEBT_VEBT),Smry: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst,Smry) )
                     => ( ( Y = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst,Smry) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst,Smry)),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr: list(vEBT_VEBT),Sm: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) )
                       => ( ( Y = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                                  | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                                  vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary)))))),Xaa),
                                      $let(
                                        minn: nat,
                                        minn:= 
                                          $ite(Xaa = Mi2,xn,Mi2),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                            $let(
                                              newnode: vEBT_VEBT,
                                              newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                                              $let(
                                                newlist: list(vEBT_VEBT),
                                                newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                                $ite(
                                                  vEBT_VEBT_minNull(newnode),
                                                  $let(
                                                    sn: vEBT_VEBT,
                                                    sn:= vEBT_vebt_delete(Summary,h),
                                                    vEBT_Node(some(product_prod(nat,nat),
                                                        aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                          $ite(
                                                            xn = Ma2,
                                                            $let(
                                                              maxs: option(nat),
                                                              maxs:= vEBT_vebt_maxt(sn),
                                                              $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) ),
                                                            Ma2 ))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),newlist,sn) ),
                                                  vEBT_Node(some(product_prod(nat,nat),
                                                      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                        $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),newlist,Summary) ) ) ),
                                            vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) ) ) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_4913_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(Xa,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu2: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(B6)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(B6))),zero_zero(nat))) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv2),(Uw2))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux,Uy2,Uz2)),Xaa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2)),Xaa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( ( Y = $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                one_one(nat),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                      $let(
                                        maxlow: option(nat),
                                        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(some(nat,l),maxlow) ),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary,h)),one_one(nat)) ) ),
                                      one_one(nat) ) ) ) ) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_4914_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(Xa,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),Xaa)) ) )
         => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info,zero_zero(nat),Ts2,S2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info,zero_zero(nat),Ts2,S2)),Xaa)) ) )
           => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),Xaa)) ) )
             => ( ! [V4: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2)))))),
                              $let(
                                xn: nat,
                                xn:= 
                                  $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $ite(
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                    & ~ ( ( Xaa = Mi2 )
                                        | ( Xaa = Ma2 ) ) ),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                      $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t(Summary,h),one_one(nat))),
                                    one_one(nat) ) ) )) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_4915_vebt__insert_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xa,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero(nat),
                      vEBT_Leaf($true,(B6)),
                      $ite(Xaa = one_one(nat),vEBT_Leaf((A5),$true),vEBT_Leaf((A5),(B6))) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),Xaa)) ) )
         => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info,zero_zero(nat),Ts2,S2) )
               => ( ( Y = vEBT_Node(Info,zero_zero(nat),Ts2,S2) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info,zero_zero(nat),Ts2,S2)),Xaa)) ) )
           => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                 => ( ( Y = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),Xaa)) ) )
             => ( ! [V4: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary) )
                   => ( ( Y = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                  & ~ ( ( Xaa = Mi2 )
                                      | ( Xaa = Ma2 ) ) ),
                                  vEBT_Node(some(product_prod(nat,nat),
                                      aa(nat,product_prod(nat,nat),
                                        aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Xaa,Mi2)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),list_update(vEBT_VEBT,TreeList2,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_vebt_insert(Summary,h),Summary)),
                                  vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_4916_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d2(Xa,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),zero_zero(nat))) ) ) )
         => ( ! [A5: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((A5),(Uw2)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A5: $o,B6: $o] :
                  ( ( Xa = vEBT_Leaf((A5),(B6)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Xaa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve)),Xaa)) ) )
                 => ( ! [V4: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi) )
                       => ( ( Y = one_one(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  one_one(nat),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          $ite(
                                            ( ( minlow != none(nat) )
                                            & vEBT_VEBT_greater(some(nat,l),minlow) ),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary,h)),one_one(nat)) ) ),
                                        one_one(nat) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_4917_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(Xa,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( Y = one_one(nat) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),Xaa)) ) )
         => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info,zero_zero(nat),Ts2,S2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info,zero_zero(nat),Ts2,S2)),Xaa)) ) )
           => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),Xaa)) ) )
             => ( ! [V4: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V4)),TreeList2,Summary)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                  & ~ ( ( Xaa = Mi2 )
                                      | ( Xaa = Ma2 ) ) ),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t2(Summary,h),one_one(nat))),
                                  one_one(nat) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_4918_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(Xa,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                      $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy2,Uz2) )
                 => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              $ite(
                                Xaa = Mi2,
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    Xaa = Ma2,
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                        one_one(nat),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                            one_one(nat),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),
                                              $let(
                                                h: nat,
                                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat)) )) )) )) )) )) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_4919_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(Xa,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( Y = one_one(nat) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xaa = Mi2,
                                zero_zero(nat),
                                $ite(
                                  Xaa = Ma2,
                                  zero_zero(nat),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                    zero_zero(nat),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                      zero_zero(nat),
                                      $ite(
                                        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xaa)
                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Ma2) ),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)) ),
                                        zero_zero(nat) ) ) ) ) )) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_4920_VEBT__internal_Oheight_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,Xa) = Y )
     => ( ( ? [A5: $o,B6: $o] : Xa = vEBT_Leaf((A5),(B6))
         => ( Y != zero_zero(nat) ) )
       => ~ ! [Uu2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Uu2,Deg2,TreeList2,Summary) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),aa(vEBT_VEBT,fun(set(vEBT_VEBT),set(vEBT_VEBT)),insert(vEBT_VEBT),Summary),set2(vEBT_VEBT,TreeList2))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_4921_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Uu,Deg,TreeList,Summary2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),aa(vEBT_VEBT,fun(set(vEBT_VEBT),set(vEBT_VEBT)),insert(vEBT_VEBT),Summary2),set2(vEBT_VEBT,TreeList))))) ).

% VEBT_internal.height.simps(2)
tff(fact_4922_VEBT__internal_Oinsert_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_insert(Xa,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( Y = vEBT_vebt_insert(vEBT_Leaf((A5),(B6)),Xaa) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),Xaa)) ) )
         => ~ ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info,Deg2,TreeList2,Summary) )
               => ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)),Xaa),vEBT_Node(Info,Deg2,TreeList2,Summary),vEBT_vebt_insert(vEBT_Node(Info,Deg2,TreeList2,Summary),Xaa)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info,Deg2,TreeList2,Summary)),Xaa)) ) ) ) ) ) ).

% VEBT_internal.insert'.pelims
tff(fact_4923_vebt__member_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A5),
                    $ite(Xaa = one_one(nat),(B6),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) )
           => ( ! [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy2,Uz2) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy2,Uz2)),Xaa)) )
             => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa))
                       => $ite(
                            Xaa = Mi2,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                $false,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_4924_vebt__member_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A5),
                      $ite(Xaa = one_one(nat),(B6),$false) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy2,Uz2) )
                 => ( ~ (Y)
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ~ (Y)
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi2,
                              $true,
                              $ite(
                                Xaa = Ma2,
                                $true,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                  $false,
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_4925_vebt__member_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A5),
                      $ite(Xaa = one_one(nat),(B6),$false) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa))
                 => ~ $ite(
                        Xaa = Mi2,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_4926_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xa,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A5),
                    $ite(Xaa = one_one(nat),(B6),$false) ) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xaa)) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2)),Xaa))
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_4927_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xa,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A5),
                      $ite(Xaa = one_one(nat),(B6),$false) ) ) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2)),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_4928_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xa,Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A5),
                      $ite(Xaa = one_one(nat),(B6),$false) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),Xaa)) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xaa)) ) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList2,S2)),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_4929_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xa,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Xaa)) )
         => ( ! [Ux: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy2)),Xaa)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa))
                   => ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2) )
                   => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)),Xaa))
                     => ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
               => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)),Xaa))
                       => $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_4930_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xa,Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y)
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Xaa)) ) )
         => ( ! [Ux: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy2) )
               => ( ~ (Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy2)),Xaa)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 ) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi2 )
                          | ( Xaa = Ma2 )
                          | $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)),Xaa)) ) )
               => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_4931_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xa,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa))
               => ~ ( ( Xaa = Mi2 )
                    | ( Xaa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V4),TreeList2,Vc2)),Xaa))
                 => ~ ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 )
                      | $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
           => ~ ! [V4: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList2,Vd2)),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V4)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_4932_pred__lesseq__max,axiom,
    ! [Deg: nat,Xa: nat,Ma: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma)
       => ( vEBT_vebt_pred(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = $let(
              l: nat,
              l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( minlow != none(nat) )
                      & vEBT_VEBT_greater(some(nat,l),minlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),some(nat,h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                      $let(
                        pr: option(nat),
                        pr:= vEBT_vebt_pred(Summary2,h),
                        $ite(
                          pr = none(nat),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa),some(nat,Mi),none(nat)),
                          aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,pr)))) ) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_4933_pred__less__length__list,axiom,
    ! [Deg: nat,Xa: nat,Ma: nat,TreeList: list(vEBT_VEBT),Mi: nat,Summary2: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
         => ( vEBT_vebt_pred(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = $let(
                l: nat,
                l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( minlow != none(nat) )
                      & vEBT_VEBT_greater(some(nat,l),minlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),some(nat,h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                      $let(
                        pr: option(nat),
                        pr:= vEBT_vebt_pred(Summary2,h),
                        $ite(
                          pr = none(nat),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa),some(nat,Mi),none(nat)),
                          aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_4934_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,Xa: nat,TreeList: list(vEBT_VEBT),Ma: nat,Summary2: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Xa)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
         => ( vEBT_vebt_succ(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = $let(
                l: nat,
                l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $let(
                    maxlow: option(nat),
                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( maxlow != none(nat) )
                      & vEBT_VEBT_less(some(nat,l),maxlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),some(nat,h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                      $let(
                        sc: option(nat),
                        sc:= vEBT_vebt_succ(Summary2,h),
                        $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_4935_add__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,some(nat,Xa)),some(nat,Y)) = some(nat,Z) ) ) ).

% add_shift
tff(fact_4936_mul__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Xa),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,Xa)),some(nat,Y)) = some(nat,Z) ) ) ).

% mul_shift
tff(fact_4937_add__def,axiom,
    vEBT_VEBT_add = vEBT_V2048590022279873568_shift(nat,plus_plus(nat)) ).

% add_def
tff(fact_4938_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V2048590022279873568_shift(nat,times_times(nat)) ).

% mul_def
tff(fact_4939_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,Xa: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Xa)
       => ( vEBT_vebt_succ(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary2),Xa) = $let(
              l: nat,
              l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  $let(
                    maxlow: option(nat),
                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( maxlow != none(nat) )
                      & vEBT_VEBT_less(some(nat,l),maxlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),some(nat,h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                      $let(
                        sc: option(nat),
                        sc:= vEBT_vebt_succ(Summary2,h),
                        $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,sc))))) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_4940_vebt__succ_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_succ(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xa) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
        some(nat,Mi),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( maxlow != none(nat) )
                  & vEBT_VEBT_less(some(nat,l),maxlow) ),
                  aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),some(nat,h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  $let(
                    sc: option(nat),
                    sc:= vEBT_vebt_succ(Summary2,h),
                    $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,sc))))) ) ) ),
              none(nat) ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_4941_vebt__pred_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_pred(vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xa) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
        some(nat,Ma),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
              $let(
                minlow: option(nat),
                minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( minlow != none(nat) )
                  & vEBT_VEBT_greater(some(nat,l),minlow) ),
                  aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),some(nat,h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  $let(
                    pr: option(nat),
                    pr:= vEBT_vebt_pred(Summary2,h),
                    $ite(
                      pr = none(nat),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa),some(nat,Mi),none(nat)),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,pr)))) ) ) ) ),
              none(nat) ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_4942_vebt__succ_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xa,Xaa) = Y )
     => ( ! [Uu2: $o,B6: $o] :
            ( ( Xa = vEBT_Leaf((Uu2),(B6)) )
           => ( ( Xaa = zero_zero(nat) )
             => ( Y != $ite((B6),some(nat,one_one(nat)),none(nat)) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xa = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa(nat,nat,suc,N)
             => ( Y != none(nat) ) ) )
         => ( ( ? [Ux: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy2,Uz2)
             => ( Y != none(nat) ) )
           => ( ( ? [V4: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2)
               => ( Y != none(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)
                 => ( Y != none(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( Y != $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                            some(nat,Mi2),
                            $let(
                              l: nat,
                              l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                  $let(
                                    maxlow: option(nat),
                                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                    $ite(
                                      ( ( maxlow != none(nat) )
                                      & vEBT_VEBT_less(some(nat,l),maxlow) ),
                                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),some(nat,h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                      $let(
                                        sc: option(nat),
                                        sc:= vEBT_vebt_succ(Summary,h),
                                        $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,sc))))) ) ) ),
                                  none(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_4943_vebt__pred_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(Xa,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != none(nat) ) ) )
       => ( ! [A5: $o] :
              ( ? [Uw2: $o] : Xa = vEBT_Leaf((A5),(Uw2))
             => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != $ite((A5),some(nat,zero_zero(nat)),none(nat)) ) ) )
         => ( ! [A5: $o,B6: $o] :
                ( ( Xa = vEBT_Leaf((A5),(B6)) )
               => ( ? [Va2: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va2))
                 => ( Y != $ite(
                        (B6),
                        some(nat,one_one(nat)),
                        $ite((A5),some(nat,zero_zero(nat)),none(nat)) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)
               => ( Y != none(nat) ) )
             => ( ( ? [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve)
                 => ( Y != none(nat) ) )
               => ( ( ? [V4: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] : Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)
                   => ( Y != none(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( Y != $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                              some(nat,Ma2),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                    $let(
                                      minlow: option(nat),
                                      minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                      $ite(
                                        ( ( minlow != none(nat) )
                                        & vEBT_VEBT_greater(some(nat,l),minlow) ),
                                        aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),some(nat,h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                        $let(
                                          pr: option(nat),
                                          pr:= vEBT_vebt_pred(Summary,h),
                                          $ite(
                                            pr = none(nat),
                                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xaa),some(nat,Mi2),none(nat)),
                                            aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,pr)))) ) ) ) ),
                                    none(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_4944_vebt__succ_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xa,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu2: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(B6)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = $ite((B6),some(nat,one_one(nat)),none(nat)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(B6))),zero_zero(nat))) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv2),(Uw2))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy2,Uz2) )
                 => ( ( Y = none(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux,Uy2,Uz2)),Xaa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vc2,Vd2)),Xaa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh) )
                     => ( ( Y = none(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( ( Y = $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                some(nat,Mi2),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                      $let(
                                        maxlow: option(nat),
                                        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(some(nat,l),maxlow) ),
                                          aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),some(nat,h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                          $let(
                                            sc: option(nat),
                                            sc:= vEBT_vebt_succ(Summary,h),
                                            $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,sc))))) ) ) ),
                                      none(nat) ) ) ) ) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_4945_vebt__pred_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(Xa,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = none(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),zero_zero(nat))) ) ) )
         => ( ! [A5: $o,Uw2: $o] :
                ( ( Xa = vEBT_Leaf((A5),(Uw2)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = $ite((A5),some(nat,zero_zero(nat)),none(nat)) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A5: $o,B6: $o] :
                  ( ( Xa = vEBT_Leaf((A5),(B6)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                     => ( ( Y = $ite(
                              (B6),
                              some(nat,one_one(nat)),
                              $ite((A5),some(nat,zero_zero(nat)),none(nat)) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A5),(B6))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Xaa)) ) )
               => ( ! [V4: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve) )
                     => ( ( Y = none(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),zero_zero(nat),Vd2,Ve)),Xaa)) ) )
                 => ( ! [V4: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi) )
                       => ( ( Y = none(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),V4),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  some(nat,Ma2),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          $ite(
                                            ( ( minlow != none(nat) )
                                            & vEBT_VEBT_greater(some(nat,l),minlow) ),
                                            aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),some(nat,h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                            $let(
                                              pr: option(nat),
                                              pr:= vEBT_vebt_pred(Summary,h),
                                              $ite(
                                                pr = none(nat),
                                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xaa),some(nat,Mi2),none(nat)),
                                                aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,pr)))) ) ) ) ),
                                        none(nat) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_4946_foldr__zero,axiom,
    ! [Xs: list(nat),D2: nat] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(nat),nat,size_size(list(nat)),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,nth(nat,Xs),I3)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(nat),nat,size_size(list(nat)),Xs)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),foldr(nat,nat,plus_plus(nat),Xs,D2)),D2)) ) ).

% foldr_zero
tff(fact_4947_foldr__one,axiom,
    ! [D2: nat,Ys2: list(nat)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),D2),foldr(nat,nat,plus_plus(nat),Ys2,D2)) ).

% foldr_one
tff(fact_4948_foldr__same__int,axiom,
    ! [Xs: list(nat),Y: nat] :
      ( ! [X: nat,Y3: nat] :
          ( member(nat,X,set2(nat,Xs))
         => ( member(nat,Y3,set2(nat,Xs))
           => ( X = Y3 ) ) )
     => ( ! [X: nat] :
            ( member(nat,X,set2(nat,Xs))
           => ( X = Y ) )
       => ( foldr(nat,nat,plus_plus(nat),Xs,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(nat),nat,size_size(list(nat)),Xs)),Y) ) ) ) ).

% foldr_same_int
tff(fact_4949_foldr__mono,axiom,
    ! [Xs: list(nat),Ys2: list(nat),C2: nat,D2: nat] :
      ( ( aa(list(nat),nat,size_size(list(nat)),Xs) = aa(list(nat),nat,size_size(list(nat)),Ys2) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(nat),nat,size_size(list(nat)),Xs))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,nth(nat,Xs),I3)),aa(nat,nat,nth(nat,Ys2),I3)) )
       => ( 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),foldr(nat,nat,plus_plus(nat),Xs,C2)),aa(list(nat),nat,size_size(list(nat)),Ys2))),foldr(nat,nat,plus_plus(nat),Ys2,D2)) ) ) ) ).

% foldr_mono
tff(fact_4950_foldr__length,axiom,
    ! [A: $tType,L: list(A)] : foldr(A,nat,aTP_Lamp_mz(A,fun(nat,nat)),L,zero_zero(nat)) = aa(list(A),nat,size_size(list(A)),L) ).

% foldr_length
tff(fact_4951_foldr__length__aux,axiom,
    ! [A: $tType,L: list(A),A2: nat] : foldr(A,nat,aTP_Lamp_mz(A,fun(nat,nat)),L,A2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),aa(list(A),nat,size_size(list(A)),L)) ).

% foldr_length_aux
tff(fact_4952_foldr__same,axiom,
    ! [Xs: list(real),Y: real] :
      ( ! [X: real,Y3: real] :
          ( member(real,X,set2(real,Xs))
         => ( member(real,Y3,set2(real,Xs))
           => ( X = Y3 ) ) )
     => ( ! [X: real] :
            ( member(real,X,set2(real,Xs))
           => ( X = Y ) )
       => ( foldr(real,real,plus_plus(real),Xs,zero_zero(real)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(list(real),nat,size_size(list(real)),Xs))),Y) ) ) ) ).

% foldr_same
tff(fact_4953_vebt__delete__code_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] : vEBT_vebt_delete(vEBT_Node(Info2,Deg,TreeList,Summary2),Xa) = case_option(vEBT_VEBT,product_prod(nat,nat),vEBT_Node(Info2,Deg,TreeList,Summary2),aa(nat,fun(product_prod(nat,nat),vEBT_VEBT),aa(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT)))),aTP_Lamp_nb(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT))))),Info2),Deg),TreeList),Summary2),Xa),Info2) ).

% vebt_delete_code(2)
tff(fact_4954_list__every__elemnt__bound__sum__bound,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,nat),Bound: nat,I: nat] :
      ( ! [X: A] :
          ( member(A,X,set2(A,Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,X)),Bound) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),foldr(nat,nat,plus_plus(nat),map(A,nat,F2,Xs),I)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_4955_foldr0,axiom,
    ! [Xs: list(real),C2: real,D2: real] : foldr(real,real,plus_plus(real),Xs,aa(real,real,aa(real,fun(real,real),plus_plus(real),C2),D2)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),foldr(real,real,plus_plus(real),Xs,D2)),C2) ).

% foldr0
tff(fact_4956_map__ident,axiom,
    ! [A: $tType,X5: list(A)] : map(A,A,aTP_Lamp_ma(A,A),X5) = X5 ).

% map_ident
tff(fact_4957_nth__map,axiom,
    ! [B: $tType,A: $tType,Na: nat,Xs: list(A),F2: fun(A,B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,B,nth(B,map(A,B,F2,Xs)),Na) = aa(A,B,F2,aa(nat,A,nth(A,Xs),Na)) ) ) ).

% nth_map
tff(fact_4958_list_Omap__ident,axiom,
    ! [A: $tType,T: list(A)] : map(A,A,aTP_Lamp_ma(A,A),T) = T ).

% list.map_ident
tff(fact_4959_nested__case__prod__simp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F2: fun(B,fun(C,fun(D,A))),Xa: product_prod(B,C),Y: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),product_case_prod(B,C,fun(D,A),F2),Xa),Y) = aa(product_prod(B,C),A,product_case_prod(B,C,A,aa(D,fun(B,fun(C,A)),aTP_Lamp_nc(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F2),Y)),Xa) ).

% nested_case_prod_simp
tff(fact_4960_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_nd(nat,fun(nat,fun(nat,$o)),Na)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_nf(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,Na)) ) ).

% sum.triangle_reindex_eq
tff(fact_4961_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_nd(nat,fun(nat,fun(nat,$o)),Na)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_nh(fun(nat,fun(nat,A)),fun(nat,A),G),set_ord_atMost(nat,Na)) ) ).

% prod.triangle_reindex_eq
tff(fact_4962_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_ni(nat,fun(nat,fun(nat,$o)),Na)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_nf(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,Na)) ) ).

% sum.triangle_reindex
tff(fact_4963_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_ni(nat,fun(nat,fun(nat,$o)),Na)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_nh(fun(nat,fun(nat,A)),fun(nat,A),G),set_ord_lessThan(nat,Na)) ) ).

% prod.triangle_reindex
tff(fact_4964_map__upd__eq,axiom,
    ! [B: $tType,A: $tType,I: nat,L: list(A),F2: fun(A,B),Xa: A] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
       => ( aa(A,B,F2,aa(nat,A,nth(A,L),I)) = aa(A,B,F2,Xa) ) )
     => ( map(A,B,F2,list_update(A,L,I,Xa)) = map(A,B,F2,L) ) ) ).

% map_upd_eq
tff(fact_4965_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_Node(Info2,Deg,TreeList,Summary2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Summary2))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2,TreeList),zero_zero(nat))) ).

% VEBT_internal.cnt'.simps(2)
tff(fact_4966_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Xa) = Y )
     => ( ( ? [A5: $o,B6: $o] : Xa = vEBT_Leaf((A5),(B6))
         => ( Y != one_one(nat) ) )
       => ~ ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info,Deg2,TreeList2,Summary) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Summary))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2,TreeList2),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.cnt'.elims
tff(fact_4967_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_space2,vEBT_Node(Info2,Deg,TreeList,Summary2)) = 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(aa(num,num,bit1,one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Summary2))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2,TreeList),zero_zero(nat))) ).

% VEBT_internal.space'.simps(2)
tff(fact_4968_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Xa) = Y )
     => ( ( ? [A5: $o,B6: $o] : Xa = vEBT_Leaf((A5),(B6))
         => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
       => ~ ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info,Deg2,TreeList2,Summary) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Summary))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2,TreeList2),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.space'.elims
tff(fact_4969_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_space,vEBT_Node(Info2,Deg,TreeList,Summary2)) = 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),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary2))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space,TreeList),zero_zero(nat))) ).

% VEBT_internal.space.simps(2)
tff(fact_4970_vebt__insert__code_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_insert(vEBT_Node(Info2,Deg,TreeList,Summary2),Xa) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Deg),one_one(nat)),vEBT_Node(Info2,Deg,TreeList,Summary2),case_option(vEBT_VEBT,product_prod(nat,nat),vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Xa)),Deg,TreeList,Summary2),product_case_prod(nat,nat,vEBT_VEBT,aa(nat,fun(nat,fun(nat,vEBT_VEBT)),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT)))),aTP_Lamp_nj(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))))),Deg),TreeList),Summary2),Xa)),Info2)) ).

% vebt_insert_code(2)
tff(fact_4971_VEBT__internal_Ospace_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,Xa) = Y )
     => ( ( ? [A5: $o,B6: $o] : Xa = vEBT_Leaf((A5),(B6))
         => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ~ ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info,Deg2,TreeList2,Summary) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space,TreeList2),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.space.elims
tff(fact_4972_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,fun(C,A)),A2: B,B2: 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),A2),B2)) = aa(C,A,aa(B,fun(C,A),F2,A2),B2) ).

% case_prod_conv
tff(fact_4973_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: 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),A2),B2),A3)
     => member(C,aa(B,C,aa(A,fun(B,C),F2,A2),B2),aa(set(product_prod(A,B)),set(C),image(product_prod(A,B),C,product_case_prod(A,B,C,F2)),A3)) ) ).

% pair_imageI
tff(fact_4974_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,real),Bound: real,I: real] :
      ( ! [X: A] :
          ( member(A,X,set2(A,Xs))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F2,X)),Bound) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),foldr(real,real,plus_plus(real),map(A,real,F2,Xs),I)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(list(A),nat,size_size(list(A)),Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_4975_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: product_prod(A,B),Z: C,C2: fun(A,fun(B,set(C)))] :
      ( ! [A5: A,B6: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B6) )
         => member(C,Z,aa(B,set(C),aa(A,fun(B,set(C)),C2,A5),B6)) )
     => member(C,Z,aa(product_prod(A,B),set(C),product_case_prod(A,B,set(C),C2),P2)) ) ).

% mem_case_prodI2
tff(fact_4976_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z: A,C2: fun(B,fun(C,set(A))),A2: B,B2: C] :
      ( member(A,Z,aa(C,set(A),aa(B,fun(C,set(A)),C2,A2),B2))
     => 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),A2),B2))) ) ).

% mem_case_prodI
tff(fact_4977_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,fun(C,$o))),Xa: C] :
      ( ! [A5: A,B6: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B6) = P2 )
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,A5),B6),Xa) )
     => aa(C,$o,aa(product_prod(A,B),fun(C,$o),product_case_prod(A,B,fun(C,$o),C2),P2),Xa) ) ).

% case_prodI2'
tff(fact_4978_split__part,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: fun(A,fun(B,$o)),X5: 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_nk($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(P)),Q)),X5)
    <=> ( (P)
        & aa(product_prod(A,B),$o,product_case_prod(A,B,$o,Q),X5) ) ) ).

% split_part
tff(fact_4979_listsum__bound,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,real),Y: real] :
      ( ! [X: A] :
          ( member(A,X,set2(A,Xs))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,F2,X)) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),foldr(real,real,plus_plus(real),map(A,real,F2,Xs),Y)) ) ).

% listsum_bound
tff(fact_4980_f__g__map__foldr__bound,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,real),C2: real,G: fun(A,real),D2: real] :
      ( ! [X: A] :
          ( member(A,X,set2(A,Xs))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F2,X)),aa(real,real,aa(real,fun(real,real),times_times(real),C2),aa(A,real,G,X))) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),foldr(real,real,plus_plus(real),map(A,real,F2,Xs),D2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),C2),foldr(real,real,plus_plus(real),map(A,real,G,Xs),zero_zero(real)))),D2)) ) ).

% f_g_map_foldr_bound
tff(fact_4981_real__nat__list,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A),C2: nat] : aa(nat,real,semiring_1_of_nat(real),foldr(nat,nat,plus_plus(nat),map(A,nat,F2,Xs),C2)) = foldr(real,real,plus_plus(real),map(A,real,aTP_Lamp_nl(fun(A,nat),fun(A,real),F2),Xs),aa(nat,real,semiring_1_of_nat(real),C2)) ).

% real_nat_list
tff(fact_4982_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,$o))] :
      ( ! [A5: A,B6: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B6) )
         => aa(B,$o,aa(A,fun(B,$o),C2,A5),B6) )
     => aa(product_prod(A,B),$o,product_case_prod(A,B,$o,C2),P2) ) ).

% case_prodI2
tff(fact_4983_case__prodI,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,$o)),A2: A,B2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),F2,A2),B2)
     => 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),A2),B2)) ) ).

% case_prodI
tff(fact_4984_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,$o,aTP_Lamp_nm(A,fun(B,$o))),Prod) ).

% prod.disc_eq_case
tff(fact_4985_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))] :
      ( ! [A5: A,B6: B] :
          ( ( T = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B6) )
         => time_TBOUND(C,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F2,A5),B6),aa(B,nat,aa(A,fun(B,nat),Bnd,A5),B6)) )
     => 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_4986_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,$o),C2),P2),Z)
     => ~ ! [X: A,Y3: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3) )
           => ~ aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,X),Y3),Z) ) ) ).

% case_prodE'
tff(fact_4987_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: fun(A,fun(B,fun(C,$o))),A2: A,B2: B,C2: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),product_case_prod(A,B,fun(C,$o),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),C2)
     => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),R2,A2),B2),C2) ) ).

% case_prodD'
tff(fact_4988_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)
     => ~ ! [X: A,Y3: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3) )
           => ~ aa(B,$o,aa(A,fun(B,$o),C2,X),Y3) ) ) ).

% case_prodE
tff(fact_4989_case__prodD,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,$o)),A2: A,B2: 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),A2),B2))
     => aa(B,$o,aa(A,fun(B,$o),F2,A2),B2) ) ).

% case_prodD
tff(fact_4990_case__prod__rule,axiom,
    ! [A: $tType,B: $tType,C: $tType,Xa: product_prod(A,B),P: assn,F2: fun(A,fun(B,heap_Time_Heap(C))),Q: fun(C,assn)] :
      ( ! [A5: A,B6: B] :
          ( ( Xa = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B6) )
         => hoare_hoare_triple(C,P,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F2,A5),B6),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),Xa),Q) ) ).

% case_prod_rule
tff(fact_4991_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,$o)),aTP_Lamp_nn(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),M),P))) ).

% rel_of_def
tff(fact_4992_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_no(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),H),F2)),Prod) ).

% prod.case_distrib
tff(fact_4993_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_np(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_4994_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_nq(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_4995_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,$o,Q,aa(product_prod(B,C),A,product_case_prod(B,C,A,P),Z))
     => ~ ! [X: B,Y3: C] :
            ( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y3) )
           => ~ aa(A,$o,Q,aa(C,A,aa(B,fun(C,A),P,X),Y3)) ) ) ).

% case_prodE2
tff(fact_4996_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_nr(fun(product_prod(A,B),C),fun(A,fun(B,C)),F2)) = F2 ).

% case_prod_eta
tff(fact_4997_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
      ( ! [X: A,Y3: B] : aa(B,C,aa(A,fun(B,C),F2,X),Y3) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3))
     => ( product_case_prod(A,B,C,F2) = G ) ) ).

% cond_case_prod_eta
tff(fact_4998_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_ns(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_4999_vebt__succ__code_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_succ(vEBT_Node(Info2,Deg,TreeList,Summary2),Xa) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Deg),one_one(nat)),none(nat),case_option(option(nat),product_prod(nat,nat),none(nat),product_case_prod(nat,nat,option(nat),aa(nat,fun(nat,fun(nat,option(nat))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat))))),aTP_Lamp_nt(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat)))))),Deg),TreeList),Summary2),Xa)),Info2)) ).

% vebt_succ_code(2)
tff(fact_5000_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_nu(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_5001_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M),bit0(Na)) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_nv(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,Na)) ) ).

% divmod_algorithm_code(6)
tff(fact_5002_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_5003_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_5004_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num] : unique8689654367752047608divmod(A,one2,aa(num,num,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_5005_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_nw(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,Na)) ) ).

% divmod_algorithm_code(5)
tff(fact_5006_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,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),aa(num,num,bit1,M))),unique1321980374590559556d_step(A,aa(num,num,bit1,Na),unique8689654367752047608divmod(A,aa(num,num,bit1,M),bit0(aa(num,num,bit1,Na))))) ) ).

% divmod_algorithm_code(8)
tff(fact_5007_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,bit0(M),aa(num,num,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,aa(num,num,bit1,Na),unique8689654367752047608divmod(A,bit0(M),bit0(aa(num,num,bit1,Na))))) ) ).

% divmod_algorithm_code(7)
tff(fact_5008_divmod__integer_H__def,axiom,
    ! [M: num,Na: num] : unique8689654367752047608divmod(code_integer,M,Na) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(num,code_integer,numeral_numeral(code_integer),M)),aa(num,code_integer,numeral_numeral(code_integer),Na))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),aa(num,code_integer,numeral_numeral(code_integer),M)),aa(num,code_integer,numeral_numeral(code_integer),Na))) ).

% divmod_integer'_def
tff(fact_5009_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_5010_inf__option__def,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Xa: option(A),Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),Xa),Y) = case_option(option(A),A,none(A),aTP_Lamp_ny(option(A),fun(A,option(A)),Y),Xa) ) ).

% inf_option_def
tff(fact_5011_sup__option__def,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Xa: option(A),Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),Xa),Y) = case_option(option(A),A,Y,aa(option(A),fun(A,option(A)),aTP_Lamp_oa(option(A),fun(option(A),fun(A,option(A))),Xa),Y),Xa) ) ).

% sup_option_def
tff(fact_5012_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_5013_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_5014_less__integer__code_I1_J,axiom,
    ~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_integer_code(1)
tff(fact_5015_abs__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,code_integer,abs_abs(code_integer),K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),K),K) ).

% abs_integer_code
tff(fact_5016_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_5017_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_5018_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_5019_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_5020_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_5021_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_5022_integer__of__int__code,axiom,
    ! [K: int] :
      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),code_integer_of_int(aa(int,int,uminus_uminus(int),K))),
        $ite(
          K = zero_zero(int),
          zero_zero(code_integer),
          $let(
            l: code_integer,
            l:= aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),code_integer_of_int(aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))))),
            $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int),l,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),l),one_one(code_integer))) ) ) ) ).

% integer_of_int_code
tff(fact_5023_set__union,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : set2(A,union(A,Xs,Ys2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set2(A,Xs)),set2(A,Ys2)) ).

% set_union
tff(fact_5024_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),zero_zero(int))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),one_one(int)),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_5025_less__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),code_integer_of_int(Xa)),code_integer_of_int(Xb))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),Xb) ) ).

% less_integer.abs_eq
tff(fact_5026_plus__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_integer_of_int(Xa)),code_integer_of_int(Xb)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Xb)) ).

% plus_integer.abs_eq
tff(fact_5027_times__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),code_integer_of_int(Xa)),code_integer_of_int(Xb)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),times_times(int),Xa),Xb)) ).

% times_integer.abs_eq
tff(fact_5028_divide__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),code_integer_of_int(Xa)),code_integer_of_int(Xb)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),Xb)) ).

% divide_integer.abs_eq
tff(fact_5029_modulo__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),code_integer_of_int(Xa)),code_integer_of_int(Xb)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xa),Xb)) ).

% modulo_integer.abs_eq
tff(fact_5030_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = R ) ) ).

% mod_int_unique
tff(fact_5031_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = Q2 ) ) ).

% div_int_unique
tff(fact_5032_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero(int) )
     => ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L) )
       => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_5033_eucl__rel__int,axiom,
    ! [K: int,L: int] : eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L))) ).

% eucl_rel_int
tff(fact_5034_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
    <=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q2)),R) )
        & $ite(
            aa(int,$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)) ),
              Q2 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_5035_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_5036_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_ob(nat,fun(nat,A))),divmod_nat(Na,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% of_nat_code_if
tff(fact_5037_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_oc(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na),Na)) ) ).

% divmod_nat_if
tff(fact_5038_divmod__nat__def,axiom,
    ! [M: nat,Na: nat] : divmod_nat(M,Na) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)) ).

% divmod_nat_def
tff(fact_5039_product__nth,axiom,
    ! [A: $tType,B: $tType,Na: nat,Xs: list(A),Ys2: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys2)))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys2)),Na) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(list(B),nat,size_size(list(B)),Ys2)))),aa(nat,B,nth(B,Ys2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),aa(list(B),nat,size_size(list(B)),Ys2)))) ) ) ).

% product_nth
tff(fact_5040_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,$o,sums(A,aTP_Lamp_od(A,fun(nat,A),Xa)),aa(A,A,sinh(A),Xa)) ) ).

% sinh_converges
tff(fact_5041_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,T: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_oe(A,fun(A,A),A2)),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_oe(A,fun(A,A),A2)),T)) ) ).

% translation_subtract_Compl
tff(fact_5042_sinh__real__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa(real,real,sinh(real),Xa) = zero_zero(real) )
    <=> ( Xa = zero_zero(real) ) ) ).

% sinh_real_zero_iff
tff(fact_5043_sinh__real__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sinh(real),Xa)),aa(real,real,sinh(real),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ).

% sinh_real_less_iff
tff(fact_5044_sinh__real__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sinh(real),Xa)),aa(real,real,sinh(real),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ).

% sinh_real_le_iff
tff(fact_5045_sinh__real__abs,axiom,
    ! [Xa: real] : aa(real,real,sinh(real),aa(real,real,abs_abs(real),Xa)) = aa(real,real,abs_abs(real),aa(real,real,sinh(real),Xa)) ).

% sinh_real_abs
tff(fact_5046_sinh__real__neg__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sinh(real),Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% sinh_real_neg_iff
tff(fact_5047_sinh__real__pos__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sinh(real),Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% sinh_real_pos_iff
tff(fact_5048_sinh__real__nonneg__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sinh(real),Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% sinh_real_nonneg_iff
tff(fact_5049_sinh__real__nonpos__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sinh(real),Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% sinh_real_nonpos_iff
tff(fact_5050_sinh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,sinh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sinh_0
tff(fact_5051_sinh__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,A,sinh(A),aa(A,A,uminus_uminus(A),Xa)) = aa(A,A,uminus_uminus(A),aa(A,A,sinh(A),Xa)) ) ).

% sinh_minus
tff(fact_5052_length__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),product(A,B,Xs,Ys2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys2)) ).

% length_product
tff(fact_5053_arsinh__sinh__real,axiom,
    ! [Xa: real] : aa(real,real,arsinh(real),aa(real,real,sinh(real),Xa)) = Xa ).

% arsinh_sinh_real
tff(fact_5054_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( aa(A,A,sinh(A),Xa) = zero_zero(A) )
        <=> member(A,aa(A,A,exp(A),Xa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),one_one(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(A,A,uminus_uminus(A),one_one(A))),bot_bot(set(A))))) ) ) ).

% sinh_zero_iff
tff(fact_5055_sinh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : aa(A,A,sinh(A),Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,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_5056_sinh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,A,sinh(A),Xa) = 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),minus_minus(A),aa(A,A,exp(A),Xa)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xa)))) ) ).

% sinh_def
tff(fact_5057_sinh__ln__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,sinh(real),aa(real,real,ln_ln(real),Xa)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),aa(real,real,inverse_inverse(real),Xa))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% sinh_ln_real
tff(fact_5058_translation__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,Sa: set(A),T: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Sa),T)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),Sa)),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),T)) ) ).

% translation_Int
tff(fact_5059_translation__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,Sa: set(A),T: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Sa),T)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),Sa)),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),T)) ) ).

% translation_diff
tff(fact_5060_translation__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,T: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),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),A2)),T)) ) ).

% translation_Compl
tff(fact_5061_translation__subtract__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,Sa: set(A),T: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_oe(A,fun(A,A),A2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Sa),T)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image(A,A,aTP_Lamp_oe(A,fun(A,A),A2)),Sa)),aa(set(A),set(A),image(A,A,aTP_Lamp_oe(A,fun(A,A),A2)),T)) ) ).

% translation_subtract_Int
tff(fact_5062_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,Sa: set(A),T: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_oe(A,fun(A,A),A2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Sa),T)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),image(A,A,aTP_Lamp_oe(A,fun(A,A),A2)),Sa)),aa(set(A),set(A),image(A,A,aTP_Lamp_oe(A,fun(A,A),A2)),T)) ) ).

% translation_subtract_diff
tff(fact_5063_list__assn__aux__ineq__len,axiom,
    ! [B: $tType,A: $tType,L: list(A),Li2: list(B),A3: fun(A,fun(B,assn))] :
      ( ( aa(list(A),nat,size_size(list(A)),L) != aa(list(B),nat,size_size(list(B)),Li2) )
     => ( aa(list(B),assn,vEBT_List_list_assn(A,B,A3,L),Li2) = bot_bot(assn) ) ) ).

% list_assn_aux_ineq_len
tff(fact_5064_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,$o,sums(A,aTP_Lamp_of(A,fun(nat,A),Xa)),aa(A,A,cosh(A),Xa)) ) ).

% cosh_converges
tff(fact_5065_vebt__buildup_Oelims,axiom,
    ! [Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va2: nat] :
                ( ( Xa = 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))),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_5066_intind,axiom,
    ! [A: $tType,I: nat,Na: nat,P: fun(A,$o),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Na)
     => ( aa(A,$o,P,Xa)
       => aa(A,$o,P,aa(nat,A,nth(A,replicate(A,Na,Xa)),I)) ) ) ).

% intind
tff(fact_5067_cosh__real__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa(real,real,cosh(real),Xa) = aa(real,real,cosh(real),Y) )
    <=> ( aa(real,real,abs_abs(real),Xa) = aa(real,real,abs_abs(real),Y) ) ) ).

% cosh_real_eq_iff
tff(fact_5068_cosh__real__abs,axiom,
    ! [Xa: real] : aa(real,real,cosh(real),aa(real,real,abs_abs(real),Xa)) = aa(real,real,cosh(real),Xa) ).

% cosh_real_abs
tff(fact_5069_sinh__real__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa(real,real,sinh(real),Xa) = aa(real,real,sinh(real),Y) )
    <=> ( Xa = Y ) ) ).

% sinh_real_eq_iff
tff(fact_5070_repli__cons__repl,axiom,
    ! [B: $tType,A: $tType,Q: assn,Xa: heap_Time_Heap(A),A3: fun(B,fun(A,assn)),Y: B,Na: nat] :
      ( hoare_hoare_triple(A,Q,Xa,aa(B,fun(A,assn),aa(fun(B,fun(A,assn)),fun(B,fun(A,assn)),aTP_Lamp_og(assn,fun(fun(B,fun(A,assn)),fun(B,fun(A,assn))),Q),A3),Y))
     => hoare_hoare_triple(list(A),Q,vEBT_VEBT_replicatei(A,Na,Xa),aa(nat,fun(list(A),assn),aa(B,fun(nat,fun(list(A),assn)),aa(fun(B,fun(A,assn)),fun(B,fun(nat,fun(list(A),assn))),aTP_Lamp_oh(assn,fun(fun(B,fun(A,assn)),fun(B,fun(nat,fun(list(A),assn)))),Q),A3),Y),Na)) ) ).

% repli_cons_repl
tff(fact_5071_repli__emp,axiom,
    ! [A: $tType,B: $tType,Xa: heap_Time_Heap(A),A3: fun(B,fun(A,assn)),Y: B,Na: nat] :
      ( hoare_hoare_triple(A,one_one(assn),Xa,aa(B,fun(A,assn),A3,Y))
     => hoare_hoare_triple(list(A),one_one(assn),vEBT_VEBT_replicatei(A,Na,Xa),vEBT_List_list_assn(B,A,A3,replicate(B,Na,Y))) ) ).

% repli_emp
tff(fact_5072_cosh__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,A,cosh(A),aa(A,A,uminus_uminus(A),Xa)) = aa(A,A,cosh(A),Xa) ) ).

% cosh_minus
tff(fact_5073_cosh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,cosh(A),zero_zero(A)) = one_one(A) ) ) ).

% cosh_0
tff(fact_5074_nth__replicate,axiom,
    ! [A: $tType,I: nat,Na: nat,Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Na)
     => ( aa(nat,A,nth(A,replicate(A,Na,Xa)),I) = Xa ) ) ).

% nth_replicate
tff(fact_5075_set__replicate,axiom,
    ! [A: $tType,Na: nat,Xa: A] :
      ( ( Na != zero_zero(nat) )
     => ( set2(A,replicate(A,Na,Xa)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_5076_sinh__less__cosh__real,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sinh(real),Xa)),aa(real,real,cosh(real),Xa)) ).

% sinh_less_cosh_real
tff(fact_5077_sinh__le__cosh__real,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sinh(real),Xa)),aa(real,real,cosh(real),Xa)) ).

% sinh_le_cosh_real
tff(fact_5078_cosh__real__nonzero,axiom,
    ! [Xa: real] : aa(real,real,cosh(real),Xa) != zero_zero(real) ).

% cosh_real_nonzero
tff(fact_5079_cosh__real__pos,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,cosh(real),Xa)) ).

% cosh_real_pos
tff(fact_5080_cosh__real__nonpos__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),zero_zero(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,cosh(real),Xa)),aa(real,real,cosh(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_5081_cosh__real__nonneg__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,cosh(real),Xa)),aa(real,real,cosh(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_5082_cosh__real__nonneg,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,cosh(real),Xa)) ).

% cosh_real_nonneg
tff(fact_5083_cosh__real__ge__1,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,cosh(real),Xa)) ).

% cosh_real_ge_1
tff(fact_5084_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K: A,Lst: list(B)] : map(B,A,aTP_Lamp_mm(A,fun(B,A),K),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K) ).

% map_replicate_const
tff(fact_5085_cosh__real__strict__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cosh(real),Xa)),aa(real,real,cosh(real),Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_5086_cosh__real__nonneg__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cosh(real),Xa)),aa(real,real,cosh(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_5087_cosh__real__nonpos__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),zero_zero(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cosh(real),Xa)),aa(real,real,cosh(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_5088_arcosh__cosh__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,real,arcosh(real),aa(real,real,cosh(real),Xa)) = Xa ) ) ).

% arcosh_cosh_real
tff(fact_5089_cosh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,A,cosh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cosh(A),Xa)),aa(A,A,cosh(A),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sinh(A),Xa)),aa(A,A,sinh(A),Y))) ) ).

% cosh_add
tff(fact_5090_sinh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,A,sinh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sinh(A),Xa)),aa(A,A,cosh(A),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cosh(A),Xa)),aa(A,A,sinh(A),Y))) ) ).

% sinh_add
tff(fact_5091_cosh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,A,cosh(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cosh(A),Xa)),aa(A,A,cosh(A),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sinh(A),Xa)),aa(A,A,sinh(A),Y))) ) ).

% cosh_diff
tff(fact_5092_sinh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : aa(A,A,sinh(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sinh(A),Xa)),aa(A,A,cosh(A),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cosh(A),Xa)),aa(A,A,sinh(A),Y))) ) ).

% sinh_diff
tff(fact_5093_sinh__plus__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,sinh(A),Xa)),aa(A,A,cosh(A),Xa)) = aa(A,A,exp(A),Xa) ) ).

% sinh_plus_cosh
tff(fact_5094_cosh__plus__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,cosh(A),Xa)),aa(A,A,sinh(A),Xa)) = aa(A,A,exp(A),Xa) ) ).

% cosh_plus_sinh
tff(fact_5095_tanh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,tanh(A),Xa) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,sinh(A),Xa)),aa(A,A,cosh(A),Xa)) ) ).

% tanh_def
tff(fact_5096_set__replicate__Suc,axiom,
    ! [A: $tType,Na: nat,Xa: A] : set2(A,replicate(A,aa(nat,nat,suc,Na),Xa)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ).

% set_replicate_Suc
tff(fact_5097_set__replicate__conv__if,axiom,
    ! [A: $tType,Na: nat,Xa: A] :
      set2(A,replicate(A,Na,Xa)) = $ite(Na = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) ).

% set_replicate_conv_if
tff(fact_5098_sinh__minus__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,sinh(A),Xa)),aa(A,A,cosh(A),Xa)) = aa(A,A,uminus_uminus(A),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xa))) ) ).

% sinh_minus_cosh
tff(fact_5099_cosh__minus__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,cosh(A),Xa)),aa(A,A,sinh(A),Xa)) = aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xa)) ) ).

% cosh_minus_sinh
tff(fact_5100_sinh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,sinh(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,sinh(A),Xa))),aa(A,A,cosh(A),Xa)) ) ).

% sinh_double
tff(fact_5101_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,cosh(A),Xa) != zero_zero(A) )
         => ( ( aa(A,A,cosh(A),Y) != zero_zero(A) )
           => ( aa(A,A,tanh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tanh(A),Xa)),aa(A,A,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),aa(A,A,tanh(A),Xa)),aa(A,A,tanh(A),Y)))) ) ) ) ) ).

% tanh_add
tff(fact_5102_cosh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : aa(A,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_5103_cosh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cosh(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sinh(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ).

% cosh_square_eq
tff(fact_5104_hyperbolic__pythagoras,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cosh(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sinh(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% hyperbolic_pythagoras
tff(fact_5105_sinh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sinh(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cosh(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ).

% sinh_square_eq
tff(fact_5106_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( aa(A,A,cosh(A),Xa) = zero_zero(A) )
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_5107_cosh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,cosh(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cosh(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sinh(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cosh_double
tff(fact_5108_cosh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(A,A,cosh(A),Xa) = 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),Xa)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xa)))) ) ).

% cosh_def
tff(fact_5109_cosh__ln__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,cosh(real),aa(real,real,ln_ln(real),Xa)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,inverse_inverse(real),Xa))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% cosh_ln_real
tff(fact_5110_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      vEBT_vebt_buildup(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))),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_5111_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_5112_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( Z != complex2(aa(real,real,cos(real),T4),aa(real,real,sin(real),T4)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_5113_or_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),A2) = A2 ) ).

% or.idem
tff(fact_5114_or_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ).

% or.left_idem
tff(fact_5115_or_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: 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),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ).

% or.right_idem
tff(fact_5116_or_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),zero_zero(A)),A2) = A2 ) ).

% or.left_neutral
tff(fact_5117_or_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),zero_zero(A)) = A2 ) ).

% or.right_neutral
tff(fact_5118_bit_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,uminus_uminus(A),one_one(A))),Xa) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_left
tff(fact_5119_bit_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xa),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_right
tff(fact_5120_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_5121_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_5122_or__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xa))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xa)) ) ).

% or_numerals(8)
tff(fact_5123_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),aa(num,num,bit1,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% or_numerals(2)
tff(fact_5124_or__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y))) ) ).

% or_numerals(3)
tff(fact_5125_or__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(Xa))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xa)) ) ).

% or_numerals(5)
tff(fact_5126_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),aa(num,num,bit1,Y)) ) ).

% or_numerals(1)
tff(fact_5127_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),aa(num,num,bit1,Na)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na))) ).

% or_minus_numerals(2)
tff(fact_5128_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),aa(num,num,bit1,Na)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na))) ).

% or_minus_numerals(6)
tff(fact_5129_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xa))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),Xa)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(7)
tff(fact_5130_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(6)
tff(fact_5131_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(Xa))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),Xa)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(4)
tff(fact_5132_or__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% or_eq_0_iff
tff(fact_5133_bit_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xa),zero_zero(A)) = Xa ) ).

% bit.disj_zero_right
tff(fact_5134_or_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),C2)) ) ).

% or.assoc
tff(fact_5135_or_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),A2) ) ).

% or.commute
tff(fact_5136_or_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),C2)) ) ).

% or.left_commute
tff(fact_5137_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_5138_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_5139_complex__scaleR,axiom,
    ! [R: real,A2: real,B2: real] : aa(complex,complex,real_V8093663219630862766scaleR(complex,R),complex2(A2,B2)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R),A2),aa(real,real,aa(real,fun(real,real),times_times(real),R),B2)) ).

% complex_scaleR
tff(fact_5140_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_5141_OR__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xa),Y)) ) ) ).

% OR_lower
tff(fact_5142_Complex__eq__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( complex2(A2,B2) = aa(num,complex,numeral_numeral(complex),W) )
    <=> ( ( A2 = aa(num,real,numeral_numeral(real),W) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_numeral
tff(fact_5143_Complex__mult__complex__of__real,axiom,
    ! [Xa: real,Y: real,R: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(Xa,Y)),aa(real,complex,real_Vector_of_real(complex),R)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),Xa),R),aa(real,real,aa(real,fun(real,real),times_times(real),Y),R)) ).

% Complex_mult_complex_of_real
tff(fact_5144_complex__of__real__mult__Complex,axiom,
    ! [R: real,Xa: real,Y: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),R)),complex2(Xa,Y)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R),Xa),aa(real,real,aa(real,fun(real,real),times_times(real),R),Y)) ).

% complex_of_real_mult_Complex
tff(fact_5145_Complex__eq__neg__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( complex2(A2,B2) = aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W)) )
    <=> ( ( A2 = aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),W)) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_5146_complex__mult,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(A2,B2)),complex2(C2,D2)) = complex2(aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),A2),C2)),aa(real,real,aa(real,fun(real,real),times_times(real),B2),D2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),A2),D2)),aa(real,real,aa(real,fun(real,real),times_times(real),B2),C2))) ).

% complex_mult
tff(fact_5147_even__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            & aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ) ).

% even_or_iff
tff(fact_5148_Complex__sum_H,axiom,
    ! [A: $tType,F2: fun(A,real),Sa: set(A)] : aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,aTP_Lamp_oi(fun(A,real),fun(A,complex),F2)),Sa) = complex2(aa(set(A),real,groups7311177749621191930dd_sum(A,real,F2),Sa),zero_zero(real)) ).

% Complex_sum'
tff(fact_5149_complex__norm,axiom,
    ! [Xa: real,Y: real] : real_V7770717601297561774m_norm(complex,complex2(Xa,Y)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% complex_norm
tff(fact_5150_OR__upper,axiom,
    ! [Xa: int,Na: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xa),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ) ).

% OR_upper
tff(fact_5151_complex__inverse,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,inverse_inverse(complex),complex2(A2,B2)) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),B2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% complex_inverse
tff(fact_5152_cot__less__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cot(real),Xa)),zero_zero(real)) ) ) ).

% cot_less_zero
tff(fact_5153_sint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),W)),aa(nat,nat,suc,zero_zero(nat)))))),ring_1_signed(A,int,W))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),W)),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% sint_range_size
tff(fact_5154_cot__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,cot(A),zero_zero(A)) = zero_zero(A) ) ) ).

% cot_zero
tff(fact_5155_cot__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,cot(A),aa(A,A,uminus_uminus(A),Xa)) = aa(A,A,uminus_uminus(A),aa(A,A,cot(A),Xa)) ) ).

% cot_minus
tff(fact_5156_More__Word_Osint__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( ring_1_signed(A,int,Xa) = zero_zero(int) )
        <=> ( Xa = zero_zero(word(A)) ) ) ) ).

% More_Word.sint_0
tff(fact_5157_cot__pi,axiom,
    aa(real,real,cot(real),pi) = zero_zero(real) ).

% cot_pi
tff(fact_5158_word__no__log__defs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2)),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(7)
tff(fact_5159_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),aa(num,num,bit1,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% or_nat_numerals(2)
tff(fact_5160_or__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xa))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xa)) ).

% or_nat_numerals(4)
tff(fact_5161_sint__minus1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( ring_1_signed(A,int,Xa) = aa(int,int,uminus_uminus(int),one_one(int)) )
        <=> ( Xa = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% sint_minus1
tff(fact_5162_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_5163_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),aa(num,num,bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_5164_or__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit0(Xa))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xa)) ).

% or_nat_numerals(3)
tff(fact_5165_word__bitwise__1__simps_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2)) = 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),B2))) ) ).

% word_bitwise_1_simps(6)
tff(fact_5166_word__bitwise__1__simps_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: 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)),A2)),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),A2)),one_one(int))) ) ).

% word_bitwise_1_simps(8)
tff(fact_5167_word__no__log__defs_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(8)
tff(fact_5168_word__no__log__defs_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2))),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(9)
tff(fact_5169_word__no__log__defs_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(10)
tff(fact_5170_cot__periodic,axiom,
    ! [Xa: real] : aa(real,real,cot(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),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),Xa) ).

% cot_periodic
tff(fact_5171_word__bitwise__1__simps_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: 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)),A2))),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),A2))),one_one(int))) ) ).

% word_bitwise_1_simps(9)
tff(fact_5172_word__bitwise__1__simps_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2))) = 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),B2)))) ) ).

% word_bitwise_1_simps(7)
tff(fact_5173_word__or__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),A2),B2) = zero_zero(word(A)) )
        <=> ( ( A2 = zero_zero(word(A)) )
            & ( B2 = zero_zero(word(A)) ) ) ) ) ).

% word_or_zero
tff(fact_5174_cot__of__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: real] : aa(real,A,real_Vector_of_real(A),aa(real,real,cot(real),Xa)) = aa(A,A,cot(A),aa(real,A,real_Vector_of_real(A),Xa)) ) ).

% cot_of_real
tff(fact_5175_More__Word_Oof__int__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(int,word(A),ring_1_of_int(word(A)),ring_1_signed(A,int,A2)) = A2 ) ).

% More_Word.of_int_sint
tff(fact_5176_divide__complex__def,axiom,
    ! [Xa: complex,Y: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Xa),Y) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Xa),aa(complex,complex,inverse_inverse(complex),Y)) ).

% divide_complex_def
tff(fact_5177_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_5178_tan__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,tan(A),Xa) = aa(A,A,inverse_inverse(A),aa(A,A,cot(A),Xa)) ) ).

% tan_altdef
tff(fact_5179_cot__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : aa(A,A,cot(A),Xa) = aa(A,A,inverse_inverse(A),aa(A,A,tan(A),Xa)) ) ).

% cot_altdef
tff(fact_5180_cot__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X5: A] : aa(A,A,cot(A),X5) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,cos(A),X5)),aa(A,A,sin(A),X5)) ) ).

% cot_def
tff(fact_5181_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_5182_sint__above__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),W)),one_one(nat)))),Xa)
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,W)),Xa) ) ) ).

% sint_above_size
tff(fact_5183_sint__below__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,W: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),W)),one_one(nat)))))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),ring_1_signed(A,int,W)) ) ) ).

% sint_below_size
tff(fact_5184_cot__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,cot(real),Xa)) ) ) ).

% cot_gt_zero
tff(fact_5185_tan__cot_H,axiom,
    ! [Xa: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(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)))),Xa)) = aa(real,real,cot(real),Xa) ).

% tan_cot'
tff(fact_5186_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_5187_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_5188_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique8689654367752047608divmod(int,bitM(M),bit0(one2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M)),one_one(int))),one_one(int)) ).

% divmod_BitM_2_eq
tff(fact_5189_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_5190_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_5191_divide__i,axiom,
    ! [Xa: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Xa),imaginary_unit) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,uminus_uminus(complex),imaginary_unit)),Xa) ).

% divide_i
tff(fact_5192_divide__numeral__i,axiom,
    ! [Z: complex,Na: num] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(num,complex,numeral_numeral(complex),Na)),imaginary_unit)) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,uminus_uminus(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Z))),aa(num,complex,numeral_numeral(complex),Na)) ).

% divide_numeral_i
tff(fact_5193_power2__i,axiom,
    aa(nat,complex,aa(complex,fun(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_5194_i__even__power,axiom,
    ! [Na: nat] : aa(nat,complex,aa(complex,fun(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,aa(complex,fun(nat,complex),power_power(complex),aa(complex,complex,uminus_uminus(complex),one_one(complex))),Na) ).

% i_even_power
tff(fact_5195_semiring__norm_I26_J,axiom,
    bitM(one2) = one2 ).

% semiring_norm(26)
tff(fact_5196_semiring__norm_I28_J,axiom,
    ! [Na: num] : bitM(aa(num,num,bit1,Na)) = aa(num,num,bit1,bit0(Na)) ).

% semiring_norm(28)
tff(fact_5197_semiring__norm_I27_J,axiom,
    ! [Na: num] : bitM(bit0(Na)) = aa(num,num,bit1,bitM(Na)) ).

% semiring_norm(27)
tff(fact_5198_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_5199_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_5200_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_5201_numeral__BitM,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(num,A,numeral_numeral(A),bitM(Na)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),bit0(Na))),one_one(A)) ) ).

% numeral_BitM
tff(fact_5202_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_5203_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_5204_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_5205_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_5206_power2__csqrt,axiom,
    ! [Z: complex] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),csqrt(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Z ).

% power2_csqrt
tff(fact_5207_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_5208_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_5209_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_5210_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_5211_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),aa(num,num,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_5212_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),aa(num,num,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_5213_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_5214_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_5215_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_5216_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_5217_or__not__num__neg_Osimps_I1_J,axiom,
    bit_or_not_num_neg(one2,one2) = one2 ).

% or_not_num_neg.simps(1)
tff(fact_5218_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_5219_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(bit0(Na),aa(num,num,bit1,M)) = bit0(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(6)
tff(fact_5220_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Na: num] : bit_or_not_num_neg(aa(num,num,bit1,Na),one2) = one2 ).

% or_not_num_neg.simps(7)
tff(fact_5221_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one2,aa(num,num,bit1,M)) = aa(num,num,bit1,M) ).

% or_not_num_neg.simps(3)
tff(fact_5222_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_5223_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(aa(num,num,bit1,Na),aa(num,num,bit1,M)) = bitM(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(9)
tff(fact_5224_cis__divide,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),cis(A2)),cis(B2)) = cis(aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2)) ).

% cis_divide
tff(fact_5225_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one2,bit0(M)) = aa(num,num,bit1,M) ).

% or_not_num_neg.simps(2)
tff(fact_5226_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(aa(num,num,bit1,Na),bit0(M)) = bitM(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(8)
tff(fact_5227_DeMoivre,axiom,
    ! [A2: real,Na: nat] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A2)),Na) = cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),A2)) ).

% DeMoivre
tff(fact_5228_or__not__num__neg_Oelims,axiom,
    ! [Xa: num,Xaa: num,Y: num] :
      ( ( bit_or_not_num_neg(Xa,Xaa) = Y )
     => ( ( ( Xa = one2 )
         => ( ( Xaa = one2 )
           => ( Y != one2 ) ) )
       => ( ( ( Xa = one2 )
           => ! [M4: num] :
                ( ( Xaa = bit0(M4) )
               => ( Y != aa(num,num,bit1,M4) ) ) )
         => ( ( ( Xa = one2 )
             => ! [M4: num] :
                  ( ( Xaa = aa(num,num,bit1,M4) )
                 => ( Y != aa(num,num,bit1,M4) ) ) )
           => ( ( ? [N: num] : Xa = bit0(N)
               => ( ( Xaa = one2 )
                 => ( Y != bit0(one2) ) ) )
             => ( ! [N: num] :
                    ( ( Xa = bit0(N) )
                   => ! [M4: num] :
                        ( ( Xaa = bit0(M4) )
                       => ( Y != bitM(bit_or_not_num_neg(N,M4)) ) ) )
               => ( ! [N: num] :
                      ( ( Xa = bit0(N) )
                     => ! [M4: num] :
                          ( ( Xaa = aa(num,num,bit1,M4) )
                         => ( Y != bit0(bit_or_not_num_neg(N,M4)) ) ) )
                 => ( ( ? [N: num] : Xa = aa(num,num,bit1,N)
                     => ( ( Xaa = one2 )
                       => ( Y != one2 ) ) )
                   => ( ! [N: num] :
                          ( ( Xa = aa(num,num,bit1,N) )
                         => ! [M4: num] :
                              ( ( Xaa = bit0(M4) )
                             => ( Y != bitM(bit_or_not_num_neg(N,M4)) ) ) )
                     => ~ ! [N: num] :
                            ( ( Xa = aa(num,num,bit1,N) )
                           => ! [M4: num] :
                                ( ( Xaa = aa(num,num,bit1,M4) )
                               => ( Y != bitM(bit_or_not_num_neg(N,M4)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_5229_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_oj(nat,fun(nat,complex),Na),set_ord_lessThan(nat,Na),collect(complex,aTP_Lamp_gc(nat,fun(complex,$o),Na))) ) ).

% bij_betw_roots_unity
tff(fact_5230_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = collect(product_prod(int,int),product_case_prod(int,int,$o,aTP_Lamp_ok(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than_def
tff(fact_5231_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = collect(product_prod(int,int),product_case_prod(int,int,$o,aTP_Lamp_ol(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than2_def
tff(fact_5232_bij__betw__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A),B3: set(A)] :
          ( bij_betw(A,A,aa(A,fun(A,A),plus_plus(A),A2),A3,B3)
        <=> ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),A3) = B3 ) ) ) ).

% bij_betw_add
tff(fact_5233_bij__betw__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N3: set(nat),A3: set(A)] :
          ( bij_betw(nat,A,semiring_1_of_nat(A),N3,A3)
        <=> ( aa(set(nat),set(A),image(nat,A,semiring_1_of_nat(A)),N3) = A3 ) ) ) ).

% bij_betw_of_nat
tff(fact_5234_bij__betw__empty2,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
      ( bij_betw(A,B,F2,A3,bot_bot(set(B)))
     => ( A3 = bot_bot(set(A)) ) ) ).

% bij_betw_empty2
tff(fact_5235_bij__betw__empty1,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] :
      ( bij_betw(A,B,F2,bot_bot(set(A)),A3)
     => ( A3 = bot_bot(set(B)) ) ) ).

% bij_betw_empty1
tff(fact_5236_prod_Oreindex__bij__betw,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [H: fun(A,B),S: set(A),T3: set(B),G: fun(B,C)] :
          ( bij_betw(A,B,H,S,T3)
         => ( groups7121269368397514597t_prod(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_om(fun(A,B),fun(fun(B,C),fun(A,C)),H),G),S) = groups7121269368397514597t_prod(B,C,G,T3) ) ) ) ).

% prod.reindex_bij_betw
tff(fact_5237_sum_Oreindex__bij__betw,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [H: fun(A,B),S: set(A),T3: set(B),G: fun(B,C)] :
          ( bij_betw(A,B,H,S,T3)
         => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_on(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),T3) ) ) ) ).

% sum.reindex_bij_betw
tff(fact_5238_notIn__Un__bij__betw3,axiom,
    ! [A: $tType,B: $tType,B2: A,A3: set(A),F2: fun(A,B),A11: set(B)] :
      ( ~ member(A,B2,A3)
     => ( ~ member(B,aa(A,B,F2,B2),A11)
       => ( bij_betw(A,B,F2,A3,A11)
        <=> bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A11),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),aa(A,B,F2,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw3
tff(fact_5239_notIn__Un__bij__betw,axiom,
    ! [A: $tType,B: $tType,B2: A,A3: set(A),F2: fun(A,B),A11: set(B)] :
      ( ~ member(A,B2,A3)
     => ( ~ member(B,aa(A,B,F2,B2),A11)
       => ( bij_betw(A,B,F2,A3,A11)
         => bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A11),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),aa(A,B,F2,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw
tff(fact_5240_bij__betw__partition,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),C3: set(A),B3: set(B),D4: set(B)] :
      ( bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B3),D4))
     => ( bij_betw(A,B,F2,C3,D4)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B3),D4) = bot_bot(set(B)) )
           => bij_betw(A,B,F2,A3,B3) ) ) ) ) ).

% bij_betw_partition
tff(fact_5241_bij__betw__combine,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B3: set(B),C3: set(A),D4: set(B)] :
      ( bij_betw(A,B,F2,A3,B3)
     => ( bij_betw(A,B,F2,C3,D4)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B3),D4) = bot_bot(set(B)) )
         => bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B3),D4)) ) ) ) ).

% bij_betw_combine
tff(fact_5242_bij__betw__disjoint__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),C3: set(B),G: fun(A,B),B3: set(A),D4: set(B)] :
      ( bij_betw(A,B,F2,A3,C3)
     => ( bij_betw(A,B,G,B3,D4)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),C3),D4) = bot_bot(set(B)) )
           => bij_betw(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_oo(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),A3),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),C3),D4)) ) ) ) ) ).

% bij_betw_disjoint_Un
tff(fact_5243_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S5: set(A),T6: set(B),H: fun(A,B),S: set(A),T3: set(B),G: fun(B,C)] :
          ( finite_finite2(A,S5)
         => ( finite_finite2(B,T6)
           => ( bij_betw(A,B,H,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),T6))
             => ( ! [A5: A] :
                    ( member(A,A5,S5)
                   => ( aa(B,C,G,aa(A,B,H,A5)) = zero_zero(C) ) )
               => ( ! [B6: B] :
                      ( member(B,B6,T6)
                     => ( aa(B,C,G,B6) = zero_zero(C) ) )
                 => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_on(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),T3) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_5244_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S5: set(A),T6: set(B),H: fun(A,B),S: set(A),T3: set(B),G: fun(B,C)] :
          ( finite_finite2(A,S5)
         => ( finite_finite2(B,T6)
           => ( bij_betw(A,B,H,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),T6))
             => ( ! [A5: A] :
                    ( member(A,A5,S5)
                   => ( aa(B,C,G,aa(A,B,H,A5)) = one_one(C) ) )
               => ( ! [B6: B] :
                      ( member(B,B6,T6)
                     => ( aa(B,C,G,B6) = one_one(C) ) )
                 => ( groups7121269368397514597t_prod(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_om(fun(A,B),fun(fun(B,C),fun(A,C)),H),G),S) = groups7121269368397514597t_prod(B,C,G,T3) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_5245_infinite__imp__bij__betw2,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ~ finite_finite2(A,A3)
     => ? [H2: fun(A,A)] : bij_betw(A,A,H2,A3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw2
tff(fact_5246_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ~ finite_finite2(A,A3)
     => ? [H2: fun(A,A)] : bij_betw(A,A,H2,A3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw
tff(fact_5247_set__decode__0,axiom,
    ! [Xa: nat] :
      ( member(nat,zero_zero(nat),nat_set_decode(Xa))
    <=> ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Xa) ) ).

% set_decode_0
tff(fact_5248_bij__betw__Suc,axiom,
    ! [M7: set(nat),N3: set(nat)] :
      ( bij_betw(nat,nat,suc,M7,N3)
    <=> ( aa(set(nat),set(nat),image(nat,nat,suc),M7) = N3 ) ) ).

% bij_betw_Suc
tff(fact_5249_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_5250_set__decode__Suc,axiom,
    ! [Na: nat,Xa: nat] :
      ( member(nat,aa(nat,nat,suc,Na),nat_set_decode(Xa))
    <=> member(nat,Na,nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% set_decode_Suc
tff(fact_5251_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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),Z)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Na),nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_5252_set__decode__def,axiom,
    ! [Xa: nat] : nat_set_decode(Xa) = collect(nat,aTP_Lamp_op(nat,fun(nat,$o),Xa)) ).

% set_decode_def
tff(fact_5253_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_gc(nat,fun(complex,$o),Na)),collect(complex,aa(nat,fun(complex,$o),aTP_Lamp_oq(complex,fun(nat,fun(complex,$o)),C2),Na))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_5254_forall__finite_I3_J,axiom,
    ! [Xa: nat,P: fun(nat,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,suc,aa(nat,nat,suc,Xa)))
         => aa(nat,$o,P,I2) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,suc,Xa))
           => aa(nat,$o,P,aa(nat,nat,suc,I2)) ) ) ) ).

% forall_finite(3)
tff(fact_5255_forall__finite_I2_J,axiom,
    ! [P: fun(nat,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,suc,zero_zero(nat)))
         => aa(nat,$o,P,I2) )
    <=> aa(nat,$o,P,zero_zero(nat)) ) ).

% forall_finite(2)
tff(fact_5256_real__root__zero,axiom,
    ! [Na: nat] : aa(real,real,root(Na),zero_zero(real)) = zero_zero(real) ).

% real_root_zero
tff(fact_5257_real__root__Suc__0,axiom,
    ! [Xa: real] : aa(real,real,root(aa(nat,nat,suc,zero_zero(nat))),Xa) = Xa ).

% real_root_Suc_0
tff(fact_5258_real__root__eq__iff,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(real,real,root(Na),Xa) = aa(real,real,root(Na),Y) )
      <=> ( Xa = Y ) ) ) ).

% real_root_eq_iff
tff(fact_5259_root__0,axiom,
    ! [Xa: real] : aa(real,real,root(zero_zero(nat)),Xa) = zero_zero(real) ).

% root_0
tff(fact_5260_real__root__eq__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(real,real,root(Na),Xa) = zero_zero(real) )
      <=> ( Xa = zero_zero(real) ) ) ) ).

% real_root_eq_0_iff
tff(fact_5261_real__root__less__iff,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),Xa)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ).

% real_root_less_iff
tff(fact_5262_real__root__le__iff,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),Xa)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ).

% real_root_le_iff
tff(fact_5263_real__root__eq__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(real,real,root(Na),Xa) = one_one(real) )
      <=> ( Xa = one_one(real) ) ) ) ).

% real_root_eq_1_iff
tff(fact_5264_real__root__one,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),one_one(real)) = one_one(real) ) ) ).

% real_root_one
tff(fact_5265_real__root__lt__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),Xa)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ) ).

% real_root_lt_0_iff
tff(fact_5266_real__root__gt__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_5267_real__root__le__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),Xa)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ) ).

% real_root_le_0_iff
tff(fact_5268_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_5269_real__root__lt__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),Xa)),one_one(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real)) ) ) ).

% real_root_lt_1_iff
tff(fact_5270_real__root__gt__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_5271_real__root__le__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),Xa)),one_one(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real)) ) ) ).

% real_root_le_1_iff
tff(fact_5272_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_5273_real__root__pow__pos2,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),Na) = Xa ) ) ) ).

% real_root_pow_pos2
tff(fact_5274_real__root__divide,axiom,
    ! [Na: nat,Xa: real,Y: real] : aa(real,real,root(Na),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,root(Na),Xa)),aa(real,real,root(Na),Y)) ).

% real_root_divide
tff(fact_5275_real__root__commute,axiom,
    ! [M: nat,Na: nat,Xa: real] : aa(real,real,root(M),aa(real,real,root(Na),Xa)) = aa(real,real,root(Na),aa(real,real,root(M),Xa)) ).

% real_root_commute
tff(fact_5276_real__root__minus,axiom,
    ! [Na: nat,Xa: real] : aa(real,real,root(Na),aa(real,real,uminus_uminus(real),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,root(Na),Xa)) ).

% real_root_minus
tff(fact_5277_real__root__inverse,axiom,
    ! [Na: nat,Xa: real] : aa(real,real,root(Na),aa(real,real,inverse_inverse(real),Xa)) = aa(real,real,inverse_inverse(real),aa(real,real,root(Na),Xa)) ).

% real_root_inverse
tff(fact_5278_real__root__mult,axiom,
    ! [Na: nat,Xa: real,Y: real] : aa(real,real,root(Na),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,root(Na),Xa)),aa(real,real,root(Na),Y)) ).

% real_root_mult
tff(fact_5279_real__root__mult__exp,axiom,
    ! [M: nat,Na: nat,Xa: real] : aa(real,real,root(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)),Xa) = aa(real,real,root(M),aa(real,real,root(Na),Xa)) ).

% real_root_mult_exp
tff(fact_5280_real__root__pos__pos__le,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Na),Xa)) ) ).

% real_root_pos_pos_le
tff(fact_5281_real__root__less__mono,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),Xa)),aa(real,real,root(Na),Y)) ) ) ).

% real_root_less_mono
tff(fact_5282_real__root__le__mono,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),Xa)),aa(real,real,root(Na),Y)) ) ) ).

% real_root_le_mono
tff(fact_5283_real__root__power,axiom,
    ! [Na: nat,Xa: real,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),K)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),K) ) ) ).

% real_root_power
tff(fact_5284_real__root__abs,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),aa(real,real,abs_abs(real),Xa)) = aa(real,real,abs_abs(real),aa(real,real,root(Na),Xa)) ) ) ).

% real_root_abs
tff(fact_5285_real__root__gt__zero,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,root(Na),Xa)) ) ) ).

% real_root_gt_zero
tff(fact_5286_real__root__strict__decreasing,axiom,
    ! [Na: nat,N3: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(N3),Xa)),aa(real,real,root(Na),Xa)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_5287_sqrt__def,axiom,
    sqrt = root(aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% sqrt_def
tff(fact_5288_root__abs__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,abs_abs(real),aa(real,real,root(Na),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),Na))) = aa(real,real,abs_abs(real),Y) ) ) ).

% root_abs_power
tff(fact_5289_real__root__pos__pos,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Na),Xa)) ) ) ).

% real_root_pos_pos
tff(fact_5290_real__root__strict__increasing,axiom,
    ! [Na: nat,N3: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),Xa)),aa(real,real,root(N3),Xa)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_5291_real__root__decreasing,axiom,
    ! [Na: nat,N3: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(N3),Xa)),aa(real,real,root(Na),Xa)) ) ) ) ).

% real_root_decreasing
tff(fact_5292_odd__real__root__power__cancel,axiom,
    ! [Na: nat,Xa: real] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( aa(real,real,root(Na),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na)) = Xa ) ) ).

% odd_real_root_power_cancel
tff(fact_5293_odd__real__root__unique,axiom,
    ! [Na: nat,Y: real,Xa: real] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),Na) = Xa )
       => ( aa(real,real,root(Na),Xa) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_5294_odd__real__root__pow,axiom,
    ! [Na: nat,Xa: real] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),Na) = Xa ) ) ).

% odd_real_root_pow
tff(fact_5295_real__root__pow__pos,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),Na) = Xa ) ) ) ).

% real_root_pow_pos
tff(fact_5296_real__root__power__cancel,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(real,real,root(Na),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na)) = Xa ) ) ) ).

% real_root_power_cancel
tff(fact_5297_real__root__pos__unique,axiom,
    ! [Na: nat,Y: real,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),Na) = Xa )
         => ( aa(real,real,root(Na),Xa) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_5298_real__root__increasing,axiom,
    ! [Na: nat,N3: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),Xa)),aa(real,real,root(N3),Xa)) ) ) ) ) ).

% real_root_increasing
tff(fact_5299_log__root,axiom,
    ! [Na: nat,A2: real,B2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ( aa(real,real,log(B2),aa(real,real,root(Na),A2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(B2),A2)),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_root
tff(fact_5300_log__base__root,axiom,
    ! [Na: nat,B2: real,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
       => ( aa(real,real,log(aa(real,real,root(Na),B2)),Xa) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B2),Xa)) ) ) ) ).

% log_base_root
tff(fact_5301_ln__root,axiom,
    ! [Na: nat,B2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
       => ( aa(real,real,ln_ln(real),aa(real,real,root(Na),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B2)),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% ln_root
tff(fact_5302_forall__finite_I1_J,axiom,
    ! [P: fun(nat,$o),I4: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),zero_zero(nat))
     => aa(nat,$o,P,I4) ) ).

% forall_finite(1)
tff(fact_5303_root__powr__inverse,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,real,root(Na),Xa) = powr(real,Xa,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Na))) ) ) ) ).

% root_powr_inverse
tff(fact_5304_Comparator__Generator_OAll__less__Suc,axiom,
    ! [Xa: nat,P: fun(nat,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,suc,Xa))
         => aa(nat,$o,P,I2) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Xa)
           => aa(nat,$o,P,aa(nat,nat,suc,I2)) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_5305_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),zero_zero(nat)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% flip_bit_0
tff(fact_5306_take__bit__word__Bit1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,M: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se2584673776208193580ke_bit(word(A)),aa(num,nat,numeral_numeral(nat),Na)),aa(num,word(A),numeral_numeral(word(A)),aa(num,num,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),aa(nat,fun(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_5307_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_5308_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_5309_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_5310_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_5311_take__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2)),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),B2)) ) ).

% take_bit_or
tff(fact_5312_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_5313_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_5314_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_5315_take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),zero_zero(nat)),A2) = zero_zero(A) ) ).

% take_bit_0
tff(fact_5316_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,Na)),one_one(A)) = one_one(A) ) ).

% take_bit_Suc_1
tff(fact_5317_take__bit__numeral__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num] : aa(A,A,aa(nat,fun(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_5318_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_5319_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_5320_Suc__0__mod__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,zero_zero(nat))),Na) = aa($o,nat,zero_neq_one_of_bool(nat),Na != aa(nat,nat,suc,zero_zero(nat))) ).

% Suc_0_mod_eq
tff(fact_5321_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_5322_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] :
          ( ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),one_one(A)) = zero_zero(A) )
        <=> ( Na = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_5323_mod__word__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),one_one(word(A))),W) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa($o,word(A),zero_neq_one_of_bool(word(A)),W = one_one(word(A))))) ) ).

% mod_word_one
tff(fact_5324_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q2),aa($o,int,zero_neq_one_of_bool(int),R != zero_zero(int))) ).

% Divides.adjust_div_eq
tff(fact_5325_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_5326_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ) ).

% take_bit_of_1
tff(fact_5327_sum__of__bool__mult__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A3: set(A),P: fun(A,$o),F2: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_or(fun(A,$o),fun(fun(A,B),fun(A,B)),P),F2)),A3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,P))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_5328_sum__mult__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A3: set(A),F2: fun(A,B),P: fun(A,$o)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_os(fun(A,B),fun(fun(A,$o),fun(A,B)),F2),P)),A3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,P))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_5329_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_5330_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2))
        <=> ( ( Na = zero_zero(nat) )
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% even_take_bit_eq
tff(fact_5331_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: $o] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa($o,A,zero_neq_one_of_bool(A),(B2))),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ).

% of_bool_half_eq_0
tff(fact_5332_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,zero_zero(nat))),A2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_0
tff(fact_5333_take__bit__word__Bit0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,M: num] : aa(word(A),word(A),aa(nat,fun(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),aa(nat,fun(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_5334_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa($o,A,zero_neq_one_of_bool(A),Na = zero_zero(nat)) ) ).

% bits_1_div_exp
tff(fact_5335_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa($o,A,zero_neq_one_of_bool(A),Na = zero_zero(nat)) ) ).

% one_div_2_pow_eq
tff(fact_5336_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% take_bit_of_exp
tff(fact_5337_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_of_2
tff(fact_5338_take__bit__word__minus__Bit0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,M: num] : aa(word(A),word(A),aa(nat,fun(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),aa(nat,fun(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_5339_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ) ).

% one_mod_2_pow_eq
tff(fact_5340_take__bit__add,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,A2: A,B2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2)),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),B2))) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ).

% take_bit_add
tff(fact_5341_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_5342_take__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,K: int] : aa(A,A,aa(nat,fun(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,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)) ) ).

% take_bit_of_int
tff(fact_5343_take__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat] : aa(A,A,aa(nat,fun(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,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),M)) ) ).

% take_bit_of_nat
tff(fact_5344_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: A,M: nat] :
          ( ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),B2) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),A2) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),B2) ) ) ) ) ).

% take_bit_tightened
tff(fact_5345_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( ( aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),Na),A2) = aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),Na),B2) )
        <=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,Na)),A2) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,Na)),B2) ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_5346_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat,A2: A] :
          aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2)) = aa(A,A,
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),M)),
            A2) ) ).

% signed_take_bit_take_bit
tff(fact_5347_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat,A2: A] :
          aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2))) ) ).

% take_bit_unset_bit_eq
tff(fact_5348_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat,A2: A] :
          aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2))) ) ).

% take_bit_set_bit_eq
tff(fact_5349_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat,A2: A] :
          aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),M),A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2),aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2))) ) ).

% take_bit_flip_bit_eq
tff(fact_5350_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),Na),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),A2) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_5351_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_ot(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_5352_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,Na)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_bit0
tff(fact_5353_take__bit__eq__mod,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% take_bit_eq_mod
tff(fact_5354_of__bool__odd__eq__mod__2,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] : aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% of_bool_odd_eq_mod_2
tff(fact_5355_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          ( ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) = zero_zero(A) )
        <=> aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)),A2) ) ) ).

% take_bit_eq_0_iff
tff(fact_5356_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,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),L),Na))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),L)
            & ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ) ) ).

% bin_last_bintrunc
tff(fact_5357_take__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,aa(nat,fun(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,aa(nat,fun(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_5358_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [A5: A] :
              ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A5),aa(num,A,numeral_numeral(A),bit0(one2))) = A5 )
             => aa(A,$o,P,A5) )
         => ( ! [A5: A,B6: $o] :
                ( aa(A,$o,P,A5)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B6))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A5))),aa(num,A,numeral_numeral(A),bit0(one2))) = A5 )
                 => aa(A,$o,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B6))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A5))) ) )
           => aa(A,$o,P,A2) ) ) ) ).

% bits_induct
tff(fact_5359_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ).

% exp_mod_exp
tff(fact_5360_one__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ).

% one_or_eq
tff(fact_5361_or__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),one_one(A)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ).

% or_one_eq
tff(fact_5362_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,Na)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_5363_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,Na)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,suc,Na))),one_one(A)) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_5364_take__bit__numeral__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(A,A,aa(nat,fun(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,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,nat,numeral_numeral(nat),K))),one_one(A)) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_5365_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,Na)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% take_bit_Suc
tff(fact_5366_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_5367_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_5368_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_5369_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_5370_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) = $ite(aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2),zero_zero(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A))) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5371_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(nat,fun(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_5372_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M) != zero_zero(A) )
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ))),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na))) ) ).

% exp_div_exp_eq
tff(fact_5373_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) = $ite(Na = zero_zero(nat),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% take_bit_rec
tff(fact_5374_take__bit__word__minus__Bit1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,M: num] : aa(word(A),word(A),aa(nat,fun(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)),aa(num,num,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),aa(nat,fun(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_5375_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( member(int,K,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
        & member(int,L,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
        aa(int,int,uminus_uminus(int),
          aa($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_5376_and__int_Oelims,axiom,
    ! [Xa: int,Xaa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Xaa) = Y )
     => ( Y = $ite(
            ( member(int,Xa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & member(int,Xaa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xa)
                & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xaa) ))),
            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))),Xa)
                  & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xaa) ))),
              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),Xa),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xaa),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.elims
tff(fact_5377_AND__twice,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [W: A,M: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),W),M)),M) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),W),M) ) ).

% AND_twice
tff(fact_5378_and_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: 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),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) ) ).

% and.right_idem
tff(fact_5379_and_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) ) ).

% and.left_idem
tff(fact_5380_and_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),A2) = A2 ) ).

% and.idem
tff(fact_5381_bit_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),zero_zero(A)) = zero_zero(A) ) ).

% bit.conj_zero_right
tff(fact_5382_bit_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),Xa) = zero_zero(A) ) ).

% bit.conj_zero_left
tff(fact_5383_zero__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% zero_and_eq
tff(fact_5384_and__zero__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% and_zero_eq
tff(fact_5385_take__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2)),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),B2)) ) ).

% take_bit_and
tff(fact_5386_bit_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),aa(A,A,uminus_uminus(A),one_one(A))) = Xa ) ).

% bit.conj_one_right
tff(fact_5387_and_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,uminus_uminus(A),one_one(A))) = A2 ) ).

% and.right_neutral
tff(fact_5388_and_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))),A2) = A2 ) ).

% and.left_neutral
tff(fact_5389_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_5390_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_5391_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).

% pred_numeral_inc
tff(fact_5392_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),aa(num,num,bit1,Y))) = one_one(A) ) ).

% and_numerals(2)
tff(fact_5393_and__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xa))),one_one(A)) = one_one(A) ) ).

% and_numerals(8)
tff(fact_5394_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,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K))) = aa(int,A,ring_1_of_int(A),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)) ) ).

% of_nat_nat_take_bit_eq
tff(fact_5395_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(Xa))),one_one(A)) = zero_zero(A) ) ).

% and_numerals(5)
tff(fact_5396_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_5397_and__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(3)
tff(fact_5398_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_5399_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_5400_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(M)) ) ).

% diff_numeral_special(6)
tff(fact_5401_diff__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),Na)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Na))) ) ).

% diff_numeral_special(5)
tff(fact_5402_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),aa(num,num,bit1,Na)))),one_one(int)) = one_one(int) ).

% and_minus_numerals(6)
tff(fact_5403_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),aa(num,num,bit1,Na)))) = one_one(int) ).

% and_minus_numerals(2)
tff(fact_5404_take__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ).

% take_bit_of_Suc_0
tff(fact_5405_and__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(Xa))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Xa)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(4)
tff(fact_5406_and__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(6)
tff(fact_5407_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_5408_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_5409_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xa))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Xa)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% and_numerals(7)
tff(fact_5410_take__bit__mult,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),L))) = aa(int,int,aa(nat,fun(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_5411_take__bit__minus,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),aa(int,int,uminus_uminus(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K))) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),aa(int,int,uminus_uminus(int),K)) ).

% take_bit_minus
tff(fact_5412_take__bit__diff,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),L))) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)) ).

% take_bit_diff
tff(fact_5413_take__bit__nat__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)) ) ) ).

% take_bit_nat_eq
tff(fact_5414_nat__take__bit__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),aa(int,nat,nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_5415_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,Na: nat,Q2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),M),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_5416_take__bit__nat__less__eq__self,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_5417_bit_Oconj__disj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),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),Xa),Y)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),Z)) ) ).

% bit.conj_disj_distrib
tff(fact_5418_bit_Odisj__conj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xa),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),Xa),Y)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xa),Z)) ) ).

% bit.disj_conj_distrib
tff(fact_5419_bit_Oconj__disj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,Xa: 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)),Xa) = 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),Xa)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z),Xa)) ) ).

% bit.conj_disj_distrib2
tff(fact_5420_bit_Odisj__conj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,Xa: 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)),Xa) = 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),Xa)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Z),Xa)) ) ).

% bit.disj_conj_distrib2
tff(fact_5421_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_5422_num__induct,axiom,
    ! [P: fun(num,$o),Xa: num] :
      ( aa(num,$o,P,one2)
     => ( ! [X: num] :
            ( aa(num,$o,P,X)
           => aa(num,$o,P,inc(X)) )
       => aa(num,$o,P,Xa) ) ) ).

% num_induct
tff(fact_5423_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_5424_and_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),C2)) ) ).

% and.left_commute
tff(fact_5425_and_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),A2) ) ).

% and.commute
tff(fact_5426_and_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),C2)) ) ).

% and.assoc
tff(fact_5427_and__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( A2 = aa(A,A,uminus_uminus(A),one_one(A)) )
            & ( B2 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_5428_add__inc,axiom,
    ! [Xa: num,Y: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),Xa),inc(Y)) = inc(aa(num,num,aa(num,fun(num,num),plus_plus(num),Xa),Y)) ).

% add_inc
tff(fact_5429_AND__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Y)) ) ).

% AND_lower
tff(fact_5430_AND__upper1,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Y)),Xa) ) ).

% AND_upper1
tff(fact_5431_AND__upper2,axiom,
    ! [Y: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Y)),Y) ) ).

% AND_upper2
tff(fact_5432_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_5433_AND__upper2_H,axiom,
    ! [Y: int,Z: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_5434_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,Na: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),M),K)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_5435_take__bit__nonnegative,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)) ).

% take_bit_nonnegative
tff(fact_5436_take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_5437_not__take__bit__negative,axiom,
    ! [Na: nat,K: int] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)),zero_zero(int)) ).

% not_take_bit_negative
tff(fact_5438_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_5439_plus__and__or,axiom,
    ! [Xa: 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),Xa),Y)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xa),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Y) ).

% plus_and_or
tff(fact_5440_inc_Osimps_I1_J,axiom,
    inc(one2) = bit0(one2) ).

% inc.simps(1)
tff(fact_5441_inc_Osimps_I2_J,axiom,
    ! [Xa: num] : inc(bit0(Xa)) = aa(num,num,bit1,Xa) ).

% inc.simps(2)
tff(fact_5442_inc_Osimps_I3_J,axiom,
    ! [Xa: num] : inc(aa(num,num,bit1,Xa)) = bit0(inc(Xa)) ).

% inc.simps(3)
tff(fact_5443_add__One,axiom,
    ! [Xa: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),Xa),one2) = inc(Xa) ).

% add_One
tff(fact_5444_inc__BitM__eq,axiom,
    ! [Na: num] : inc(bitM(Na)) = bit0(Na) ).

% inc_BitM_eq
tff(fact_5445_BitM__inc__eq,axiom,
    ! [Na: num] : bitM(inc(Na)) = aa(num,num,bit1,Na) ).

% BitM_inc_eq
tff(fact_5446_mult__inc,axiom,
    ! [Xa: num,Y: num] : aa(num,num,aa(num,fun(num,num),times_times(num),Xa),inc(Y)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),times_times(num),Xa),Y)),Xa) ).

% mult_inc
tff(fact_5447_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_5448_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_5449_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_5450_take__bit__decr__eq,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K) != zero_zero(int) )
     => ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)),one_one(int)) ) ) ).

% take_bit_decr_eq
tff(fact_5451_numeral__inc,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Xa: num] : aa(num,A,numeral_numeral(A),inc(Xa)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Xa)),one_one(A)) ) ).

% numeral_inc
tff(fact_5452_even__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ) ).

% even_and_iff
tff(fact_5453_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),Xa) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Xa) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),Y) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( Xa = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_5454_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_5455_one__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),A2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% one_and_eq
tff(fact_5456_and__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),one_one(A)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% and_one_eq
tff(fact_5457_take__bit__nat__eq__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),M) = M )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_5458_take__bit__nat__less__exp,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% take_bit_nat_less_exp
tff(fact_5459_take__bit__nat__eq__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_5460_take__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),M) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% take_bit_nat_def
tff(fact_5461_take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% take_bit_Suc_minus_bit1
tff(fact_5462_take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% take_bit_int_less_exp
tff(fact_5463_take__bit__int__def,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% take_bit_int_def
tff(fact_5464_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,aa(nat,fun(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),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),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_5465_take__bit__nat__less__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),M)),M)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_5466_take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% take_bit_Suc_minus_bit0
tff(fact_5467_take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_5468_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_5469_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_5470_take__bit__int__eq__self,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_5471_take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_5472_take__bit__incr__eq,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K) != aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),one_one(int)) )
     => ( aa(int,int,aa(nat,fun(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,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)) ) ) ).

% take_bit_incr_eq
tff(fact_5473_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,aa(nat,fun(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,aa(nat,fun(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_5474_take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ) ) ).

% take_bit_int_less_eq
tff(fact_5475_take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)) ) ).

% take_bit_int_greater_eq
tff(fact_5476_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,suc,Na)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_5477_take__bit__minus__small__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),aa(int,int,uminus_uminus(int),K)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_5478_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_5479_and__int_Opelims,axiom,
    ! [Xa: int,Xaa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Xaa) = 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),Xa),Xaa))
       => ~ ( ( Y = $ite(
                  ( member(int,Xa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                  & member(int,Xaa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
                  aa(int,int,uminus_uminus(int),
                    aa($o,int,zero_neq_one_of_bool(int),
                      ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xa)
                      & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xaa) ))),
                  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))),Xa)
                        & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xaa) ))),
                    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),Xa),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xaa),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),Xa),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_5480_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) = $ite(
            ( member(int,K,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & member(int,L,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($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_5481_slice__nth,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A),I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From))
         => ( aa(nat,A,nth(A,slice(A,From,To,Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),From),I)) ) ) ) ) ).

% slice_nth
tff(fact_5482_word__no__log__defs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2)),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(3)
tff(fact_5483_and__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit0(Xa))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% and_nat_numerals(3)
tff(fact_5484_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_5485_and__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xa))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% and_nat_numerals(4)
tff(fact_5486_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),aa(num,num,bit1,Y))) = one_one(nat) ).

% and_nat_numerals(2)
tff(fact_5487_word__bitwise__1__simps_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: 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)),A2)),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),A2)),one_one(int))) ) ).

% word_bitwise_1_simps(4)
tff(fact_5488_word__bitwise__1__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2)) = 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),B2))) ) ).

% word_bitwise_1_simps(2)
tff(fact_5489_word__no__log__defs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(6)
tff(fact_5490_word__no__log__defs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2))),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(5)
tff(fact_5491_word__no__log__defs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(4)
tff(fact_5492_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_5493_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_5494_word__bitwise__1__simps_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: 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)),A2))),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),A2))),one_one(int))) ) ).

% word_bitwise_1_simps(5)
tff(fact_5495_word__bitwise__1__simps_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2))) = 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),B2)))) ) ).

% word_bitwise_1_simps(3)
tff(fact_5496_word__combine__masks,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: word(A),Z: word(A),M6: word(A),Z4: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),M) = Z )
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),M6) = Z4 )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),M),M6)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Z),Z4) ) ) ) ) ).

% word_combine_masks
tff(fact_5497_word__and__max__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),A2: word(A)] :
          ( ( Xa = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),Xa) = A2 ) ) ) ).

% word_and_max_word
tff(fact_5498_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_5499_word__plus__and__or__coroll,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y) = zero_zero(word(A)) )
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y) ) ) ) ).

% word_plus_and_or_coroll
tff(fact_5500_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_5501_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_5502_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [K2: int,L2: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K2),L2))
           => ( ( ~ ( member(int,K2,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                    & member(int,L2,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L2),aa(num,int,numeral_numeral(int),bit0(one2)))) )
             => aa(int,$o,aa(int,fun(int,$o),P,K2),L2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_5503_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [I3: int,J3: int] :
            ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I3),J3))
           => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I3),J3)
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))),J3) )
             => aa(int,$o,aa(int,fun(int,$o),P,I3),J3) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_5504_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),$o),ord_less(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
          <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ) ).

% word_2p_lem
tff(fact_5505_arctan__def,axiom,
    ! [Y: real] : aa(real,real,arctan,Y) = the(real,aTP_Lamp_ou(real,fun(real,$o),Y)) ).

% arctan_def
tff(fact_5506_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_5507_uint__lt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),zero_zero(int)) ) ).

% uint_lt_0
tff(fact_5508_word__le__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B2))) ) ) ).

% word_le_no
tff(fact_5509_word__less__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B2))) ) ) ).

% word_less_no
tff(fact_5510_word__div__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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)),A2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B2)))) ) ).

% word_div_no
tff(fact_5511_word__mod__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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)),A2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B2)))) ) ).

% word_mod_no
tff(fact_5512_More__Word_Oof__int__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(int,word(A),ring_1_of_int(word(A)),aa(word(A),int,semiring_1_unsigned(A,int),Xa)) = Xa ) ).

% More_Word.of_int_uint
tff(fact_5513_word__less__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)) ) ) ).

% word_less_def
tff(fact_5514_uint__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ).

% uint_div
tff(fact_5515_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_5516_uint__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ).

% uint_mod
tff(fact_5517_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_5518_word__less__iff__unsigned,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & linordered_semidom(B) )
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(word(A),B,semiring_1_unsigned(A,B),A2)),aa(word(A),B,semiring_1_unsigned(A,B),B2)) ) ) ).

% word_less_iff_unsigned
tff(fact_5519_ln__real__def,axiom,
    ! [Xa: real] : aa(real,real,ln_ln(real),Xa) = the(real,aTP_Lamp_ov(real,fun(real,$o),Xa)) ).

% ln_real_def
tff(fact_5520_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_5521_uint__add__ge0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(A),Xb: word(B)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(B),int,semiring_1_unsigned(B,int),Xb))) ) ).

% uint_add_ge0
tff(fact_5522_uint__mult__ge0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(A),Xb: word(B)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(B),int,semiring_1_unsigned(B,int),Xb))) ) ).

% uint_mult_ge0
tff(fact_5523_nat__uint__less__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: nat,Xa: word(A)] :
          ( ( aa(int,nat,nat2,aa(word(A),int,semiring_1_unsigned(A,int),Y)) = Z )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,aa(word(A),int,semiring_1_unsigned(A,int),Xa))),Z) ) ) ) ).

% nat_uint_less_helper
tff(fact_5524_uint__add__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))) ) ).

% uint_add_le
tff(fact_5525_uint__plus__simple__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_plus_simple_iff
tff(fact_5526_uint__plus__simple,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
         => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_plus_simple
tff(fact_5527_word__add__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2) = 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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_add_def
tff(fact_5528_word__arith__power__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Na: nat] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),A2),Na) = aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),Na)) ) ).

% word_arith_power_alt
tff(fact_5529_word__mult__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2) = 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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_mult_def
tff(fact_5530_word__div__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),B2) = 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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_div_def
tff(fact_5531_word__mod__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),A2),B2) = 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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_mod_def
tff(fact_5532_ln__neg__is__const,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(real,real,ln_ln(real),Xa) = the(real,aTP_Lamp_ow(real,$o)) ) ) ).

% ln_neg_is_const
tff(fact_5533_udvd__incr__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Up: int,Uq: int,Ua: int,Na: int,K6: word(A),N6: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Up),Uq)
         => ( ( Up = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
           => ( ( Uq = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
             => aa(int,$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),K6))),Uq) ) ) ) ) ).

% udvd_incr_lem
tff(fact_5534_udvd__incr__lem0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Up: int,Uq: int,Na: int,K6: word(A),N6: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Up),Uq)
         => ( ( Up = aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
           => ( ( Uq = aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
             => aa(int,$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),K6))),Uq) ) ) ) ) ).

% udvd_incr_lem0
tff(fact_5535_udvd__incr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q2: word(A),Na: int,K6: word(A),N6: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P2),Q2)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
             => aa(word(A),$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),K6)),Q2) ) ) ) ) ).

% udvd_incr0
tff(fact_5536_udvd__decr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q2: word(A),Na: int,K6: word(A),N6: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P2),Q2)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
             => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
               => aa(word(A),$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)),minus_minus(word(A)),Q2),K6)) ) ) ) ) ) ).

% udvd_decr0
tff(fact_5537_the__elem__def,axiom,
    ! [A: $tType,X7: set(A)] : the_elem(A,X7) = the(A,aTP_Lamp_ox(set(A),fun(A,$o),X7)) ).

% the_elem_def
tff(fact_5538_udvd__incr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q2: word(A),Ua: int,Na: int,K6: word(A),N6: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P2),Q2)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
             => aa(word(A),$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),K6)),Q2) ) ) ) ) ).

% udvd_incr'
tff(fact_5539_udvd__decr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q2: word(A),Ua: int,Na: int,K6: word(A),N6: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P2),Q2)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
             => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
               => aa(word(A),$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)),minus_minus(word(A)),Q2),K6)) ) ) ) ) ) ).

% udvd_decr'
tff(fact_5540_arccos__def,axiom,
    ! [Y: real] : aa(real,real,arccos,Y) = the(real,aTP_Lamp_oy(real,fun(real,$o),Y)) ).

% arccos_def
tff(fact_5541_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,aa(int,fun(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_5542_uint__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
         => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na) ) ) ) ).

% uint_2p
tff(fact_5543_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_oz(real,$o)) ).

% pi_half
tff(fact_5544_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_oz(real,$o))) ).

% pi_def
tff(fact_5545_no__plus__overflow__uint__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa))) ) ) ).

% no_plus_overflow_uint_size
tff(fact_5546_uint__plus__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa)))) ) ).

% uint_plus_if_size
tff(fact_5547_uint__sub__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa)))) ) ).

% uint_sub_if_size
tff(fact_5548_arcsin__def,axiom,
    ! [Y: real] : aa(real,real,arcsin,Y) = the(real,aTP_Lamp_pa(real,fun(real,$o),Y)) ).

% arcsin_def
tff(fact_5549_the__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( aa(A,$o,P,A2)
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
           => ( X = A2 ) )
       => ( the(A,P) = A2 ) ) ) ).

% the_equality
tff(fact_5550_the__eq__trivial,axiom,
    ! [A: $tType,A2: A] : the(A,aTP_Lamp_aa(A,fun(A,$o),A2)) = A2 ).

% the_eq_trivial
tff(fact_5551_the__sym__eq__trivial,axiom,
    ! [A: $tType,Xa: A] : the(A,aa(A,fun(A,$o),fequal(A),Xa)) = Xa ).

% the_sym_eq_trivial
tff(fact_5552_The__split__eq,axiom,
    ! [A: $tType,B: $tType,Xa: A,Y: B] : the(product_prod(A,B),product_case_prod(A,B,$o,aa(B,fun(A,fun(B,$o)),aTP_Lamp_pb(A,fun(B,fun(A,fun(B,$o))),Xa),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y) ).

% The_split_eq
tff(fact_5553_Suc__unat__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( Xa != zero_zero(word(A)) )
         => ( aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A))))) = aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) ) ) ) ).

% Suc_unat_minus_one
tff(fact_5554_ucast__or__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(B),Y: word(B)] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se1065995026697491101ons_or(word(B)),Xa),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y)) ) ).

% ucast_or_distrib
tff(fact_5555_le__ucast__ucast__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(A),Y: word(B)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y))
         => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),Y) ) ) ).

% le_ucast_ucast_le
tff(fact_5556_word__unat__Rep__inject1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) = aa(word(B),nat,semiring_1_unsigned(B,nat),one_one(word(B))) )
        <=> ( Xa = one_one(word(A)) ) ) ) ).

% word_unat_Rep_inject1
tff(fact_5557_ucast__nat__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(B)] : aa(nat,word(A),semiring_1_of_nat(word(A)),aa(word(B),nat,semiring_1_unsigned(B,nat),Xa)) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa) ) ).

% ucast_nat_def
tff(fact_5558_word__eq__unatI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),V) = aa(word(A),nat,semiring_1_unsigned(A,nat),W) )
         => ( V = W ) ) ) ).

% word_eq_unatI
tff(fact_5559_ucast__0__I,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(A)] :
          ( ( Xa = zero_zero(word(A)) )
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa) = zero_zero(word(B)) ) ) ) ).

% ucast_0_I
tff(fact_5560_word__less__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) ) ).

% word_less_nat_alt
tff(fact_5561_unat__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) ) ).

% unat_mono
tff(fact_5562_word__unat__and__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat,Y: word(A)] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),Na)
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),Na) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y))),Na) ) ) ).

% word_unat_and_lt
tff(fact_5563_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_5564_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_5565_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_5566_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_5567_unat__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ).

% unat_div
tff(fact_5568_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_5569_max__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: 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)),A2),B2)),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)),A2),B2))),aa(word(A),nat,semiring_1_unsigned(A,nat),C2)) ) ).

% max_lt
tff(fact_5570_unat__of__nat__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2)
         => ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),B2) = C2 )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),aa(nat,word(A),semiring_1_of_nat(word(A)),C2)) ) ) ) ).

% unat_of_nat_less
tff(fact_5571_unat__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ).

% unat_mod
tff(fact_5572_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_5573_unat__eq__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) = aa(nat,nat,suc,zero_zero(nat)) )
        <=> ( Xa = one_one(word(A)) ) ) ) ).

% unat_eq_1
tff(fact_5574_unat__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
        <=> ( Xa != zero_zero(word(A)) ) ) ) ).

% unat_gt_0
tff(fact_5575_unat__plus__simple,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_plus_simple
tff(fact_5576_word__of__nat__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)),Xa) ) ) ).

% word_of_nat_less
tff(fact_5577_unat__less__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),semiring_1_of_nat(word(A)),Na))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),Na) ) ) ).

% unat_less_helper
tff(fact_5578_word__of__nat__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)),Xa) ) ) ).

% word_of_nat_le
tff(fact_5579_word__unat__less__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(nat,word(A),semiring_1_of_nat(word(A)),B2))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),B2) ) ) ).

% word_unat_less_le
tff(fact_5580_word__arith__nat__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2) = 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),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))) ) ).

% word_arith_nat_add
tff(fact_5581_word__arith__nat__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2) = 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),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))) ) ).

% word_arith_nat_mult
tff(fact_5582_word__arith__nat__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),B2) = 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),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))) ) ).

% word_arith_nat_div
tff(fact_5583_word__arith__nat__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),A2),B2) = 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),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))) ) ).

% word_arith_nat_mod
tff(fact_5584_unat__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),Xa)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)) ) ) ).

% unat_1_0
tff(fact_5585_unat__max__word__pos,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))) ) ).

% unat_max_word_pos
tff(fact_5586_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_5587_Suc__unat__diff__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),Xa)
         => ( aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A))))) = aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) ) ) ) ).

% Suc_unat_diff_1
tff(fact_5588_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_5589_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_5590_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),aa(word(A),fun(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_5591_word__overflow__unat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A)))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),one_one(nat)) )
          | ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% word_overflow_unat
tff(fact_5592_the1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( ? [X5: A] :
          ( aa(A,$o,P,X5)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X5 ) ) )
     => ( aa(A,$o,P,A2)
       => ( the(A,P) = A2 ) ) ) ).

% the1_equality
tff(fact_5593_the1I2,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X5: A] :
          ( aa(A,$o,P,X5)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X5 ) ) )
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
           => aa(A,$o,Q,X) )
       => aa(A,$o,Q,the(A,P)) ) ) ).

% the1I2
tff(fact_5594_If__def,axiom,
    ! [A: $tType,P: $o,Xa: A,Y: A] :
      $ite((P),Xa,Y) = the(A,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_pc($o,fun(A,fun(A,fun(A,$o))),(P)),Xa),Y)) ).

% If_def
tff(fact_5595_theI2,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,A2)
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
           => ( X = A2 ) )
       => ( ! [X: A] :
              ( aa(A,$o,P,X)
             => aa(A,$o,Q,X) )
         => aa(A,$o,Q,the(A,P)) ) ) ) ).

% theI2
tff(fact_5596_theI_H,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X5: A] :
          ( aa(A,$o,P,X5)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X5 ) ) )
     => aa(A,$o,P,the(A,P)) ) ).

% theI'
tff(fact_5597_theI,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( aa(A,$o,P,A2)
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
           => ( X = A2 ) )
       => aa(A,$o,P,the(A,P)) ) ) ).

% theI
tff(fact_5598_lt__plus__1__le__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,MaxBound: word(A),Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,semiring_1_unsigned(A,nat),MaxBound))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)) ) ) ) ).

% lt_plus_1_le_word
tff(fact_5599_odd__word__imp__even__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))) = zero_zero(word(A)) )
            | aa(nat,$o,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)),Xa),one_one(word(A))))) ) ) ) ).

% odd_word_imp_even_next
tff(fact_5600_even__word__imp__odd__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))) = zero_zero(word(A)) )
            | ~ aa(nat,$o,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)),Xa),one_one(word(A))))) ) ) ) ).

% even_word_imp_odd_next
tff(fact_5601_floor__real__def,axiom,
    ! [Xa: real] : archim6421214686448440834_floor(real,Xa) = the(int,aTP_Lamp_pd(real,fun(int,$o),Xa)) ).

% floor_real_def
tff(fact_5602_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_5603_of__nat__eq__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Na) = W )
        <=> ? [Q5: nat] : Na = aa(nat,nat,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),Q5),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),W)))) ) ) ).

% of_nat_eq_size
tff(fact_5604_unat__plus__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa)))) ) ).

% unat_plus_if_size
tff(fact_5605_unat__sub__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa)))),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))) ) ).

% unat_sub_if_size
tff(fact_5606_no__plus__overflow__unat__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa))) ) ) ).

% no_plus_overflow_unat_size
tff(fact_5607_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),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),W)),Na)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),W)),Na))) ) ) ) ).

% and_mask_arith'
tff(fact_5608_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,
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),L)),aa($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_5609_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),
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(int,$o,dvd_dvd(int,L),K)))))) ) ).

% divide_int_def
tff(fact_5610_sgn__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: A] : sgn_sgn(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sgn_sgn(A,A2)),sgn_sgn(A,B2)) ) ).

% sgn_divide
tff(fact_5611_power__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] : sgn_sgn(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),sgn_sgn(A,A2)),Na) ) ).

% power_sgn
tff(fact_5612_sgn__inverse,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),sgn_sgn(A,A2)) ) ).

% sgn_inverse
tff(fact_5613_inverse__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),sgn_sgn(A,A2)) = sgn_sgn(A,A2) ) ).

% inverse_sgn
tff(fact_5614_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),sgn_sgn(A,A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% sgn_greater
tff(fact_5615_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),sgn_sgn(A,A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% sgn_less
tff(fact_5616_mask__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] :
          ( ( aa(nat,A,bit_se2239418461657761734s_mask(A),Na) = zero_zero(A) )
        <=> ( Na = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_5617_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( aa(nat,A,bit_se2239418461657761734s_mask(A),zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_5618_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),sgn_sgn(A,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),sgn_sgn(A,B2)) ) ).

% divide_sgn
tff(fact_5619_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( sgn_sgn(A,A2) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_5620_take__bit__minus__one__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,uminus_uminus(A),one_one(A))) = aa(nat,A,bit_se2239418461657761734s_mask(A),Na) ) ).

% take_bit_minus_one_eq_mask
tff(fact_5621_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A2)),sgn_sgn(A,A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% sgn_mult_self_eq
tff(fact_5622_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R: int] :
      ( aa(int,$o,dvd_dvd(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),K),sgn_sgn(int,R)))
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        | ( R = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_5623_dvd__sgn__mult__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,R)),K))
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        | ( R = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_5624_mult__sgn__dvd__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),L),sgn_sgn(int,R))),K)
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        & ( ( R = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_5625_sgn__mult__dvd__iff,axiom,
    ! [R: int,L: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,R)),L)),K)
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        & ( ( R = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_5626_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( sgn_sgn(A,A2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_5627_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat] : sgn_sgn(A,aa(nat,A,semiring_1_of_nat(A),Na)) = aa($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_5628_ucast__and__mask,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(B),Na: nat] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),Xa),aa(nat,word(B),bit_se2239418461657761734s_mask(word(B)),Na))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) ) ).

% ucast_and_mask
tff(fact_5629_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L2: int] : K != aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L2)),aa(nat,int,semiring_1_of_nat(int),N)) ).

% int_sgnE
tff(fact_5630_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( sgn_sgn(A,B2) = sgn_sgn(A,A2) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% same_sgn_abs_add
tff(fact_5631_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),sgn_sgn(A,K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_5632_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),sgn_sgn(A,A2)) = A2 ) ).

% abs_mult_sgn
tff(fact_5633_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A2)),aa(A,A,abs_abs(A),A2)) = A2 ) ).

% sgn_mult_abs
tff(fact_5634_mult__sgn__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,Xa)),aa(A,A,abs_abs(A),Xa)) = Xa ) ).

% mult_sgn_abs
tff(fact_5635_ex__mask__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ? [X: nat] : aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),X) = one_one(word(A)) ) ).

% ex_mask_1
tff(fact_5636_of__nat__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se2239418461657761734s_mask(nat),Na)) = aa(nat,A,bit_se2239418461657761734s_mask(A),Na) ) ).

% of_nat_mask_eq
tff(fact_5637_of__int__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,bit_se2239418461657761734s_mask(int),Na)) = aa(nat,A,bit_se2239418461657761734s_mask(A),Na) ) ).

% of_int_mask_eq
tff(fact_5638_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xa: A,Y: A] : sgn_sgn(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,Xa)),sgn_sgn(A,Y)) ) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_5639_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: A] : sgn_sgn(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A2)),sgn_sgn(A,B2)) ) ).

% sgn_mult
tff(fact_5640_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( sgn_sgn(A,B2) = sgn_sgn(A,A2) )
         => ( sgn_sgn(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = sgn_sgn(A,A2) ) ) ) ).

% same_sgn_sgn_add
tff(fact_5641_mask__twice2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) ) ) ) ).

% mask_twice2
tff(fact_5642_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( sgn_sgn(int,K) = sgn_sgn(int,L) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) ) ) ).

% div_eq_sgn_abs
tff(fact_5643_and__mask__eq__iff__le__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = W )
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) ) ) ).

% and_mask_eq_iff_le_mask
tff(fact_5644_le__mask__imp__and__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = Xa ) ) ) ).

% le_mask_imp_and_mask
tff(fact_5645_mask__power__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: 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),aa(word(A),fun(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)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))),K)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),Xa),K)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) ) ).

% mask_power_eq
tff(fact_5646_less__eq__mask__iff__take__bit__eq__self,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),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))
        <=> ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se2584673776208193580ke_bit(word(A)),Na),W) = W ) ) ) ).

% less_eq_mask_iff_take_bit_eq_self
tff(fact_5647_take__bit__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(nat,A,bit_se2239418461657761734s_mask(A),Na)) ) ).

% take_bit_eq_mask
tff(fact_5648_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = one_one(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% sgn_1_pos
tff(fact_5649_More__Word_Omask__Suc__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),aa(nat,nat,suc,zero_zero(nat))) = one_one(word(A)) ) ) ).

% More_Word.mask_Suc_0
tff(fact_5650_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ~ aa(int,$o,dvd_dvd(int,L),K)
       => ( sgn_sgn(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)) = sgn_sgn(int,L) ) ) ) ).

% sgn_mod
tff(fact_5651_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          sgn_sgn(A,Xa) = $ite(
            Xa = zero_zero(A),
            zero_zero(A),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% sgn_if
tff(fact_5652_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% sgn_1_neg
tff(fact_5653_zsgn__def,axiom,
    ! [I: 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_5654_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),sgn_sgn(int,V)),aa(int,int,abs_abs(int),K))),aa(int,int,aa(int,fun(int,int),times_times(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_5655_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),sgn_sgn(int,K)),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_5656_word__1FF__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(num,word(A),numeral_numeral(word(A)),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))) = aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))) ) ) ).

% word_1FF_is_mask
tff(fact_5657_word__FF__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(num,word(A),numeral_numeral(word(A)),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))))) = aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))) ) ) ).

% word_FF_is_mask
tff(fact_5658_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)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),aa(num,int,numeral_numeral(int),I))) ) ).

% and_mask_no
tff(fact_5659_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))),aa(nat,A,bit_se2239418461657761734s_mask(A),Na))
        <=> ( Na = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_5660_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)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)),one_one(word(A))) = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) ) ).

% mask_plus_1
tff(fact_5661_less__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = Xa ) ) ) ).

% less_mask_eq
tff(fact_5662_mask__eq__decr__exp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A))) ) ).

% mask_eq_decr_exp
tff(fact_5663_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K: int,Q2: int] :
      ( ( sgn_sgn(int,R) = 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),Q2),L)),R) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_5664_mask__eq__exp__minus__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(nat,A,bit_se2239418461657761734s_mask(A),Na) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A)) ) ).

% mask_eq_exp_minus_1
tff(fact_5665_mask__Suc__exp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(nat,A,bit_se2239418461657761734s_mask(A),aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)),aa(nat,A,bit_se2239418461657761734s_mask(A),Na)) ) ).

% mask_Suc_exp
tff(fact_5666_mask__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = W )
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% mask_eq_iff
tff(fact_5667_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),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ).

% and_mask_lt_2p
tff(fact_5668_mask__Suc__rec,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),aa(nat,nat,suc,Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))),one_one(word(A))) ) ).

% mask_Suc_rec
tff(fact_5669_is__aligned__AND__less__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),Na: nat,V: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),U),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),V),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),U),V) = zero_zero(word(A)) ) ) ) ) ).

% is_aligned_AND_less_0
tff(fact_5670_add__mask__fold,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),one_one(word(A))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) ) ).

% add_mask_fold
tff(fact_5671_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ( sgn_sgn(int,K) != sgn_sgn(int,L) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)))),aa($o,int,zero_neq_one_of_bool(int),~ aa(int,$o,dvd_dvd(int,L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_5672_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero(int) )
         => ( A32 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A22) ) ) )
         => ~ ! [R4: int,Q3: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R4) )
               => ( ( sgn_sgn(int,R4) = sgn_sgn(int,A22) )
                 => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R4)),aa(int,int,abs_abs(int),A22))
                   => ( A1 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A22)),R4) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_5673_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero(int) )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K3) ) )
        | ? [L4: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),zero_zero(int)) )
            & ( L4 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R5) )
            & ( sgn_sgn(int,R5) = sgn_sgn(int,L4) )
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L4))
            & ( K3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_5674_and__mask__dvd__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(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),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = zero_zero(word(A)) ) ) ) ).

% and_mask_dvd_nat
tff(fact_5675_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),aa(nat,word(A),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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ) ).

% and_mask_mod_2p
tff(fact_5676_and__mask__dvd,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(int,$o,dvd_dvd(int,aa(nat,int,aa(int,fun(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),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = zero_zero(word(A)) ) ) ) ).

% and_mask_dvd
tff(fact_5677_mask__Suc__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(nat,A,bit_se2239418461657761734s_mask(A),aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,bit_se2239418461657761734s_mask(A),Na))) ) ).

% mask_Suc_double
tff(fact_5678_mask__eq__iff__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),W))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = W )
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% mask_eq_iff_w2p
tff(fact_5679_and__mask__less__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),Xa))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ).

% and_mask_less_size
tff(fact_5680_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),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A)))) ) ).

% word_and_mask_le_2pm1
tff(fact_5681_word__mod__2p__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) ) ) ) ).

% word_mod_2p_is_mask
tff(fact_5682_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),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% word_unat_mask_lt
tff(fact_5683_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),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),W)),Na)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),W)),Na))) ) ).

% and_mask_arith
tff(fact_5684_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: num] : aa(nat,A,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))),aa(nat,A,bit_se2239418461657761734s_mask(A),pred_numeral(Na)))) ) ).

% mask_numeral
tff(fact_5685_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),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),Na))) = $ite(
        ( ( sgn_sgn(int,L) = zero_zero(int) )
        | ( sgn_sgn(int,K) = zero_zero(int) )
        | ( Na = zero_zero(nat) ) ),
        zero_zero(int),
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,dvd_dvd(nat,Na),M)))))) ) ).

% divide_int_unfold
tff(fact_5686_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),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),Na))) = $ite(
        ( ( sgn_sgn(int,L) = zero_zero(int) )
        | ( sgn_sgn(int,K) = zero_zero(int) )
        | ( Na = zero_zero(nat) ) ),
        aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M)),
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,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_5687_floor__rat__def,axiom,
    ! [Xa: rat] : archim6421214686448440834_floor(rat,Xa) = the(int,aTP_Lamp_pe(rat,fun(int,$o),Xa)) ).

% floor_rat_def
tff(fact_5688_sgn__div__eq__sgn__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) != zero_zero(int) )
     => ( sgn_sgn(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) = sgn_sgn(int,aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2)) ) ) ).

% sgn_div_eq_sgn_mult
tff(fact_5689_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)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% neg_mask_is_div'
tff(fact_5690_bit_Odouble__compl,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)) = Xa ) ).

% bit.double_compl
tff(fact_5691_bit_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,bit_ri4277139882892585799ns_not(A),Xa) = aa(A,A,bit_ri4277139882892585799ns_not(A),Y) )
        <=> ( Xa = Y ) ) ) ).

% bit.compl_eq_compl_iff
tff(fact_5692_bit_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)) = zero_zero(A) ) ).

% bit.conj_cancel_right
tff(fact_5693_bit_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)),Xa) = zero_zero(A) ) ).

% bit.conj_cancel_left
tff(fact_5694_bit_Ode__Morgan__disj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A,Y: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) ) ).

% bit.de_Morgan_disj
tff(fact_5695_bit_Ode__Morgan__conj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A,Y: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) ) ).

% bit.de_Morgan_conj
tff(fact_5696_and__and__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: 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)),A2),B2)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),B2)) = zero_zero(word(A)) ) ).

% and_and_not
tff(fact_5697_mask__nat__positive__iff,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,bit_se2239418461657761734s_mask(nat),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% mask_nat_positive_iff
tff(fact_5698_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_5699_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_5700_bit_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)),Xa) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_left
tff(fact_5701_bit_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xa),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_right
tff(fact_5702_NOT__mask__AND__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [W: A,Na: nat] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),W),aa(nat,A,bit_se2239418461657761734s_mask(A),Na))),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,bit_se2239418461657761734s_mask(A),Na))) = zero_zero(A) ) ).

% NOT_mask_AND_mask
tff(fact_5703_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_5704_even__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4277139882892585799ns_not(A),A2))
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% even_not_iff
tff(fact_5705_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_5706_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_5707_real__sgn__eq,axiom,
    ! [Xa: real] : sgn_sgn(real,Xa) = aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),aa(real,real,abs_abs(real),Xa)) ).

% real_sgn_eq
tff(fact_5708_less__eq__mask,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,bit_se2239418461657761734s_mask(nat),Na)) ).

% less_eq_mask
tff(fact_5709_nat__mask__eq,axiom,
    ! [Na: nat] : aa(int,nat,nat2,aa(nat,int,bit_se2239418461657761734s_mask(int),Na)) = aa(nat,nat,bit_se2239418461657761734s_mask(nat),Na) ).

% nat_mask_eq
tff(fact_5710_sgn__rat__def,axiom,
    ! [A2: rat] :
      sgn_sgn(rat,A2) = $ite(
        A2 = zero_zero(rat),
        zero_zero(rat),
        $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),A2),one_one(rat),aa(rat,rat,uminus_uminus(rat),one_one(rat))) ) ).

% sgn_rat_def
tff(fact_5711_mask__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat,Y: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) )
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) )
           => ( Xa = Y ) ) ) ) ).

% mask_eqI
tff(fact_5712_split__word__eq__on__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),M: word(A)] :
          ( ( Xa = Y )
        <=> ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),M) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),M) )
            & ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),M)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),M)) ) ) ) ) ).

% split_word_eq_on_mask
tff(fact_5713_take__bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,bit_ri4277139882892585799ns_not(A),B2)) )
        <=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),B2) ) ) ) ).

% take_bit_not_iff
tff(fact_5714_take__bit__not__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2))) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) ) ).

% take_bit_not_take_bit
tff(fact_5715_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)
           => ! [T4: rat] :
                ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),T4)
               => ( R != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S2),T4) ) ) ) ) ).

% obtain_pos_sum
tff(fact_5716_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_5717_abs__rat__def,axiom,
    ! [A2: rat] :
      aa(rat,rat,abs_abs(rat),A2) = $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),A2),zero_zero(rat)),aa(rat,rat,uminus_uminus(rat),A2),A2) ).

% abs_rat_def
tff(fact_5718_not__switch,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Xa: A] :
          ( ( aa(A,A,bit_ri4277139882892585799ns_not(A),A2) = Xa )
         => ( A2 = aa(A,A,bit_ri4277139882892585799ns_not(A),Xa) ) ) ) ).

% not_switch
tff(fact_5719_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_5720_less__eq__rat__def,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),Xa),Y)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Xa),Y)
        | ( Xa = Y ) ) ) ).

% less_eq_rat_def
tff(fact_5721_not__add__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),B2) ) ).

% not_add_distrib
tff(fact_5722_not__diff__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),B2) ) ).

% not_diff_distrib
tff(fact_5723_and__eq__not__not__or,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) = 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),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),B2))) ) ).

% and_eq_not_not_or
tff(fact_5724_or__eq__not__not__and,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = 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),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),B2))) ) ).

% or_eq_not_not_and
tff(fact_5725_word__plus__and__or__coroll2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: 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)),Xa),W)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),W))) = Xa ) ).

% word_plus_and_or_coroll2
tff(fact_5726_mask__eq__x__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),W: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),W) = Xa )
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),W)) = zero_zero(word(A)) ) ) ) ).

% mask_eq_x_eq_0
tff(fact_5727_mask__eq__0__eq__x,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),W: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),W) = zero_zero(word(A)) )
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),W)) = Xa ) ) ) ).

% mask_eq_0_eq_x
tff(fact_5728_mask__nonnegative__int,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(nat,int,bit_se2239418461657761734s_mask(int),Na)) ).

% mask_nonnegative_int
tff(fact_5729_not__mask__negative__int,axiom,
    ! [Na: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,bit_se2239418461657761734s_mask(int),Na)),zero_zero(int)) ).

% not_mask_negative_int
tff(fact_5730_minus__eq__not__plus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),one_one(A)) ) ).

% minus_eq_not_plus_1
tff(fact_5731_not__eq__complement,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A2)),one_one(A)) ) ).

% not_eq_complement
tff(fact_5732_minus__eq__not__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),A2) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))) ) ).

% minus_eq_not_minus_1
tff(fact_5733_sgn__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( sgn_sgn(real,aa(real,real,root(Na),Xa)) = sgn_sgn(real,Xa) ) ) ).

% sgn_root
tff(fact_5734_take__bit__not__eq__mask__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,bit_se2239418461657761734s_mask(A),Na)),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2)) ) ).

% take_bit_not_eq_mask_diff
tff(fact_5735_mask__out__first__mask__some,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat,Y: word(A),M: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) = Y )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))) ) ) ) ) ).

% mask_out_first_mask_some
tff(fact_5736_mask__lower__twice,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))) ) ) ) ).

% mask_lower_twice
tff(fact_5737_NOT__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),Xa)),one_one(word(A))) ) ).

% NOT_eq
tff(fact_5738_mask__AND__NOT__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = zero_zero(word(A)) ) ).

% mask_AND_NOT_mask
tff(fact_5739_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)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) = W ) ).

% AND_NOT_mask_plus_AND_mask_eq
tff(fact_5740_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_5741_mask__out__sub__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) ) ).

% mask_out_sub_mask
tff(fact_5742_subtract__mask_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),P2),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),P2),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) ) ).

% subtract_mask(1)
tff(fact_5743_subtract__mask_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),P2),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),P2),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) ) ).

% subtract_mask(2)
tff(fact_5744_ucast__and__neg__mask,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(B),Na: nat] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),Xa),aa(word(B),word(B),bit_ri4277139882892585799ns_not(word(B)),aa(nat,word(B),bit_se2239418461657761734s_mask(word(B)),Na)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) ) ).

% ucast_and_neg_mask
tff(fact_5745_or__not__mask__nop,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) ) ).

% or_not_mask_nop
tff(fact_5746_mask__or__not__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))) = Xa ) ).

% mask_or_not_mask
tff(fact_5747_neg__mask__twice,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat,M: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),M)))) ) ).

% neg_mask_twice
tff(fact_5748_neg__mask__combine,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: nat,B2: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),A2))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),B2))) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),B2))) ) ).

% neg_mask_combine
tff(fact_5749_mask__lower__twice2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Na: nat,M: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),M)))) ) ).

% mask_lower_twice2
tff(fact_5750_and__not__mask__twice,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat,M: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Na)))) ) ).

% and_not_mask_twice
tff(fact_5751_less__mask,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,bit_se2239418461657761734s_mask(nat),Na)) ) ).

% less_mask
tff(fact_5752_sgn__eq,axiom,
    ! [Z: complex] : sgn_sgn(complex,Z) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),aa(real,complex,real_Vector_of_real(complex),real_V7770717601297561774m_norm(complex,Z))) ).

% sgn_eq
tff(fact_5753_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),aa(num,num,bit1,Na))) ) ).

% not_numeral_Bit0_eq
tff(fact_5754_sgn__real__def,axiom,
    ! [A2: real] :
      sgn_sgn(real,A2) = $ite(
        A2 = zero_zero(real),
        zero_zero(real),
        $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ) ).

% sgn_real_def
tff(fact_5755_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,bit_se2239418461657761734s_mask(A),Na))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_5756_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_5757_multiple__mask__trivia,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))) ) ) ) ).

% multiple_mask_trivia
tff(fact_5758_and__mask__0__iff__le__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) = zero_zero(word(A)) )
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) ) ) ).

% and_mask_0_iff_le_mask
tff(fact_5759_mask__subsume,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))) ) ) ) ).

% mask_subsume
tff(fact_5760_take__bit__eq__mask__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K) = aa(nat,int,bit_se2239418461657761734s_mask(int),Na) )
    <=> ( aa(int,int,aa(nat,fun(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_5761_sgn__integer__code,axiom,
    ! [K: code_integer] :
      sgn_sgn(code_integer,K) = $ite(
        K = zero_zero(code_integer),
        zero_zero(code_integer),
        $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),one_one(code_integer)) ) ).

% sgn_integer_code
tff(fact_5762_sgn__power__injE,axiom,
    ! [A2: real,Na: nat,Xa: real,B2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,A2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),A2)),Na)) = Xa )
     => ( ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,B2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),B2)),Na)) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_5763_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),Y) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xa),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),Xa) = Y ) ) ) ) ).

% bit.compl_unique
tff(fact_5764_Suc__mask__eq__exp,axiom,
    ! [Na: nat] : aa(nat,nat,suc,aa(nat,nat,bit_se2239418461657761734s_mask(nat),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ).

% Suc_mask_eq_exp
tff(fact_5765_mask__nat__less__exp,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,bit_se2239418461657761734s_mask(nat),Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% mask_nat_less_exp
tff(fact_5766_minus__exp__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,bit_se2239418461657761734s_mask(A),Na)) ) ).

% minus_exp_eq_not_mask
tff(fact_5767_root__sgn__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Y)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y)),Na))) = Y ) ) ).

% root_sgn_power
tff(fact_5768_sgn__power__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,aa(real,real,root(Na),Xa))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),aa(real,real,root(Na),Xa))),Na)) = Xa ) ) ).

% sgn_power_root
tff(fact_5769_NOT__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = aa(word(A),word(A),uminus_uminus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ).

% NOT_mask
tff(fact_5770_mask__half__int,axiom,
    ! [Na: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,bit_se2239418461657761734s_mask(int),Na)),aa(num,int,numeral_numeral(int),bit0(one2))) = aa(nat,int,bit_se2239418461657761734s_mask(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) ).

% mask_half_int
tff(fact_5771_mask__int__def,axiom,
    ! [Na: nat] : aa(nat,int,bit_se2239418461657761734s_mask(int),Na) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),one_one(int)) ).

% mask_int_def
tff(fact_5772_mask__nat__def,axiom,
    ! [Na: nat] : aa(nat,nat,bit_se2239418461657761734s_mask(nat),Na) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),one_one(nat)) ).

% mask_nat_def
tff(fact_5773_cis__Arg__unique,axiom,
    ! [Z: complex,Xa: real] :
      ( ( sgn_sgn(complex,Z) = cis(Xa) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
         => ( arg(Z) = Xa ) ) ) ) ).

% cis_Arg_unique
tff(fact_5774_split__root,axiom,
    ! [P: fun(real,$o),Na: nat,Xa: real] :
      ( aa(real,$o,P,aa(real,real,root(Na),Xa))
    <=> ( ( ( Na = zero_zero(nat) )
         => aa(real,$o,P,zero_zero(real)) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ! [Y5: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Y5)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y5)),Na)) = Xa )
             => aa(real,$o,P,Y5) ) ) ) ) ).

% split_root
tff(fact_5775_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(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_5776_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)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ).

% neg_mask_is_div
tff(fact_5777_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K) = aa(nat,int,bit_se2239418461657761734s_mask(int),Na) )
    <=> aa(int,$o,dvd_dvd(int,aa(nat,int,aa(int,fun(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_5778_arctan__inverse,axiom,
    ! [Xa: real] :
      ( ( Xa != zero_zero(real) )
     => ( aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Xa)) = aa(real,real,aa(real,fun(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),sgn_sgn(real,Xa)),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(real,real,arctan,Xa)) ) ) ).

% arctan_inverse
tff(fact_5779_neg__mask__add__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: 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)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) ) ).

% neg_mask_add_mask
tff(fact_5780_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Na),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,suc,Na)),K)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,Na))),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_5781_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : size_num(aa(num,num,bit1,X32)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X32)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(3)
tff(fact_5782_hash__code__prod__simps,axiom,
    ! [A: $tType,B: $tType,H_a: fun(A,uint32),H_b: fun(B,uint32),Xa: A,Xaa: B] : hash_hash_code_prod(A,B,H_a,H_b,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xaa)) = aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(A,uint32,H_a,Xa)),aa(num,uint32,numeral_numeral(uint32),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))))),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(B,uint32,H_b,Xaa)),aa(num,uint32,numeral_numeral(uint32),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))))))))))))))))))))))))))))),aa(num,uint32,numeral_numeral(uint32),bit0(bit0(bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,one2))))))))))))))))))))))))))))))))) ).

% hash_code_prod_simps
tff(fact_5783_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_5784_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_5785_bit__numeral__Bit1__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),aa(nat,nat,suc,Na))
        <=> aa(nat,$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_5786_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_5787_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_5788_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,aa(nat,fun(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_5789_signed__take__bit__negative__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_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_5790_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_5791_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_5792_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),aa(num,num,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_5793_bit__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Na: num] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,W))),aa(num,nat,numeral_numeral(nat),Na))
        <=> aa(nat,$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_5794_word__no__log__defs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),A2))) ) ).

% word_no_log_defs(1)
tff(fact_5795_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),zero_zero(nat))
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% bit_0
tff(fact_5796_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_5797_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),aa(num,num,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_5798_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),Na)
        <=> ( ( Na = zero_zero(nat) )
            & ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% bit_mod_2_iff
tff(fact_5799_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) ) ) ).

% bin_nth_minus_Bit0
tff(fact_5800_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),aa(num,num,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,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) ) ) ).

% bin_nth_minus_Bit1
tff(fact_5801_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),aa(word(A),fun(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_5802_word__no__log__defs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: 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)),A2))) = 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),A2)))) ) ).

% word_no_log_defs(2)
tff(fact_5803_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,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Na)),one_one(int)))) ).

% and_minus_minus_numerals
tff(fact_5804_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,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Na)),one_one(int)))) ).

% or_minus_minus_numerals
tff(fact_5805_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),A2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na) ) ) ) ).

% bit_take_bit_iff
tff(fact_5806_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,aa(int,fun(int,int),minus_minus(int),K),one_one(int)))),Na) ) ).

% bit_minus_int_iff
tff(fact_5807_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_5808_signed__take__bit__eq__if__positive,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Na: nat] :
          ( ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
         => ( aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),Na),A2) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) ) ) ) ).

% signed_take_bit_eq_if_positive
tff(fact_5809_bit__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),Na)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
            | aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B2),Na) ) ) ) ).

% bit_or_iff
tff(fact_5810_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_5811_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_5812_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_5813_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_5814_test__bit__cong,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Xa: A,Y: A,Xb: nat] :
          ( ( Xa = Y )
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Xa),Xb)
          <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Y),Xb) ) ) ) ).

% test_bit_cong
tff(fact_5815_bit__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)),Na)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
            & ( M != Na ) ) ) ) ).

% bit_unset_bit_iff
tff(fact_5816_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_5817_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_5818_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( ! [N: nat] :
              ( ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),N)
              | ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B2),N) )
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),Na)
          <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
              | aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B2),Na) ) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_5819_disjunctive__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( ! [N: nat] :
              ( ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),N)
              | ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B2),N) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ) ) ).

% disjunctive_add
tff(fact_5820_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [B2: $o,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa($o,A,zero_neq_one_of_bool(A),(B2))),Na)
        <=> ( (B2)
            & ( Na = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_5821_bit__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),Na)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B2),Na) ) ) ) ).

% bit_and_iff
tff(fact_5822_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_5823_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_5824_disjunctive__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [B2: A,A2: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B2),N)
             => aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),N) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,bit_ri4277139882892585799ns_not(A),B2)) ) ) ) ).

% disjunctive_diff
tff(fact_5825_not__int__def,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),K)),one_one(int)) ).

% not_int_def
tff(fact_5826_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_5827_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_5828_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,aa(int,fun(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_5829_flip__bit__eq__if,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),Na),A2) = aa(A,A,
            aa(nat,fun(A,A),
              $ite(aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na),bit_se2638667681897837118et_bit(A),bit_se5668285175392031749et_bit(A)),
              Na),
            A2) ) ).

% flip_bit_eq_if
tff(fact_5830_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_5831_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_5832_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_5833_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_5834_bit__imp__take__bit__positive,axiom,
    ! [Na: nat,M: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),M),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_5835_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_5836_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_5837_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_5838_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_5839_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_5840_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat,A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
         => ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_5841_bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,nat,suc,Na))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),Na) ) ) ).

% bit_Suc
tff(fact_5842_stable__imp__bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
          <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_5843_bit__iff__idd__imp__stable,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),N)
            <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 ) ) ) ).

% bit_iff_idd_imp_stable
tff(fact_5844_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M2)
             => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),M2)
              <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) ) )
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))
              <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) ) ) ) ).

% int_bit_bound
tff(fact_5845_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_5846_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),aa(num,num,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_5847_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),aa(num,num,bit1,Na)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na))) ).

% or_not_numerals(3)
tff(fact_5848_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),aa(num,num,bit1,Na)))) = zero_zero(int) ).

% and_not_numerals(3)
tff(fact_5849_signed__take__bit__eq__if__negative,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
         => ( aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),Na),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,bit_se2239418461657761734s_mask(A),Na))) ) ) ) ).

% signed_take_bit_eq_if_negative
tff(fact_5850_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),aa(num,num,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_5851_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_5852_bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na))) ) ) ).

% bit_iff_odd
tff(fact_5853_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = zero_zero(A) )
        <=> ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_5854_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),Na)
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) )
            & ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na) ) ) ) ).

% bit_not_iff_eq
tff(fact_5855_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),aa(num,num,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_5856_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),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,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_5857_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),aa(num,num,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_5858_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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ) ).

% bit_int_def
tff(fact_5859_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),Na)
          <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
              | ( Na = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_5860_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
          <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),Na)
              | ( Na = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_5861_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_5862_signed__take__bit__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),Na),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na))),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,bit_se2239418461657761734s_mask(A),Na)))) ) ).

% signed_take_bit_def
tff(fact_5863_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( ! [J3: nat] : ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,nat,suc,J3))
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),Na)
          <=> $ite(Na = zero_zero(nat),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)),Na)) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_5864_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
        <=> $ite(Na = zero_zero(nat),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))) ) ) ).

% bit_rec
tff(fact_5865_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),aa(num,num,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_5866_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),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,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_5867_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),aa(num,num,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_5868_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_5869_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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ).

% Bit_Operations.set_bit_eq
tff(fact_5870_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,aa(int,fun(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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ).

% unset_bit_eq
tff(fact_5871_take__bit__Suc__from__most,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,suc,Na)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)) ).

% take_bit_Suc_from_most
tff(fact_5872_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : size_num(bit0(X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_5873_hash__code__option__simps_I2_J,axiom,
    ! [A: $tType,H_a: fun(A,uint32),Xa: A] : hash_h1887023736457453652option(A,H_a,some(A,Xa)) = aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(A,uint32,H_a,Xa)),aa(num,uint32,numeral_numeral(uint32),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(one2)))))))))))))))))))))))))))))))),aa(num,uint32,numeral_numeral(uint32),bit0(bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(one2))))))))))))))))))))))))))))))) ).

% hash_code_option_simps(2)
tff(fact_5874_hash__code__option__simps_I1_J,axiom,
    ! [A: $tType,H_a: fun(A,uint32)] : hash_h1887023736457453652option(A,H_a,none(A)) = aa(num,uint32,numeral_numeral(uint32),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))))))))))))))))))))))))))) ).

% hash_code_option_simps(1)
tff(fact_5875_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,aa(int,fun(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_5876_bit_Oxor__left__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xa),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xa),Y)) = Y ) ).

% bit.xor_left_self
tff(fact_5877_xor_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),zero_zero(A)) = A2 ) ).

% xor.right_neutral
tff(fact_5878_xor_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),zero_zero(A)),A2) = A2 ) ).

% xor.left_neutral
tff(fact_5879_xor__self__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),A2) = zero_zero(A) ) ).

% xor_self_eq
tff(fact_5880_bit_Oxor__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xa),Xa) = zero_zero(A) ) ).

% bit.xor_self
tff(fact_5881_test__bit__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),one_one(word(A))),Na)
        <=> ( Na = zero_zero(nat) ) ) ) ).

% test_bit_1
tff(fact_5882_take__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2)),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),B2)) ) ).

% take_bit_xor
tff(fact_5883_bit_Oxor__compl__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)),Y) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xa),Y)) ) ).

% bit.xor_compl_left
tff(fact_5884_bit_Oxor__compl__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xa),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),Xa),Y)) ) ).

% bit.xor_compl_right
tff(fact_5885_map__nth__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(nat)] : map(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),zero_zero(word(A))),Xs) = replicate($o,aa(list(nat),nat,size_size(list(nat)),Xs),$false) ) ).

% map_nth_0
tff(fact_5886_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_5887_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_5888_bit_Oxor__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,uminus_uminus(A),one_one(A))),Xa) = aa(A,A,bit_ri4277139882892585799ns_not(A),Xa) ) ).

% bit.xor_one_left
tff(fact_5889_bit_Oxor__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xa),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),Xa) ) ).

% bit.xor_one_right
tff(fact_5890_bit_Oxor__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)),Xa) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_left
tff(fact_5891_bit_Oxor__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xa),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_right
tff(fact_5892_xor__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(3)
tff(fact_5893_xor__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xa))),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(Xa)) ) ).

% xor_numerals(8)
tff(fact_5894_xor__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(Xa))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xa)) ) ).

% xor_numerals(5)
tff(fact_5895_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),aa(num,num,bit1,Y))) = aa(num,A,numeral_numeral(A),bit0(Y)) ) ).

% xor_numerals(2)
tff(fact_5896_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),aa(num,num,bit1,Y)) ) ).

% xor_numerals(1)
tff(fact_5897_xor__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xa))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Xa)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(7)
tff(fact_5898_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xa))),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),Xa)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(6)
tff(fact_5899_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(Xa))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Xa)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(4)
tff(fact_5900_XOR__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xa),Y)) ) ) ).

% XOR_lower
tff(fact_5901_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_5902_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_5903_word__or__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)),Na)
        <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na)
            | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Na) ) ) ) ).

% word_or_nth
tff(fact_5904_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_5905_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_5906_finite__bit__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : finite_finite2(nat,collect(nat,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W))) ) ).

% finite_bit_word
tff(fact_5907_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_5908_bit__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)),Na)
        <=> ~ ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
            <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B2),Na) ) ) ) ).

% bit_xor_iff
tff(fact_5909_word__exists__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
         => ? [X_1: nat] : aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),X_1) ) ) ).

% word_exists_nth
tff(fact_5910_nth__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),zero_zero(word(A))),Na) ) ).

% nth_0
tff(fact_5911_word__and__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)),Na)
        <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na)
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Na) ) ) ) ).

% word_and_nth
tff(fact_5912_test__bit__over,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),Xa)),Na)
         => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na) ) ) ).

% test_bit_over
tff(fact_5913_bit_Oconj__xor__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,Xa: 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)),Xa) = 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),Xa)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z),Xa)) ) ).

% bit.conj_xor_distrib2
tff(fact_5914_bit_Oconj__xor__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),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),Xa),Y)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),Z)) ) ).

% bit.conj_xor_distrib
tff(fact_5915_xor_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),C2)) ) ).

% xor.assoc
tff(fact_5916_xor_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),A2) ) ).

% xor.commute
tff(fact_5917_xor_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),C2)) ) ).

% xor.left_commute
tff(fact_5918_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_5919_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_5920_word__test__bit__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A2) = aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)) ) ).

% word_test_bit_def
tff(fact_5921_test__bit__def_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Xa: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A2),Xa)
        <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),Xa) ) ) ).

% test_bit_def'
tff(fact_5922_word__ao__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Na: nat] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)),Na)
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na)
              | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Na) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)),Na)
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na)
              & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Na) ) ) ) ) ).

% word_ao_nth
tff(fact_5923_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_5924_lsb__this__or__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A)))),zero_zero(nat))
         => aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),zero_zero(nat)) ) ) ).

% lsb_this_or_next
tff(fact_5925_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_5926_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_5927_nth__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,I: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)),I)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(word(A),nat,size_size(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) ) ) ) ).

% nth_mask
tff(fact_5928_bit_Oxor__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xa),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),Xa),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),Xa)),Y)) ) ).

% bit.xor_def
tff(fact_5929_bit_Oxor__def2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xa),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),Xa),Y)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y))) ) ).

% bit.xor_def2
tff(fact_5930_even__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
          <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ) ).

% even_xor_iff
tff(fact_5931_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_5932_overflow__imp__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))) = zero_zero(word(A)) )
         => aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),zero_zero(nat)) ) ) ).

% overflow_imp_lsb
tff(fact_5933_word__and__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] :
          aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Na),one_one(word(A))) = $ite(aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Na),zero_zero(nat)),one_one(word(A)),zero_zero(word(A))) ) ).

% word_and_1
tff(fact_5934_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_5935_bang__is__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),M)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)),Xa) ) ) ).

% bang_is_le
tff(fact_5936_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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ) ).

% bit_nat_def
tff(fact_5937_odd__iff__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),zero_zero(nat)) ) ) ).

% odd_iff_lsb
tff(fact_5938_and__neq__0__is__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Na: nat,Xa: word(A)] :
          ( ( Y = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) )
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y) != zero_zero(word(A)) )
          <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na) ) ) ) ).

% and_neq_0_is_nth
tff(fact_5939_nth__is__and__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na)
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) != zero_zero(word(A)) ) ) ) ).

% nth_is_and_neq_0
tff(fact_5940_XOR__upper,axiom,
    ! [Xa: int,Na: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xa),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ) ).

% XOR_upper
tff(fact_5941_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_5942_one__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)))),aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ).

% one_xor_eq
tff(fact_5943_xor__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),one_one(A)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)))),aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ).

% xor_one_eq
tff(fact_5944_Arg__def,axiom,
    ! [Z: complex] :
      arg(Z) = $ite(Z = zero_zero(complex),zero_zero(real),fChoice(real,aTP_Lamp_pf(complex,fun(real,$o),Z))) ).

% Arg_def
tff(fact_5945_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_5946_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( vEBT_V9176841429113362141ildupi(Xa) = Y )
     => ( accp(nat,vEBT_V3352910403632780892pi_rel,Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = one_one(int) )
             => ~ accp(nat,vEBT_V3352910403632780892pi_rel,zero_zero(nat)) ) )
         => ( ( ( Xa = 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] :
                  ( ( Xa = 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),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(bit0(one2))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))) )
                   => ~ accp(nat,vEBT_V3352910403632780892pi_rel,aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
tff(fact_5947_some__sym__eq__trivial,axiom,
    ! [A: $tType,Xa: A] : fChoice(A,aa(A,fun(A,$o),fequal(A),Xa)) = Xa ).

% some_sym_eq_trivial
tff(fact_5948_some__eq__trivial,axiom,
    ! [A: $tType,Xa: A] : fChoice(A,aTP_Lamp_aa(A,fun(A,$o),Xa)) = Xa ).

% some_eq_trivial
tff(fact_5949_some__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( aa(A,$o,P,A2)
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
           => ( X = A2 ) )
       => ( fChoice(A,P) = A2 ) ) ) ).

% some_equality
tff(fact_5950_some__insert__self,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),fChoice(A,aTP_Lamp_a(set(A),fun(A,$o),S))),S) = S ) ) ).

% some_insert_self
tff(fact_5951_word__no__log__defs_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2)),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(11)
tff(fact_5952_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),aa(num,num,bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_5953_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),aa(num,num,bit1,Y))) = aa(num,nat,numeral_numeral(nat),bit0(Y)) ).

% xor_nat_numerals(2)
tff(fact_5954_xor__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit0(Xa))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xa)) ).

% xor_nat_numerals(3)
tff(fact_5955_xor__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xa))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(Xa)) ).

% xor_nat_numerals(4)
tff(fact_5956_word__bitwise__1__simps_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2)) = 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),B2))) ) ).

% word_bitwise_1_simps(10)
tff(fact_5957_word__bitwise__1__simps_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: 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)),A2)),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),A2)),one_one(int))) ) ).

% word_bitwise_1_simps(12)
tff(fact_5958_word__no__log__defs_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(12)
tff(fact_5959_word__no__log__defs_I13_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2))),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(13)
tff(fact_5960_word__no__log__defs_I14_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(14)
tff(fact_5961_word__bitwise__1__simps_I13_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: 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)),A2))),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),A2))),one_one(int))) ) ).

% word_bitwise_1_simps(13)
tff(fact_5962_word__bitwise__1__simps_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2))) = 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),B2)))) ) ).

% word_bitwise_1_simps(11)
tff(fact_5963_test__bit__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),V: word(A)] :
          ( ( aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),U) = aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),V) )
        <=> ( U = V ) ) ) ).

% test_bit_eq_iff
tff(fact_5964_word__xor__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y)),Na)
        <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na)
            <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Na) ) ) ) ).

% word_xor_nth
tff(fact_5965_word__eqD,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),V: word(A),Xa: nat] :
          ( ( U = V )
         => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),U),Xa)
          <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),V),Xa) ) ) ) ).

% word_eqD
tff(fact_5966_bang__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( Xa = Y )
        <=> ! [N2: nat] :
              ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),N2)
            <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),N2) ) ) ) ).

% bang_eq
tff(fact_5967_some__elem,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => member(A,fChoice(A,aTP_Lamp_a(set(A),fun(A,$o),S)),S) ) ).

% some_elem
tff(fact_5968_some__in__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( member(A,fChoice(A,aTP_Lamp_a(set(A),fun(A,$o),A3)),A3)
    <=> ( A3 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_5969_some1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( ? [X5: A] :
          ( aa(A,$o,P,X5)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X5 ) ) )
     => ( aa(A,$o,P,A2)
       => ( fChoice(A,P) = A2 ) ) ) ).

% some1_equality
tff(fact_5970_some__eq__ex,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
    <=> ? [X_12: A] : aa(A,$o,P,X_12) ) ).

% some_eq_ex
tff(fact_5971_someI2__bex,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X5: A] :
          ( member(A,X5,A3)
          & aa(A,$o,P,X5) )
     => ( ! [X: A] :
            ( ( member(A,X,A3)
              & aa(A,$o,P,X) )
           => aa(A,$o,Q,X) )
       => aa(A,$o,Q,fChoice(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_al(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) ) ) ).

% someI2_bex
tff(fact_5972_someI2__ex,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X_13: A] : aa(A,$o,P,X_13)
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
           => aa(A,$o,Q,X) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2_ex
tff(fact_5973_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_5974_someI2,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,A2)
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
           => aa(A,$o,Q,X) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2
tff(fact_5975_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib2,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Y: A,Z: A,Xa: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Xor,Y),Z)),Xa) = aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,aa(A,fun(A,A),Conj,Y),Xa)),aa(A,A,aa(A,fun(A,A),Conj,Z),Xa)) ) ) ).

% abstract_boolean_algebra_sym_diff.conj_xor_distrib2
tff(fact_5976_abstract__boolean__algebra__sym__diff_Oxor__cancel__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,Xa),aa(A,A,Compl,Xa)) = One ) ) ).

% abstract_boolean_algebra_sym_diff.xor_cancel_right
tff(fact_5977_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A,Y: A,Z: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Conj,Xa),aa(A,A,aa(A,fun(A,A),Xor,Y),Z)) = aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,aa(A,fun(A,A),Conj,Xa),Y)),aa(A,A,aa(A,fun(A,A),Conj,Xa),Z)) ) ) ).

% abstract_boolean_algebra_sym_diff.conj_xor_distrib
tff(fact_5978_abstract__boolean__algebra__sym__diff_Oxor__compl__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A,Y: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,Xa),aa(A,A,Compl,Y)) = aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Xor,Xa),Y)) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_compl_right
tff(fact_5979_abstract__boolean__algebra__sym__diff_Oxor__cancel__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,Compl,Xa)),Xa) = One ) ) ).

% abstract_boolean_algebra_sym_diff.xor_cancel_left
tff(fact_5980_abstract__boolean__algebra__sym__diff_Oxor__compl__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A,Y: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,Compl,Xa)),Y) = aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Xor,Xa),Y)) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_compl_left
tff(fact_5981_abstract__boolean__algebra__sym__diff_Oxor__one__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,Xa),One) = aa(A,A,Compl,Xa) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_one_right
tff(fact_5982_abstract__boolean__algebra__sym__diff_Oxor__left__self,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A,Y: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,Xa),aa(A,A,aa(A,fun(A,A),Xor,Xa),Y)) = Y ) ) ).

% abstract_boolean_algebra_sym_diff.xor_left_self
tff(fact_5983_abstract__boolean__algebra__sym__diff_Oxor__one__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,One),Xa) = aa(A,A,Compl,Xa) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_one_left
tff(fact_5984_abstract__boolean__algebra__sym__diff_Oxor__self,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,Xa),Xa) = Zero ) ) ).

% abstract_boolean_algebra_sym_diff.xor_self
tff(fact_5985_abstract__boolean__algebra__sym__diff_Oxor__def2,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A,Y: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,Xa),Y) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,Xa),Y)),aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,Xa)),aa(A,A,Compl,Y))) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_def2
tff(fact_5986_abstract__boolean__algebra__sym__diff_Oxor__def,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A,Y: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,Xa),Y) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,Xa),aa(A,A,Compl,Y))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,Xa)),Y)) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_def
tff(fact_5987_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,Xa: A,P: fun(A,$o),P3: fun(A,$o)] :
      ( ( Xa = fChoice(A,aTP_Lamp_az(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
          <=> aa(A,$o,P3,X) )
       => ( ! [X_12: A] : aa(A,$o,P3,X_12)
        <=> aa(A,$o,P,Xa) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_5988_verit__sko__forall__indirect,axiom,
    ! [A: $tType,Xa: A,P: fun(A,$o)] :
      ( ( Xa = fChoice(A,aTP_Lamp_az(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X_12: A] : aa(A,$o,P,X_12)
      <=> aa(A,$o,P,Xa) ) ) ).

% verit_sko_forall_indirect
tff(fact_5989_verit__sko__ex__indirect2,axiom,
    ! [A: $tType,Xa: A,P: fun(A,$o),P3: fun(A,$o)] :
      ( ( Xa = fChoice(A,P) )
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
          <=> aa(A,$o,P3,X) )
       => ( ? [X_12: A] : aa(A,$o,P3,X_12)
        <=> aa(A,$o,P,Xa) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_5990_verit__sko__ex__indirect,axiom,
    ! [A: $tType,Xa: A,P: fun(A,$o)] :
      ( ( Xa = fChoice(A,P) )
     => ( ? [X_12: A] : aa(A,$o,P,X_12)
      <=> aa(A,$o,P,Xa) ) ) ).

% verit_sko_ex_indirect
tff(fact_5991_verit__sko__forall_H_H,axiom,
    ! [A: $tType,B3: A,A3: A,P: fun(A,$o)] :
      ( ( B3 = A3 )
     => ( ( fChoice(A,P) = A3 )
      <=> ( fChoice(A,P) = B3 ) ) ) ).

% verit_sko_forall''
tff(fact_5992_verit__sko__forall_H,axiom,
    ! [A: $tType,P: fun(A,$o),A3: $o] :
      ( ( aa(A,$o,P,fChoice(A,aTP_Lamp_az(fun(A,$o),fun(A,$o),P)))
      <=> (A3) )
     => ( ! [X_12: A] : aa(A,$o,P,X_12)
      <=> (A3) ) ) ).

% verit_sko_forall'
tff(fact_5993_verit__sko__forall,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X_12: A] : aa(A,$o,P,X_12)
    <=> aa(A,$o,P,fChoice(A,aTP_Lamp_az(fun(A,$o),fun(A,$o),P))) ) ).

% verit_sko_forall
tff(fact_5994_verit__sko__ex_H,axiom,
    ! [A: $tType,P: fun(A,$o),A3: $o] :
      ( ( aa(A,$o,P,fChoice(A,P))
      <=> (A3) )
     => ( ? [X_12: A] : aa(A,$o,P,X_12)
      <=> (A3) ) ) ).

% verit_sko_ex'
tff(fact_5995_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_5996_word__ops__nth__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),Xa))
         => ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)),Na)
            <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na)
                | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Na) ) )
            & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)),Na)
            <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na)
                & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Na) ) )
            & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y)),Na)
            <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na)
                <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Na) ) )
            & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)),Na)
            <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na) ) ) ) ) ).

% word_ops_nth_size
tff(fact_5997_bit__twiddle__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),
            aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y)),
              $ite(aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),zero_zero(word(A))))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),ord_max(word(A)),Xa),Y) ) ).

% bit_twiddle_max
tff(fact_5998_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_5999_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_6000_Suc__0__xor__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,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_6001_xor__Suc__0__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Na),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,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_6002_vebt__buildup_Opelims,axiom,
    ! [Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xa) = Y )
     => ( accp(nat,vEBT_v4011308405150292612up_rel,Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ accp(nat,vEBT_v4011308405150292612up_rel,zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xa = 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))),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) )
                   => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_6003_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(Xa) = Y )
     => ( accp(nat,vEBT_V5144397997797733112_d_rel,Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
             => ~ accp(nat,vEBT_V5144397997797733112_d_rel,zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
               => ~ accp(nat,vEBT_V5144397997797733112_d_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xa = 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(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) )
                   => ~ 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_6004_Eps__case__prod__eq,axiom,
    ! [A: $tType,B: $tType,Xa: A,Y: B] : fChoice(product_prod(A,B),product_case_prod(A,B,$o,aa(B,fun(A,fun(B,$o)),aTP_Lamp_pb(A,fun(B,fun(A,fun(B,$o))),Xa),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y) ).

% Eps_case_prod_eq
tff(fact_6005_swap__with__xor,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),A2: word(A),B2: word(A),Y: word(A),Z: word(A)] :
          ( ( Xa = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),A2),B2) )
         => ( ( Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),B2),Xa) )
           => ( ( Z = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y) )
             => ( ( Z = B2 )
                & ( Y = A2 ) ) ) ) ) ) ).

% swap_with_xor
tff(fact_6006_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,$o,aTP_Lamp_pg(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),P))) ).

% split_paired_Eps
tff(fact_6007_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(Xa) = Y )
     => ( accp(nat,vEBT_V1247956027447740395_p_rel,Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
             => ~ accp(nat,vEBT_V1247956027447740395_p_rel,zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ accp(nat,vEBT_V1247956027447740395_p_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xa = 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),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) )
                   => ~ 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_6008_VEBT__internal_OTb_Opelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( vEBT_VEBT_Tb(Xa) = Y )
     => ( accp(nat,vEBT_VEBT_Tb_rel2,Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) )
             => ~ accp(nat,vEBT_VEBT_Tb_rel2,zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) )
               => ~ accp(nat,vEBT_VEBT_Tb_rel2,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                 => ( ( Y = $ite(aa(nat,$o,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),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) )
                   => ~ accp(nat,vEBT_VEBT_Tb_rel2,aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
tff(fact_6009_VEBT__internal_OTb_H_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_VEBT_Tb2(Xa) = Y )
     => ( accp(nat,vEBT_VEBT_Tb_rel,Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
             => ~ accp(nat,vEBT_VEBT_Tb_rel,zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ accp(nat,vEBT_VEBT_Tb_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                 => ( ( Y = $ite(aa(nat,$o,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),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) )
                   => ~ accp(nat,vEBT_VEBT_Tb_rel,aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
tff(fact_6010_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V441764108873111860ildupi(Xa) = Y )
     => ( accp(nat,vEBT_V2957053500504383685pi_rel,Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = aa(nat,nat,suc,zero_zero(nat)) )
             => ~ accp(nat,vEBT_V2957053500504383685pi_rel,zero_zero(nat)) ) )
         => ( ( ( Xa = 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] :
                  ( ( Xa = 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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))) )
                   => ~ accp(nat,vEBT_V2957053500504383685pi_rel,aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_6011_arg__min__SOME__Min,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( lattic7623131987881927897min_on(A,B,F2,S) = fChoice(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ph(set(A),fun(fun(A,B),fun(A,$o)),S),F2)) ) ) ) ).

% arg_min_SOME_Min
tff(fact_6012_fun__of__rel__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),Xa: B] : fun_of_rel(B,A,R2,Xa) = fChoice(A,aa(B,fun(A,$o),aTP_Lamp_pi(set(product_prod(B,A)),fun(B,fun(A,$o)),R2),Xa)) ).

% fun_of_rel_def
tff(fact_6013_arg__min__if__finite_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( ( S != bot_bot(set(A)) )
           => member(A,lattic7623131987881927897min_on(A,B,F2,S),S) ) ) ) ).

% arg_min_if_finite(1)
tff(fact_6014_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),Y: A,F2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( member(A,Y,S)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S))),aa(A,B,F2,Y)) ) ) ) ) ).

% arg_min_least
tff(fact_6015_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ~ ? [X5: A] :
                  ( member(A,X5,S)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X5)),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_6016_prod__case__refines,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: product_prod(A,B),P5: product_prod(A,B),F2: fun(A,fun(B,heap_Time_Heap(C))),F4: fun(A,fun(B,heap_Time_Heap(C)))] :
      ( ( P2 = P5 )
     => ( ! [A5: A,B6: B] : refine_Imp_refines(C,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F2,A5),B6),aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F4,A5),B6))
       => refine_Imp_refines(C,aa(product_prod(A,B),heap_Time_Heap(C),product_case_prod(A,B,heap_Time_Heap(C),F2),P2),aa(product_prod(A,B),heap_Time_Heap(C),product_case_prod(A,B,heap_Time_Heap(C),F4),P5)) ) ) ).

% prod_case_refines
tff(fact_6017_refines__case__prod__right,axiom,
    ! [C: $tType,B: $tType,A: $tType,M: heap_Time_Heap(C),M6: fun(A,fun(B,heap_Time_Heap(C))),T: product_prod(A,B)] :
      ( ! [A5: A,B6: B] : refine_Imp_refines(C,M,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),M6,A5),B6))
     => refine_Imp_refines(C,M,aa(product_prod(A,B),heap_Time_Heap(C),product_case_prod(A,B,heap_Time_Heap(C),M6),T)) ) ).

% refines_case_prod_right
tff(fact_6018_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_6019_monoseq__minus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: fun(nat,A)] :
          ( topological_monoseq(A,A2)
         => topological_monoseq(A,aTP_Lamp_pj(fun(nat,A),fun(nat,A),A2)) ) ) ).

% monoseq_minus
tff(fact_6020_Ints__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => member(B,aa(A,B,F2,X),ring_1_Ints(B)) )
         => member(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3),ring_1_Ints(B)) ) ) ).

% Ints_sum
tff(fact_6021_Ints__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(B)
        & ring_1(B) )
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => member(B,aa(A,B,F2,X),ring_1_Ints(B)) )
         => member(B,groups7121269368397514597t_prod(A,B,F2,A3),ring_1_Ints(B)) ) ) ).

% Ints_prod
tff(fact_6022_frac__in__Ints__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( member(A,archimedean_frac(A,Xa),ring_1_Ints(A))
        <=> member(A,Xa,ring_1_Ints(A)) ) ) ).

% frac_in_Ints_iff
tff(fact_6023_frac__eq__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( ( archimedean_frac(A,Xa) = zero_zero(A) )
        <=> member(A,Xa,ring_1_Ints(A)) ) ) ).

% frac_eq_0_iff
tff(fact_6024_floor__add2,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( ( member(A,Xa,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),Xa),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Y)) ) ) ) ).

% floor_add2
tff(fact_6025_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),archimedean_frac(A,Xa))
        <=> ~ member(A,Xa,ring_1_Ints(A)) ) ) ).

% frac_gt_0_iff
tff(fact_6026_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_6027_Ints__induct,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q2: A,P: fun(A,$o)] :
          ( member(A,Q2,ring_1_Ints(A))
         => ( ! [Z3: int] : aa(A,$o,P,aa(int,A,ring_1_of_int(A),Z3))
           => aa(A,$o,P,Q2) ) ) ) ).

% Ints_induct
tff(fact_6028_Ints__cases,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q2: A] :
          ( member(A,Q2,ring_1_Ints(A))
         => ~ ! [Z3: int] : Q2 != aa(int,A,ring_1_of_int(A),Z3) ) ) ).

% Ints_cases
tff(fact_6029_Ints__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( member(A,B2,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),ring_1_Ints(A)) ) ) ) ).

% Ints_diff
tff(fact_6030_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_6031_Ints__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Na: nat] :
          ( member(A,A2,ring_1_Ints(A))
         => member(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na),ring_1_Ints(A)) ) ) ).

% Ints_power
tff(fact_6032_Ints__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( member(A,B2,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),ring_1_Ints(A)) ) ) ) ).

% Ints_add
tff(fact_6033_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_6034_Ints__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( member(A,B2,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),ring_1_Ints(A)) ) ) ) ).

% Ints_mult
tff(fact_6035_minus__in__Ints__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: A] :
          ( member(A,aa(A,A,uminus_uminus(A),Xa),ring_1_Ints(A))
        <=> member(A,Xa,ring_1_Ints(A)) ) ) ).

% minus_in_Ints_iff
tff(fact_6036_Ints__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => member(A,aa(A,A,uminus_uminus(A),A2),ring_1_Ints(A)) ) ) ).

% Ints_minus
tff(fact_6037_Ints__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => member(A,aa(A,A,abs_abs(A),A2),ring_1_Ints(A)) ) ) ).

% Ints_abs
tff(fact_6038_Ints__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,zero_zero(A),ring_1_Ints(A)) ) ).

% Ints_0
tff(fact_6039_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,one_one(A),ring_1_Ints(A)) ) ).

% Ints_1
tff(fact_6040_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_6041_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : finite_finite2(A,collect(A,aa(A,fun(A,$o),aTP_Lamp_pk(A,fun(A,fun(A,$o)),A2),B2))) ) ).

% finite_int_segment
tff(fact_6042_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),A2) != zero_zero(A) ) ) ) ).

% Ints_odd_nonzero
tff(fact_6043_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: int,A2: int] :
          ( aa(int,$o,dvd_dvd(int,B2),A2)
         => member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A2)),aa(int,A,ring_1_of_int(A),B2)),ring_1_Ints(A)) ) ) ).

% of_int_divide_in_Ints
tff(fact_6044_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A] : finite_finite2(A,collect(A,aTP_Lamp_pl(A,fun(A,$o),A2))) ) ).

% finite_abs_int_segment
tff(fact_6045_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),A2)),zero_zero(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% Ints_odd_less_0
tff(fact_6046_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( member(A,Xa,ring_1_Ints(A))
         => ( ( Xa != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),Xa)) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_6047_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( member(A,Xa,ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Xa)),one_one(A))
           => ( Xa = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_6048_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( member(A,Xa,ring_1_Ints(A))
         => ( member(A,Y,ring_1_Ints(A))
           => ( ( Xa = Y )
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y))),one_one(A)) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_6049_sin__times__pi__eq__0,axiom,
    ! [Xa: real] :
      ( ( aa(real,real,sin(real),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),pi)) = zero_zero(real) )
    <=> member(real,Xa,ring_1_Ints(real)) ) ).

% sin_times_pi_eq_0
tff(fact_6050_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          archimedean_frac(A,aa(A,A,uminus_uminus(A),Xa)) = $ite(member(A,Xa,ring_1_Ints(A)),zero_zero(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),archimedean_frac(A,Xa))) ) ).

% frac_neg
tff(fact_6051_le__mult__floor__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( member(A,A2,ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A2)),archim6421214686448440834_floor(A,B2)))),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_6052_mult__ceiling__le__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( member(A,A2,ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A2)),archimedean_ceiling(A,B2)))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_6053_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A2: A] :
          ( ( archimedean_frac(A,Xa) = A2 )
        <=> ( member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),A2),ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) ) ) ) ).

% frac_unique_iff
tff(fact_6054_sin__integer__2pi,axiom,
    ! [Na: real] :
      ( member(real,Na,ring_1_Ints(real))
     => ( aa(real,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_6055_cos__integer__2pi,axiom,
    ! [Na: real] :
      ( member(real,Na,ring_1_Ints(real))
     => ( aa(real,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_6056_ran__nth__set__encoding__conv,axiom,
    ! [A: $tType,L: list(A)] : ran(nat,A,aTP_Lamp_pm(list(A),fun(nat,option(A)),L)) = set2(A,L) ).

% ran_nth_set_encoding_conv
tff(fact_6057_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_pn(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_6058_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_6059_map__update__eta__repair_I2_J,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),K: B,V: A] :
      ( ( aa(B,option(A),M,K) = none(A) )
     => ( ran(B,A,aa(A,fun(B,option(A)),aa(B,fun(A,fun(B,option(A))),aTP_Lamp_po(fun(B,option(A)),fun(B,fun(A,fun(B,option(A)))),M),K),V)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V),ran(B,A,M)) ) ) ).

% map_update_eta_repair(2)
tff(fact_6060_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).

% quotient_of_number(5)
tff(fact_6061_divide__rat__def,axiom,
    ! [Q2: rat,R: rat] : aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),Q2),R) = aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),Q2),aa(rat,rat,inverse_inverse(rat),R)) ).

% divide_rat_def
tff(fact_6062_quotient__of__div,axiom,
    ! [R: rat,Na: int,D2: int] :
      ( ( quotient_of(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Na),D2) )
     => ( R = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(int,rat,ring_1_of_int(rat),Na)),aa(int,rat,ring_1_of_int(rat),D2)) ) ) ).

% quotient_of_div
tff(fact_6063_quotient__of__denom__pos,axiom,
    ! [R: rat,P2: int,Q2: int] :
      ( ( quotient_of(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q2) ) ).

% quotient_of_denom_pos
tff(fact_6064_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_pp(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_6065_rat__less__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),P2),Q2)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aTP_Lamp_pr(rat,fun(int,fun(int,$o)),Q2)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_6066_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_6067_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_ps(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_abs_code
tff(fact_6068_rat__less__eq__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),P2),Q2)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aTP_Lamp_pu(rat,fun(int,fun(int,$o)),Q2)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_6069_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_pv(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_6070_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_px(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_6071_normalize__negative,axiom,
    ! [Q2: int,P2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Q2),zero_zero(int))
     => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),P2)),aa(int,int,uminus_uminus(int),Q2))) ) ) ).

% normalize_negative
tff(fact_6072_normalize__denom__pos,axiom,
    ! [R: product_prod(int,int),P2: int,Q2: int] :
      ( ( normalize(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q2) ) ).

% normalize_denom_pos
tff(fact_6073_normalize__crossproduct,axiom,
    ! [Q2: int,Sa: int,P2: int,R: int] :
      ( ( Q2 != zero_zero(int) )
     => ( ( Sa != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R),Sa)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P2),Sa) = aa(int,int,aa(int,fun(int,int),times_times(int),R),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_6074_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_pz(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_6075_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_qb(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_times_code
tff(fact_6076_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_qd(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_6077_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_6078_listI__assn__extract,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A3: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( member(nat,I,I5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),I5),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),A3,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_6079_listI__assn__finite,axiom,
    ! [B: $tType,A: $tType,I5: set(nat),A3: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B)] :
      ( ~ finite_finite2(nat,I5)
     => ( vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) = bot_bot(assn) ) ) ).

% listI_assn_finite
tff(fact_6080_listI__assn__weak__cong,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),I6: set(nat),A3: fun(A,fun(B,assn)),A11: fun(A,fun(B,assn)),Xs: list(A),Xs4: list(A),Xsi: list(B),Xsi2: list(B)] :
      ( ( I5 = I6 )
     => ( ( A3 = A11 )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Xs4) )
         => ( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(B),nat,size_size(list(B)),Xsi2) )
           => ( ! [I3: nat] :
                  ( member(nat,I3,I5)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
                   => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Xsi) )
                     => ( ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Xs4),I3) )
                        & ( aa(nat,B,nth(B,Xsi),I3) = aa(nat,B,nth(B,Xsi2),I3) ) ) ) ) )
             => ( vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) = vEBT_List_listI_assn(A,B,I6,A11,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_6081_listI__assn__cong,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),I6: set(nat),Xs: list(A),Xs4: list(A),Xsi: list(B),Xsi2: list(B),A3: fun(A,fun(B,assn)),A11: fun(A,fun(B,assn))] :
      ( ( I5 = I6 )
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Xs4) )
       => ( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(B),nat,size_size(list(B)),Xsi2) )
         => ( ! [I3: nat] :
                ( member(nat,I3,I5)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
                 => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Xsi) )
                   => ( ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Xs4),I3) )
                      & ( aa(nat,B,nth(B,Xsi),I3) = aa(nat,B,nth(B,Xsi2),I3) )
                      & ( aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I3)),aa(nat,B,nth(B,Xsi),I3)) = aa(B,assn,aa(A,fun(B,assn),A11,aa(nat,A,nth(A,Xs4),I3)),aa(nat,B,nth(B,Xsi2),I3)) ) ) ) ) )
           => ( vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) = vEBT_List_listI_assn(A,B,I6,A11,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_6082_subst__not__in,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A3: fun(A,fun(B,assn)),X12: A,Xsi: list(B),X22: B] :
      ( ~ member(nat,I,I5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,I5,A3,list_update(A,Xs,I,X12),list_update(B,Xsi,I,X22)) = vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_6083_listI__assn__insert,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A3: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( ~ member(nat,I,I5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),I5),A3,Xs,Xsi) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_6084_listI__assn__subst,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A3: fun(A,fun(B,assn)),X12: A,Xsi: list(B),X22: B] :
      ( ~ member(nat,I,I5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),I5),A3,list_update(A,Xs,I,X12),list_update(B,Xsi,I,X22)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,X12),X22)),vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_6085_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_6086_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_6087_listI__assn__wrap__insert,axiom,
    ! [A: $tType,P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set(nat),I: nat,Xs: list(vEBT_VEBT),Xsi: list(vEBT_VEBTi),F3: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(Uu,Uua)),Xi)),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),I5),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,Xs,Xsi))),F3))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Xs))
       => ( member(nat,I,I5)
         => ( hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,I5,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,Xs,I,vEBT_vebt_insert(Uu,Uua)),list_update(vEBT_VEBTi,Xsi,I,Xi))),F3),C2,Q)
           => hoare_hoare_triple(A,P,C2,Q) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_6088_listI__assn__reinsert__upd_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,A3: fun(A,fun(B,assn)),Xa: A,Xi: B,I5: set(nat),I: nat,Xs: list(A),Xsi: list(B),F3: assn,C2: heap_Time_Heap(C),Q: fun(C,assn)] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,Xa),Xi)),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),I5),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),A3,Xs,Xsi))),F3))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( member(nat,I,I5)
         => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A3,list_update(A,Xs,I,Xa),list_update(B,Xsi,I,Xi))),F3),C2,Q)
           => hoare_hoare_triple(C,P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_6089_swappa,axiom,
    ! [B3: assn,A3: assn,C3: assn,X7: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),A3)),C3),X7)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B3)),C3),X7) ) ).

% swappa
tff(fact_6090_midextr,axiom,
    ! [P: assn,Q: assn,Q6: assn,R2: assn,X7: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q)),Q6)),R2),X7)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),R2)),Q)),Q6),X7) ) ).

% midextr
tff(fact_6091_groupy,axiom,
    ! [A3: assn,B3: assn,C3: assn,D4: assn,X7: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B3)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C3),D4)),X7)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B3)),C3)),D4),X7) ) ).

% groupy
tff(fact_6092_ent__false,axiom,
    ! [P: assn] : entails(bot_bot(assn),P) ).

% ent_false
tff(fact_6093_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_6094_htt__cons__rule,axiom,
    ! [A: $tType,P3: assn,C2: heap_Time_Heap(A),Q6: fun(A,assn),T7: nat,P: assn,Q: fun(A,assn),T: nat] :
      ( time_htt(A,P3,C2,Q6,T7)
     => ( entails(P,P3)
       => ( ! [X: A] : entails(aa(A,assn,Q6,X),aa(A,assn,Q,X))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),T7),T)
           => time_htt(A,P,C2,Q,T) ) ) ) ) ).

% htt_cons_rule
tff(fact_6095_listI__assn__reinsert,axiom,
    ! [B: $tType,A: $tType,P: assn,A3: fun(A,fun(B,assn)),Xs: list(A),I: nat,Xsi: list(B),I5: set(nat),F3: assn,Q: assn] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),I5),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),A3,Xs,Xsi))),F3))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( member(nat,I,I5)
         => ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi)),F3),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_6096_listI__assn__reinsert__upd,axiom,
    ! [B: $tType,A: $tType,P: assn,A3: fun(A,fun(B,assn)),Xa: A,Xi: B,I5: set(nat),I: nat,Xs: list(A),Xsi: list(B),F3: assn,Q: assn] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,Xa),Xi)),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),I5),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),A3,Xs,Xsi))),F3))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( member(nat,I,I5)
         => ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A3,list_update(A,Xs,I,Xa),list_update(B,Xsi,I,Xi))),F3),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_6097_listI__assn__reinsert_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,A3: fun(A,fun(B,assn)),Xs: list(A),I: nat,Xsi: list(B),I5: set(nat),F3: assn,C2: heap_Time_Heap(C),Q: fun(C,assn)] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),I5),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),A3,Xs,Xsi))),F3))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( member(nat,I,I5)
         => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi)),F3),C2,Q)
           => hoare_hoare_triple(C,P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_6098_assnle,axiom,
    ! [TreeList: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),X13: array(vEBT_VEBTi),Summary2: vEBT_VEBT,X14: vEBT_VEBTi] : entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is))) ).

% assnle
tff(fact_6099_big__assn__simp,axiom,
    ! [H: nat,TreeList: list(vEBT_VEBT),L: nat,Xa: vEBT_VEBTi,Xaa: option(nat),X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),H),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L)),Xa)),pure_assn(Xaa = vEBT_vebt_mint(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H,Xa))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),H),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H,Xa))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),pure_assn(Xaa = vEBT_vebt_mint(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))),list_update(vEBT_VEBTi,Tree_is,H,Xa)))) ) ).

% big_assn_simp
tff(fact_6100_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_6101_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_6102_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_6103_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A2,B2) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_6104_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ finite_finite2(A,set_or7035219750837199246ssThan(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ico_iff
tff(fact_6105_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_6106_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_mb(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_6107_snga__same__false,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P2: array(A),Xa: list(A),Y: list(A)] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(A),assn,snga_assn(A,P2),Xa)),aa(list(A),assn,snga_assn(A,P2),Y)) = bot_bot(assn) ) ).

% snga_same_false
tff(fact_6108_local_Oext,axiom,
    ! [Y: nat,TreeList: list(vEBT_VEBT),X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),Y)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),Y))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Y),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is)))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Y),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),Y)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),Y)))) ) ).

% local.ext
tff(fact_6109_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,M)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_6110_txe,axiom,
    ! [Y: nat,TreeList: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),X13: array(vEBT_VEBTi),Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),Y)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),Y))),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Y),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is))) ) ).

% txe
tff(fact_6111_repack,axiom,
    ! [I: nat,TreeList: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),Rest: assn,X13: array(vEBT_VEBTi),Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),I))),Rest)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Rest),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is))) ) ).

% repack
tff(fact_6112_recomp,axiom,
    ! [I: nat,TreeList: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),X13: array(vEBT_VEBTi),Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),I))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is))) ) ).

% recomp
tff(fact_6113_tcd,axiom,
    ! [A: $tType,I: nat,TreeList: list(vEBT_VEBT),TreeList4: list(A),Y: vEBT_VEBT,Xa: vEBT_VEBTi,X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(list(A),nat,size_size(list(A)),TreeList4) )
       => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Y),Xa)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,I,Xa)))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,I,Y),list_update(vEBT_VEBTi,Tree_is,I,Xa))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,I,Xa))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,I,Y)),list_update(vEBT_VEBTi,Tree_is,I,Xa)))) ) ) ).

% tcd
tff(fact_6114_big__assn__simp_H,axiom,
    ! [H: nat,TreeList: list(vEBT_VEBT),Xaa: vEBT_VEBT,L: nat,Xa: vEBT_VEBTi,Xb: option(nat),X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),H),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( ( Xaa = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
       => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Xaa),Xa)),pure_assn(Xb = vEBT_vebt_mint(Xaa)))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H,Xa))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),H),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H,Xa))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),pure_assn(Xb = vEBT_vebt_mint(Xaa)))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,H,Xaa)),list_update(vEBT_VEBTi,Tree_is,H,Xa)))) ) ) ).

% big_assn_simp'
tff(fact_6115_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_6116_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_6117_ex__nat__less__eq,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Na)
          & aa(nat,$o,P,M3) )
    <=> ? [X4: nat] :
          ( member(nat,X4,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
          & aa(nat,$o,P,X4) ) ) ).

% ex_nat_less_eq
tff(fact_6118_all__nat__less__eq,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Na)
         => aa(nat,$o,P,M3) )
    <=> ! [X4: nat] :
          ( member(nat,X4,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
         => aa(nat,$o,P,X4) ) ) ).

% all_nat_less_eq
tff(fact_6119_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ finite_finite2(A,set_or7035219750837199246ssThan(A,A2,B2)) ) ) ).

% infinite_Ico
tff(fact_6120_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
            <=> ( ( A2 = C2 )
                & ( B2 = D2 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_6121_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
             => ( A2 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_6122_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
             => ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_6123_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_6124_ivl__disj__int__two_I3_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(3)
tff(fact_6125_atLeastLessThan0,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_6126_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_es(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_6127_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_et(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_6128_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_hr(fun(nat,A),fun(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_6129_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_ht(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_6130_sum_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_add(B) )
     => ! [A2: A,C2: A,B2: A,D2: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),D2)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),set_or7035219750837199246ssThan(A,A2,B2)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),set_or7035219750837199246ssThan(A,C2,D2)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_6131_prod_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_mult(B) )
     => ! [A2: A,C2: A,B2: A,D2: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),D2)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) ) )
             => ( groups7121269368397514597t_prod(A,B,G,set_or7035219750837199246ssThan(A,A2,B2)) = groups7121269368397514597t_prod(A,B,H,set_or7035219750837199246ssThan(A,C2,D2)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_6132_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_6133_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_6134_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_6135_ivl__disj__int__two_I7_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(7)
tff(fact_6136_ivl__disj__int__one_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,L)),set_or7035219750837199246ssThan(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(2)
tff(fact_6137_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_6138_atLeast0__lessThan__Suc,axiom,
    ! [Na: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Na),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ).

% atLeast0_lessThan_Suc
tff(fact_6139_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_6140_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D2) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_6141_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_6142_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_6143_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_6144_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_6145_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_6146_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or7035219750837199246ssThan(A,A2,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_6147_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_6148_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_6149_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_6150_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_6151_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_6152_atLeastLessThanSuc,axiom,
    ! [M: nat,Na: nat] :
      set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Na),set_or7035219750837199246ssThan(nat,M,Na)),bot_bot(set(nat))) ).

% atLeastLessThanSuc
tff(fact_6153_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),$o),ord_less_eq(word(A)),W),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))
        <=> ! [X4: nat] :
              ( member(nat,X4,set_or7035219750837199246ssThan(nat,Na,aa(word(A),nat,size_size(word(A)),W)))
             => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),X4) ) ) ) ).

% le_mask_high_bits
tff(fact_6154_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_6155_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_fa(fun(nat,A),fun(nat,A),F2)),set_or7035219750837199246ssThan(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,Na)),aa(nat,A,F2,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_6156_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_qe(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M)),set_or7035219750837199246ssThan(nat,Na,M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_6157_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_qf(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jd(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Na)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% sum.nested_swap
tff(fact_6158_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_qg(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M),set_or7035219750837199246ssThan(nat,Na,M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_6159_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Na: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_qh(fun(nat,fun(nat,A)),fun(nat,A),A2),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jg(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Na),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% prod.nested_swap
tff(fact_6160_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_qi(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_6161_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_qj(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_6162_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_6163_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_6164_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_6165_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),aa(nat,fun(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_6166_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_6167_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_ey(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_6168_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_hx(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_6169_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] : comm_s3205402744901411588hammer(A,A2,Na) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_ib(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% pochhammer_prod
tff(fact_6170_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ) ).

% fact_prod_rev
tff(fact_6171_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,$o),ord_less(real),zero_zero(real)),E3)
             => ? [N10: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N10),M3)
                 => ! [N2: nat] : aa(real,$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,M3,N2)))),E3) ) ) ) ) ).

% summable_Cauchy
tff(fact_6172_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),Sa: A,K: nat] :
          ( aa(A,$o,sums(A,F2),Sa)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
           => aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_qk(fun(nat,A),fun(nat,fun(nat,A)),F2),K)),Sa) ) ) ) ).

% sums_group
tff(fact_6173_atLeast1__lessThan__eq__remove0,axiom,
    ! [Na: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),Na) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_lessThan(nat,Na)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_6174_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Xa: nat,Y: nat] :
      aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_ql(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,Xa,Y)) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),C2),Y),
        set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y),C2)),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_6175_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),K),Na)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K))) ) ) ) ).

% fact_split
tff(fact_6176_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_qm(nat,fun(nat,fun(nat,A)),K),Na),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_6177_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_qn(A,fun(nat,fun(nat,A)),A2),K),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_6178_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),semiring_char_0_fact(A,K)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_qo(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_6179_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,gbinomial(A,A2),K)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_qo(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_6180_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_lj(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_6181_sum__power2,axiom,
    ! [K: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),one_one(nat)) ).

% sum_power2
tff(fact_6182_Sum__Ico__nat,axiom,
    ! [M: nat,Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fj(nat,nat)),set_or7035219750837199246ssThan(nat,M,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Ico_nat
tff(fact_6183_rule__at__index,axiom,
    ! [B: $tType,A: $tType,C: $tType,P: assn,A3: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B),F3: assn,I: nat,C2: heap_Time_Heap(C),Q6: fun(C,assn),F7: fun(C,assn)] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,A3,Xs),Xsi)),F3))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),A3,Xs,Xsi))),F3),C2,Q6)
         => ( ! [R4: C] : entails(aa(C,assn,Q6,R4),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),A3,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),A3,Xs,Xsi))),aa(C,assn,F7,R4)))
           => 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_qp(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn)))),A3),Xs),Xsi),F7)) ) ) ) ) ).

% rule_at_index
tff(fact_6184_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Na)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,I3)),aa(nat,A,A2,J3)) ) )
         => ( ! [I3: nat,J3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Na)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,B2,J3)),aa(nat,A,B2,I3)) ) )
           => aa(A,$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_qq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_6185_Chebyshev__sum__upper__nat,axiom,
    ! [Na: nat,A2: fun(nat,nat),B2: fun(nat,nat)] :
      ( ! [I3: nat,J3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Na)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,A2,I3)),aa(nat,nat,A2,J3)) ) )
     => ( ! [I3: nat,J3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Na)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,B2,J3)),aa(nat,nat,B2,I3)) ) )
       => 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_qr(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_6186_word__to__1__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( set_or7035219750837199246ssThan(word(A),zero_zero(word(A)),one_one(word(A))) = aa(set(word(A)),set(word(A)),aa(word(A),fun(set(word(A)),set(word(A))),insert(word(A)),zero_zero(word(A))),bot_bot(set(word(A)))) ) ) ).

% word_to_1_set
tff(fact_6187_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_6188_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image(int,int,aTP_Lamp_qs(int,fun(int,int),L)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L))) = set_or7035219750837199246ssThan(int,L,U) ).

% image_add_int_atLeastLessThan
tff(fact_6189_image__int__atLeastLessThan,axiom,
    ! [A2: nat,B2: nat] : aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_or7035219750837199246ssThan(nat,A2,B2)) = set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastLessThan
tff(fact_6190_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_qt(code_integer,fun(code_integer,code_integer),L)),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),U),L))) = set_or7035219750837199246ssThan(code_integer,L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_6191_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_6192_word__range__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( ( B2 != zero_zero(word(A)) )
         => ( set_or1337092689740270186AtMost(word(A),A2,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),B2),one_one(word(A)))) = set_or7035219750837199246ssThan(word(A),A2,B2) ) ) ) ).

% word_range_minus_1
tff(fact_6193_word__count__from__top,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] :
          ( ( Na != zero_zero(word(A)) )
         => ( set_or7035219750837199246ssThan(word(A),zero_zero(word(A)),Na) = aa(set(word(A)),set(word(A)),aa(set(word(A)),fun(set(word(A)),set(word(A))),sup_sup(set(word(A))),set_or7035219750837199246ssThan(word(A),zero_zero(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Na),one_one(word(A))))),aa(set(word(A)),set(word(A)),aa(word(A),fun(set(word(A)),set(word(A))),insert(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Na),one_one(word(A)))),bot_bot(set(word(A))))) ) ) ) ).

% word_count_from_top
tff(fact_6194_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_6195_nth__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xs: list(A),A2: array(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
         => hoare_hoare_triple(A,aa(list(A),assn,snga_assn(A,A2),Xs),array_nth(A,A2,I),aa(array(A),fun(A,assn),aa(list(A),fun(array(A),fun(A,assn)),aTP_Lamp_qu(nat,fun(list(A),fun(array(A),fun(A,assn))),I),Xs),A2)) ) ) ).

% nth_rule
tff(fact_6196_fi__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Ps: assn,F3: assn] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( entails(Ps,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),F3))
       => hoare_hoare_triple(A,Ps,C2,aa(assn,fun(A,assn),aTP_Lamp_cw(fun(A,assn),fun(assn,fun(A,assn)),Q),F3)) ) ) ).

% fi_rule
tff(fact_6197_norm__assertion__simps_I6_J,axiom,
    ! [Xa: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Xa),bot_bot(assn)) = Xa ).

% norm_assertion_simps(6)
tff(fact_6198_norm__assertion__simps_I5_J,axiom,
    ! [Xa: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),bot_bot(assn)),Xa) = Xa ).

% norm_assertion_simps(5)
tff(fact_6199_norm__assertion__simps_I9_J,axiom,
    ! [Xa: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),bot_bot(assn)),Xa) = bot_bot(assn) ).

% norm_assertion_simps(9)
tff(fact_6200_norm__assertion__simps_I10_J,axiom,
    ! [Xa: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Xa),bot_bot(assn)) = bot_bot(assn) ).

% norm_assertion_simps(10)
tff(fact_6201_frame__rule__left,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),R2: assn] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R2),P),C2,aa(assn,fun(A,assn),aTP_Lamp_qv(fun(A,assn),fun(assn,fun(A,assn)),Q),R2)) ) ).

% frame_rule_left
tff(fact_6202_return__cons__rule,axiom,
    ! [A: $tType,P: assn,Q: fun(A,assn),Xa: A] :
      ( entails(P,aa(A,assn,Q,Xa))
     => hoare_hoare_triple(A,P,heap_Time_return(A,Xa),Q) ) ).

% return_cons_rule
tff(fact_6203_freeze__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A2: array(A),Xs: list(A)] : hoare_hoare_triple(list(A),aa(list(A),assn,snga_assn(A,A2),Xs),array_freeze(A,A2),aa(list(A),fun(list(A),assn),aTP_Lamp_qw(array(A),fun(list(A),fun(list(A),assn)),A2),Xs)) ) ).

% freeze_rule
tff(fact_6204_length__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A2: array(A),Xs: list(A)] : hoare_hoare_triple(nat,aa(list(A),assn,snga_assn(A,A2),Xs),array_len(A,A2),aa(list(A),fun(nat,assn),aTP_Lamp_qx(array(A),fun(list(A),fun(nat,assn)),A2),Xs)) ) ).

% length_rule
tff(fact_6205_upd__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xs: list(A),A2: array(A),Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
         => hoare_hoare_triple(array(A),aa(list(A),assn,snga_assn(A,A2),Xs),array_upd(A,I,Xa,A2),aa(A,fun(array(A),assn),aa(array(A),fun(A,fun(array(A),assn)),aa(list(A),fun(array(A),fun(A,fun(array(A),assn))),aTP_Lamp_qy(nat,fun(list(A),fun(array(A),fun(A,fun(array(A),assn)))),I),Xs),A2),Xa)) ) ) ).

% upd_rule
tff(fact_6206_listI__assn__def,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),A3: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B)] :
      vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) = aa(assn,assn,
        aa(assn,fun(assn,assn),times_times(assn),
          pure_assn(( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(A),nat,size_size(list(A)),Xs) )
            & aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),I5),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) ))),
        finite_fold(nat,assn,aa(list(B),fun(nat,fun(assn,assn)),aa(list(A),fun(list(B),fun(nat,fun(assn,assn))),aTP_Lamp_qz(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),A3),Xs),Xsi),one_one(assn),I5)) ).

% listI_assn_def
tff(fact_6207_fold__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),Z: A] : finite_fold(B,A,F2,Z,bot_bot(set(B))) = Z ).

% fold_empty
tff(fact_6208_Max_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) = finite_fold(A,A,ord_max(A),Xa,A3) ) ) ) ).

% Max.eq_fold
tff(fact_6209_new__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Na: nat,Xa: A] : hoare_hoare_triple(array(A),one_one(assn),array_new(A,Na,Xa),aa(A,fun(array(A),assn),aTP_Lamp_ra(nat,fun(A,fun(array(A),assn)),Na),Xa)) ) ).

% new_rule
tff(fact_6210_aux,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn)),A2: A,As: list(A),C2: B,Cs: list(B)] : finite_fold(nat,assn,aa(list(B),fun(nat,fun(assn,assn)),aa(B,fun(list(B),fun(nat,fun(assn,assn))),aa(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))),aa(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn))))),aTP_Lamp_rb(fun(A,fun(B,assn)),fun(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))))),P),A2),As),C2),Cs),one_one(assn),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),As)))) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),P,A2),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_qz(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),P),As),Cs),one_one(assn),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),As)))) ).

% aux
tff(fact_6211_list_Osimps_I15_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : set2(A,aa(list(A),list(A),cons(A,X21),X222)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X21),set2(A,X222)) ).

% list.simps(15)
tff(fact_6212_nth__Cons__numeral,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),V: num] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat))) ).

% nth_Cons_numeral
tff(fact_6213_nth__Cons__pos,axiom,
    ! [A: $tType,Na: nat,Xa: A,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),Na) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_6214_union__fold__insert,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( finite_finite2(A,A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = finite_fold(A,set(A),insert(A),B3,A3) ) ) ).

% union_fold_insert
tff(fact_6215_TBOUND__new,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Na: nat,Xa: A] : time_TBOUND(array(A),array_new(A,Na,Xa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))) ) ).

% TBOUND_new
tff(fact_6216_time__array__new,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Na: nat,Xa: A,H: heap_ext(product_unit)] : time_time(array(A),array_new(A,Na,Xa),H) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)) ) ).

% time_array_new
tff(fact_6217_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),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_6218_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( finite_finite2(A,A3)
     => ( aa(set(A),set(B),image(A,B,F2),A3) = finite_fold(A,set(B),aTP_Lamp_rc(fun(A,B),fun(A,fun(set(B),set(B))),F2),bot_bot(set(B)),A3) ) ) ).

% image_fold_insert
tff(fact_6219_nth__non__equal__first__eq,axiom,
    ! [A: $tType,Xa: A,Y: A,Xs: list(A),Na: nat] :
      ( ( Xa != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),Na) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) = Y )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_6220_Cons__replicate__eq,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Na: nat,Y: A] :
      ( ( aa(list(A),list(A),cons(A,Xa),Xs) = replicate(A,Na,Y) )
    <=> ( ( Xa = Y )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
        & ( Xs = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)),Xa) ) ) ) ).

% Cons_replicate_eq
tff(fact_6221_slice__Cons,axiom,
    ! [A: $tType,Begin: nat,End: nat,Xa: A,Xs: list(A)] :
      slice(A,Begin,End,aa(list(A),list(A),cons(A,Xa),Xs)) = $ite(
        ( ( Begin = zero_zero(nat) )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),End) ),
        aa(list(A),list(A),cons(A,Xa),slice(A,Begin,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),End),one_one(nat)),Xs)),
        slice(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Begin),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),End),one_one(nat)),Xs) ) ).

% slice_Cons
tff(fact_6222_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : lattic643756798349783984er_Max(A,A3) = the2(A,finite_fold(A,option(A),aTP_Lamp_rd(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Max.eq_fold'
tff(fact_6223_hash__code__list__simps_I2_J,axiom,
    ! [A: $tType,H_a: fun(A,uint32),Xa: A,Xaa: list(A)] : hash_hash_code_list(A,H_a,aa(list(A),list(A),cons(A,Xa),Xaa)) = aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(A,uint32,H_a,Xa)),aa(num,uint32,numeral_numeral(uint32),bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,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,Xaa)),aa(num,uint32,numeral_numeral(uint32),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(bit0(one2))))))))))))))))))))))))))))))))),aa(num,uint32,numeral_numeral(uint32),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))))) ).

% hash_code_list_simps(2)
tff(fact_6224_VEBT__internal_Ospace_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,Xa) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_space_rel2,Xa)
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ accp(vEBT_VEBT,vEBT_VEBT_space_rel2,vEBT_Leaf((A5),(B6))) ) )
         => ~ ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info,Deg2,TreeList2,Summary) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space,TreeList2),zero_zero(nat))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_space_rel2,vEBT_Node(Info,Deg2,TreeList2,Summary)) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_6225_TBOUND__option__case,axiom,
    ! [B: $tType,A: $tType,T: option(A),F2: heap_Time_Heap(B),Bnd: nat,F4: fun(A,heap_Time_Heap(B)),Bnd2: fun(A,nat)] :
      ( ( ( T = none(A) )
       => time_TBOUND(B,F2,Bnd) )
     => ( ! [X: A] :
            ( ( T = some(A,X) )
           => time_TBOUND(B,aa(A,heap_Time_Heap(B),F4,X),aa(A,nat,Bnd2,X)) )
       => time_TBOUND(B,case_option(heap_Time_Heap(B),A,F2,F4,T),case_option(nat,A,Bnd,Bnd2,T)) ) ) ).

% TBOUND_option_case
tff(fact_6226_less__eq__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A),Y: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),Xa),Y)
        <=> case_option($o,A,$true,aTP_Lamp_re(option(A),fun(A,$o),Y),Xa) ) ) ).

% less_eq_option_def
tff(fact_6227_refines__option,axiom,
    ! [B: $tType,A: $tType,A2: option(A),A4: option(A),M1: heap_Time_Heap(B),M12: heap_Time_Heap(B),M22: fun(A,heap_Time_Heap(B)),M23: fun(A,heap_Time_Heap(B))] :
      ( ( A2 = A4 )
     => ( refine_Imp_refines(B,M1,M12)
       => ( ! [X: A] : refine_Imp_refines(B,aa(A,heap_Time_Heap(B),M22,X),aa(A,heap_Time_Heap(B),M23,X))
         => refine_Imp_refines(B,case_option(heap_Time_Heap(B),A,M1,M22,A2),case_option(heap_Time_Heap(B),A,M12,M23,A4)) ) ) ) ).

% refines_option
tff(fact_6228_less__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A),Y: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),Xa),Y)
        <=> case_option($o,A,$false,aTP_Lamp_rg(option(A),fun(A,$o),Xa),Y) ) ) ).

% less_option_def
tff(fact_6229_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Xa) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_space_rel,Xa)
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
               => ~ accp(vEBT_VEBT,vEBT_VEBT_space_rel,vEBT_Leaf((A5),(B6))) ) )
         => ~ ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info,Deg2,TreeList2,Summary) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Summary))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2,TreeList2),zero_zero(nat))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_space_rel,vEBT_Node(Info,Deg2,TreeList2,Summary)) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_6230_VEBT__internal_Oheight_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,Xa) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_height_rel,Xa)
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( Y = zero_zero(nat) )
               => ~ accp(vEBT_VEBT,vEBT_VEBT_height_rel,vEBT_Leaf((A5),(B6))) ) )
         => ~ ! [Uu2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu2,Deg2,TreeList2,Summary) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),aa(vEBT_VEBT,fun(set(vEBT_VEBT),set(vEBT_VEBT)),insert(vEBT_VEBT),Summary),set2(vEBT_VEBT,TreeList2))))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_height_rel,vEBT_Node(Uu2,Deg2,TreeList2,Summary)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_6231_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Xa) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_cnt_rel,Xa)
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( Y = one_one(nat) )
               => ~ accp(vEBT_VEBT,vEBT_VEBT_cnt_rel,vEBT_Leaf((A5),(B6))) ) )
         => ~ ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info,Deg2,TreeList2,Summary) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Summary))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2,TreeList2),zero_zero(nat))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_cnt_rel,vEBT_Node(Info,Deg2,TreeList2,Summary)) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
tff(fact_6232_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(Xa) = Y )
     => ( accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,Xa)
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite((A5),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,vEBT_Leaf((A5),(B6))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_6233_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(Xa) = Y )
     => ( accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,Xa)
       => ( ! [A5: $o,B6: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B6)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite((B6),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,vEBT_Leaf((A5),(B6))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_6234_take__bit__numeral__minus__numeral__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Na: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se2584673776208193580ke_bit(word(A)),aa(num,nat,numeral_numeral(nat),M)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Na))) = case_option(word(A),num,zero_zero(word(A)),aTP_Lamp_rh(num,fun(num,word(A)),M),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),Na)) ) ).

% take_bit_numeral_minus_numeral_word
tff(fact_6235_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : bit_take_bit_num(zero_zero(nat),M) = none(num) ).

% take_bit_num_simps(1)
tff(fact_6236_take__bit__num__simps_I2_J,axiom,
    ! [Na: nat] : bit_take_bit_num(aa(nat,nat,suc,Na),one2) = some(num,one2) ).

% take_bit_num_simps(2)
tff(fact_6237_take__bit__num__simps_I5_J,axiom,
    ! [R: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R),one2) = some(num,one2) ).

% take_bit_num_simps(5)
tff(fact_6238_take__bit__num__simps_I3_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,Na),bit0(M)) = case_option(option(num),num,none(num),aTP_Lamp_ri(num,option(num)),bit_take_bit_num(Na,M)) ).

% take_bit_num_simps(3)
tff(fact_6239_take__bit__num__simps_I4_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,Na),aa(num,num,bit1,M)) = some(num,case_option(num,num,one2,bit1,bit_take_bit_num(Na,M))) ).

% take_bit_num_simps(4)
tff(fact_6240_take__bit__num__simps_I6_J,axiom,
    ! [R: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R),bit0(M)) = case_option(option(num),num,none(num),aTP_Lamp_ri(num,option(num)),bit_take_bit_num(pred_numeral(R),M)) ).

% take_bit_num_simps(6)
tff(fact_6241_take__bit__num__simps_I7_J,axiom,
    ! [R: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R),aa(num,num,bit1,M)) = some(num,case_option(num,num,one2,bit1,bit_take_bit_num(pred_numeral(R),M))) ).

% take_bit_num_simps(7)
tff(fact_6242_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,Na: num] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(num,nat,numeral_numeral(nat),M)),aa(num,A,numeral_numeral(A),Na)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),Na)) ) ).

% take_bit_numeral_numeral
tff(fact_6243_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(num,nat,numeral_numeral(nat),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = case_option(int,num,zero_zero(int),aTP_Lamp_rj(num,fun(num,int),M),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),Na)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_6244_take__bit__num__eq__Some__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: num,Q2: num] :
          ( ( bit_take_bit_num(M,Na) = some(num,Q2) )
         => ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(num,A,numeral_numeral(A),Na)) = aa(num,A,numeral_numeral(A),Q2) ) ) ) ).

% take_bit_num_eq_Some_imp
tff(fact_6245_take__bit__num__eq__None__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: num] :
          ( ( bit_take_bit_num(M,Na) = none(num) )
         => ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(num,A,numeral_numeral(A),Na)) = zero_zero(A) ) ) ) ).

% take_bit_num_eq_None_imp
tff(fact_6246_case__optionE,axiom,
    ! [A: $tType,P: $o,Q: fun(A,$o),Xa: option(A)] :
      ( case_option($o,A,(P),Q,Xa)
     => ( ( ( Xa = none(A) )
         => ~ (P) )
       => ~ ! [Y3: A] :
              ( ( Xa = some(A,Y3) )
             => ~ aa(A,$o,Q,Y3) ) ) ) ).

% case_optionE
tff(fact_6247_and__minus__numerals_I7_J,axiom,
    ! [Na: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Na)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(Na))) ).

% and_minus_numerals(7)
tff(fact_6248_and__minus__numerals_I8_J,axiom,
    ! [Na: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bit0(Na))) ).

% and_minus_numerals(8)
tff(fact_6249_and__minus__numerals_I4_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Na)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bit0(Na))) ).

% and_minus_numerals(4)
tff(fact_6250_and__minus__numerals_I3_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(Na))) ).

% and_minus_numerals(3)
tff(fact_6251_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one2,one2) = none(num) ).

% and_not_num.simps(1)
tff(fact_6252_and__not__num_Osimps_I2_J,axiom,
    ! [Na: num] : bit_and_not_num(one2,bit0(Na)) = some(num,one2) ).

% and_not_num.simps(2)
tff(fact_6253_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_and_not_num(bit0(M),one2) = some(num,bit0(M)) ).

% and_not_num.simps(4)
tff(fact_6254_and__not__num_Osimps_I3_J,axiom,
    ! [Na: num] : bit_and_not_num(one2,aa(num,num,bit1,Na)) = none(num) ).

% and_not_num.simps(3)
tff(fact_6255_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_and_not_num(aa(num,num,bit1,M),one2) = some(num,bit0(M)) ).

% and_not_num.simps(7)
tff(fact_6256_and__not__num__eq__Some__iff,axiom,
    ! [M: num,Na: num,Q2: num] :
      ( ( bit_and_not_num(M,Na) = some(num,Q2) )
    <=> ( 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))) = aa(num,int,numeral_numeral(int),Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_6257_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,Na: num] : bit_and_not_num(aa(num,num,bit1,M),bit0(Na)) = case_option(option(num),num,some(num,one2),aTP_Lamp_rk(num,option(num)),bit_and_not_num(M,Na)) ).

% and_not_num.simps(8)
tff(fact_6258_option_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType,H: fun(B,A),F1: B,F22: fun(C,B),Option: option(C)] : aa(B,A,H,case_option(B,C,F1,F22,Option)) = case_option(A,C,aa(B,A,H,F1),aa(fun(C,B),fun(C,A),aTP_Lamp_mh(fun(B,A),fun(fun(C,B),fun(C,A)),H),F22),Option) ).

% option.case_distrib
tff(fact_6259_and__not__num__eq__None__iff,axiom,
    ! [M: num,Na: num] :
      ( ( bit_and_not_num(M,Na) = none(num) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na))) = zero_zero(int) ) ) ).

% and_not_num_eq_None_iff
tff(fact_6260_int__numeral__and__not__num,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,Na)) ).

% int_numeral_and_not_num
tff(fact_6261_int__numeral__not__and__num,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),Na)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Na,M)) ).

% int_numeral_not_and_num
tff(fact_6262_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> case_option($o,A,$true,aTP_Lamp_ab(A,$o),Option) ) ).

% option.disc_eq_case(1)
tff(fact_6263_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> case_option($o,A,$false,aTP_Lamp_rl(A,$o),Option) ) ).

% option.disc_eq_case(2)
tff(fact_6264_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xa: fun(A,nat),X22: A] : size_option(A,Xa,some(A,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xa,X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_6265_disjE__realizer2,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: fun(A,$o),Xa: option(A),R2: fun(B,$o),F2: B,G: fun(A,B)] :
      ( case_option($o,A,(P),Q,Xa)
     => ( ( (P)
         => aa(B,$o,R2,F2) )
       => ( ! [Q3: A] :
              ( aa(A,$o,Q,Q3)
             => aa(B,$o,R2,aa(A,B,G,Q3)) )
         => aa(B,$o,R2,case_option(B,A,F2,G,Xa)) ) ) ) ).

% disjE_realizer2
tff(fact_6266_sofl__test,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),ring_1_signed(A,int,Xa)),ring_1_signed(A,int,Y)) = ring_1_signed(A,int,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) )
        <=> ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),Xa)),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)),Xa),Y)),Xa)),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)),Xa),Y)),Y))) = zero_zero(word(A)) ) ) ) ).

% sofl_test
tff(fact_6267_take__bit__num__def,axiom,
    ! [Na: nat,M: num] :
      bit_take_bit_num(Na,M) = $ite(aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),aa(num,nat,numeral_numeral(nat),M)) = zero_zero(nat),none(num),some(num,aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),aa(num,nat,numeral_numeral(nat),M))))) ).

% take_bit_num_def
tff(fact_6268_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_6269_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,A2: 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),A2)) = 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)),A2) ) ).

% drop_bit_drop_bit
tff(fact_6270_drop__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: 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),A2),B2)) = 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),A2)),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),B2)) ) ).

% drop_bit_and
tff(fact_6271_drop__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: 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),A2),B2)) = 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),A2)),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),B2)) ) ).

% drop_bit_or
tff(fact_6272_drop__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: 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),A2),B2)) = 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),A2)),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),B2)) ) ).

% drop_bit_xor
tff(fact_6273_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] : aa(nat,num,num_of_nat,aa(num,nat,numeral_numeral(nat),Q2)) = Q2 ).

% num_of_nat_numeral_eq
tff(fact_6274_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,B2: $o] :
          aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa($o,A,zero_neq_one_of_bool(A),(B2))) = aa($o,A,zero_neq_one_of_bool(A),
            ( ( Na = zero_zero(nat) )
            & (B2) )) ) ).

% drop_bit_of_bool
tff(fact_6275_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_6276_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),aa(num,num,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_6277_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_6278_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_6279_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),aa(num,num,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_6280_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          ( ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) = A2 )
        <=> ( aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A2) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_6281_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_6282_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_6283_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),A2)) ) ).

% take_bit_drop_bit
tff(fact_6284_drop__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),A2)) ) ).

% drop_bit_take_bit
tff(fact_6285_drop__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(nat,A,bit_se2239418461657761734s_mask(A),Na)) = aa(nat,A,bit_se2239418461657761734s_mask(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) ) ).

% drop_bit_mask_eq
tff(fact_6286_num__of__nat_Osimps_I1_J,axiom,
    aa(nat,num,num_of_nat,zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_6287_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A2)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_6288_numeral__num__of__nat,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(num,nat,numeral_numeral(nat),aa(nat,num,num_of_nat,Na)) = Na ) ) ).

% numeral_num_of_nat
tff(fact_6289_num__of__nat__One,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),one_one(nat))
     => ( aa(nat,num,num_of_nat,Na) = one2 ) ) ).

% num_of_nat_One
tff(fact_6290_drop__bit__half,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: 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),A2),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),A2)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% drop_bit_half
tff(fact_6291_stable__imp__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A2) = A2 ) ) ) ).

% stable_imp_drop_bit_eq
tff(fact_6292_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,Na)),A2) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% drop_bit_Suc
tff(fact_6293_drop__bit__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% drop_bit_eq_div
tff(fact_6294_bit__iff__odd__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
        <=> ~ aa(A,$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),A2)) ) ) ).

% bit_iff_odd_drop_bit
tff(fact_6295_even__drop__bit__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          ( aa(A,$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),A2))
        <=> ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na) ) ) ).

% even_drop_bit_iff_not_bit
tff(fact_6296_numeral__num__of__nat__unfold,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          aa(num,A,numeral_numeral(A),aa(nat,num,num_of_nat,Na)) = $ite(Na = zero_zero(nat),one_one(A),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% numeral_num_of_nat_unfold
tff(fact_6297_num__of__nat__double,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),Na)) = bit0(aa(nat,num,num_of_nat,Na)) ) ) ).

% num_of_nat_double
tff(fact_6298_num__of__nat__plus__distrib,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),zero_zero(nat)),Na)
       => ( aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(nat,num,num_of_nat,M)),aa(nat,num,num_of_nat,Na)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_6299_num__of__nat_Osimps_I2_J,axiom,
    ! [Na: nat] :
      aa(nat,num,num_of_nat,aa(nat,nat,suc,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na),inc(aa(nat,num,num_of_nat,Na)),one2) ).

% num_of_nat.simps(2)
tff(fact_6300_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A2) = $ite(Na = zero_zero(nat),A2,aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% drop_bit_rec
tff(fact_6301_div__half__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Xa),Y)) = $let(
                q: word(A),
                q:= aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),one_one(nat)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),one_one(nat)),Xa)),Y)),
                $let(
                  r: word(A),
                  r:= aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),q),Y)),
                  $ite(aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),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),aa(word(A),fun(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_6302_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_6303_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,A2: A] :
          ( ( aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_6304_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_6305_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_6306_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_6307_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,A2: 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),A2)) = 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)),A2) ) ).

% push_bit_push_bit
tff(fact_6308_push__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: 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),A2),B2)) = 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),A2)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),B2)) ) ).

% push_bit_and
tff(fact_6309_push__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: 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),A2),B2)) = 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),A2)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),B2)) ) ).

% push_bit_or
tff(fact_6310_push__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: 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),A2),B2)) = 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),A2)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),B2)) ) ).

% push_bit_xor
tff(fact_6311_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_6312_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_6313_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_6314_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_6315_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),aa(nat,A,bit_se2239418461657761734s_mask(A),Na)) ) ).

% push_bit_minus_one_eq_not_mask
tff(fact_6316_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_6317_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_6318_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),aa(num,num,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_6319_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,Na)),A2) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% push_bit_Suc
tff(fact_6320_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,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ).

% push_bit_of_1
tff(fact_6321_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          ( aa(A,$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),A2))
        <=> ( ( Na != zero_zero(nat) )
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% even_push_bit_iff
tff(fact_6322_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_6323_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),aa(num,num,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_6324_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,aa(A,fun(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_6325_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_6326_take__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),A2)) ) ).

% take_bit_push_bit
tff(fact_6327_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_6328_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_6329_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_6330_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),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),A2)) ) ).

% push_bit_take_bit
tff(fact_6331_push__bit__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: 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),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),A2)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),B2)) ) ).

% push_bit_add
tff(fact_6332_push__bit__minus,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),A2)) ) ).

% push_bit_minus
tff(fact_6333_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),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),A2) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_6334_bits__ident,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: 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),A2))),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2)) = A2 ) ).

% bits_ident
tff(fact_6335_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Na),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),one_one(A))) ) ).

% set_bit_eq_or
tff(fact_6336_flip__bit__eq__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),Na),A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),one_one(A))) ) ).

% flip_bit_eq_xor
tff(fact_6337_push__bit__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: 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),A2),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),A2)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% push_bit_double
tff(fact_6338_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_6339_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),aa(nat,A,bit_se2239418461657761734s_mask(A),Na)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(nat,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),aa(nat,A,bit_se2239418461657761734s_mask(A),M))) ) ).

% push_bit_mask_eq
tff(fact_6340_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Na),A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),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_6341_VEBT__internal_OperInsTrans_Ointros_I1_J,axiom,
    ! [T: vEBT_VEBT] : vEBT_V6289311342943941716sTrans(T,T) ).

% VEBT_internal.perInsTrans.intros(1)
tff(fact_6342_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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% drop_bit_int_def
tff(fact_6343_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% push_bit_eq_mult
tff(fact_6344_exp__dvdE,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)),A2)
         => ~ ! [B6: A] : A2 != aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),B6) ) ) ).

% exp_dvdE
tff(fact_6345_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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% drop_bit_nat_def
tff(fact_6346_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,M: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A2))) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(nat,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),aa(nat,A,bit_se2239418461657761734s_mask(A),Na)))) ) ).

% slice_eq_mask
tff(fact_6347_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_rm(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% take_bit_sum
tff(fact_6348_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),aa(nat,word(A),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),aa(nat,word(A),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_6349_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] :
          aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),Na),A2) = $let(
            l: A,
            l:= aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,Na)),A2),
            $ite(aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),l),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),l),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,Na)),aa(A,A,uminus_uminus(A),one_one(A)))),l) ) ) ).

% signed_take_bit_code
tff(fact_6350_VEBT__internal_OperInsTrans_Ointros_I2_J,axiom,
    ! [T: vEBT_VEBT,Sa: vEBT_VEBT,Xa: nat,U: vEBT_VEBT] :
      ( ( T = vEBT_vebt_insert(Sa,Xa) )
     => ( vEBT_V6289311342943941716sTrans(T,U)
       => vEBT_V6289311342943941716sTrans(Sa,U) ) ) ).

% VEBT_internal.perInsTrans.intros(2)
tff(fact_6351_VEBT__internal_OperInsTrans_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: vEBT_VEBT] :
      ( vEBT_V6289311342943941716sTrans(A1,A22)
     => ( ( A22 != A1 )
       => ~ ! [T4: vEBT_VEBT] :
              ( ? [X: nat] : T4 = vEBT_vebt_insert(A1,X)
             => ~ vEBT_V6289311342943941716sTrans(T4,A22) ) ) ) ).

% VEBT_internal.perInsTrans.cases
tff(fact_6352_VEBT__internal_OperInsTrans_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: vEBT_VEBT] :
      ( vEBT_V6289311342943941716sTrans(A1,A22)
    <=> ( ? [T2: vEBT_VEBT] :
            ( ( A1 = T2 )
            & ( A22 = T2 ) )
        | ? [T2: vEBT_VEBT,S6: vEBT_VEBT,X4: nat,U2: vEBT_VEBT] :
            ( ( A1 = S6 )
            & ( A22 = U2 )
            & ( T2 = vEBT_vebt_insert(S6,X4) )
            & vEBT_V6289311342943941716sTrans(T2,U2) ) ) ) ).

% VEBT_internal.perInsTrans.simps
tff(fact_6353_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_6354_shiftr__integer__conv__div__pow2,axiom,
    ! [Na: nat,Xa: code_integer] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4197421643247451524op_bit(code_integer),Na),Xa) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),Xa),aa(nat,code_integer,aa(code_integer,fun(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_6355_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_6356_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_6357_push__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),Na),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ).

% push_bit_of_Suc_0
tff(fact_6358_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,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)),K)) ).

% drop_bit_push_bit_int
tff(fact_6359_flip__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,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_6360_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_6361_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_6362_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,Na: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),M),K)),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_6363_shiftl__integer__conv__mult__pow2,axiom,
    ! [Na: nat,Xa: code_integer] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),Na),Xa) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),Xa),aa(nat,code_integer,aa(code_integer,fun(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_6364_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_6365_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: 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),Q2)),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),Q2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_6366_flip__bit__int__def,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,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_6367_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_6368_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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% push_bit_int_def
tff(fact_6369_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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% push_bit_nat_def
tff(fact_6370_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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% push_bit_minus_one
tff(fact_6371_Bit__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: $o] : bits_Bit_integer(code_integer_of_int(Xa),(Xb)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($o,int,zero_neq_one_of_bool(int),(Xb))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xa))) ).

% Bit_integer.abs_eq
tff(fact_6372_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,aa(nat,fun(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_6373_test__bit__split__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A2: word(A),B2: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A2),B2) )
        <=> ( ! [N2: nat] :
                ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),B2),N2)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(word(B),nat,size_size(word(B)),B2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),N2) ) )
            & ! [M3: nat] :
                ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A2),M3)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),aa(word(A),nat,size_size(word(A)),A2))
                  & aa(nat,$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)),B2))) ) ) ) ) ) ).

% test_bit_split_eq
tff(fact_6374_test__bit__split_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A2: word(A),B2: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A2),B2) )
         => ! [N9: nat,M2: nat] :
              ( ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),B2),N9)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N9),aa(word(B),nat,size_size(word(B)),B2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),N9) ) )
              & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A2),M2)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(word(A),nat,size_size(word(A)),A2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(word(B),nat,size_size(word(B)),B2))) ) ) ) ) ) ).

% test_bit_split'
tff(fact_6375_test__bit__split,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A2: word(A),B2: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A2),B2) )
         => ( ! [N9: nat] :
                ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),B2),N9)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N9),aa(word(B),nat,size_size(word(B)),B2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),N9) ) )
            & ! [M2: nat] :
                ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A2),M2)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(word(A),nat,size_size(word(A)),A2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(word(B),nat,size_size(word(B)),B2))) ) ) ) ) ) ).

% test_bit_split
tff(fact_6376_dup__1,axiom,
    code_dup(one_one(code_integer)) = aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)) ).

% dup_1
tff(fact_6377_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_6378_dup_Oabs__eq,axiom,
    ! [Xa: int] : code_dup(code_integer_of_int(Xa)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Xa)) ).

% dup.abs_eq
tff(fact_6379_bin__last__integer_Oabs__eq,axiom,
    ! [Xa: int] :
      ( bits_b8758750999018896077nteger(code_integer_of_int(Xa))
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xa) ) ).

% bin_last_integer.abs_eq
tff(fact_6380_bin__rest__integer_Oabs__eq,axiom,
    ! [Xa: int] : bits_b2549910563261871055nteger(code_integer_of_int(Xa)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% bin_rest_integer.abs_eq
tff(fact_6381_case__prod__app,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F2: fun(B,fun(C,fun(D,A))),Xa: product_prod(B,C),Y: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),product_case_prod(B,C,fun(D,A),F2),Xa),Y) = aa(product_prod(B,C),A,product_case_prod(B,C,A,aa(D,fun(B,fun(C,A)),aTP_Lamp_nc(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F2),Y)),Xa) ).

% case_prod_app
tff(fact_6382_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F2: fun(fun(A,B),C),G: C] :
          ( ! [X: fun(A,B)] : aa(fun(A,B),C,F2,X) = G
         => ( aa(fun(A,B),C,F2,aTP_Lamp_rn(A,B)) = G ) ) ) ).

% fun_cong_unused_0
tff(fact_6383_eq__subset,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),aTP_Lamp_ro(fun(A,fun(A,$o)),fun(A,fun(A,$o)),P)) ).

% eq_subset
tff(fact_6384_bin__rest__integer__code,axiom,
    ! [I: code_integer] : bits_b2549910563261871055nteger(I) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))) ).

% bin_rest_integer_code
tff(fact_6385_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_6386_Set__filter__fold,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] :
      ( finite_finite2(A,A3)
     => ( filter2(A,P,A3) = finite_fold(A,set(A),aTP_Lamp_rp(fun(A,$o),fun(A,fun(set(A),set(A))),P),bot_bot(set(A)),A3) ) ) ).

% Set_filter_fold
tff(fact_6387_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list(int)] :
      upto_aux(I,J,Js) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),Js,upto_aux(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)),aa(list(int),list(int),cons(int,J),Js))) ).

% upto_aux_rec
tff(fact_6388_member__filter,axiom,
    ! [A: $tType,Xa: A,P: fun(A,$o),A3: set(A)] :
      ( member(A,Xa,filter2(A,P,A3))
    <=> ( member(A,Xa,A3)
        & aa(A,$o,P,Xa) ) ) ).

% member_filter
tff(fact_6389_Set_Ofilter__def,axiom,
    ! [A: $tType,P: fun(A,$o),A3: set(A)] : filter2(A,P,A3) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_rq(fun(A,$o),fun(set(A),fun(A,$o)),P),A3)) ).

% Set.filter_def
tff(fact_6390_inter__Set__filter,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( finite_finite2(A,B3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = filter2(A,aTP_Lamp_a(set(A),fun(A,$o),A3),B3) ) ) ).

% inter_Set_filter
tff(fact_6391_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_6392_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_6393_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_6394_concat__bit__of__zero__2,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_concat_bit(Na,K),zero_zero(int)) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K) ).

% concat_bit_of_zero_2
tff(fact_6395_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_6396_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_6397_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_6398_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_6399_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_6400_concat__bit__eq__iff,axiom,
    ! [Na: nat,K: int,L: int,R: int,Sa: int] :
      ( ( aa(int,int,bit_concat_bit(Na,K),L) = aa(int,int,bit_concat_bit(Na,R),Sa) )
    <=> ( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),R) )
        & ( L = Sa ) ) ) ).

% concat_bit_eq_iff
tff(fact_6401_concat__bit__take__bit__eq,axiom,
    ! [Na: nat,B2: int] : bit_concat_bit(Na,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),B2)) = bit_concat_bit(Na,B2) ).

% concat_bit_take_bit_eq
tff(fact_6402_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,aa(nat,fun(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_6403_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,aa(nat,fun(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_6404_lsb__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X5: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),X5)
        <=> ~ aa(word(A),$o,dvd_dvd(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),X5) ) ) ).

% lsb_word_eq
tff(fact_6405_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_6406_signed__take__bit__eq__concat__bit,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(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_6407_word__lsb__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),A2)
        <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),int,semiring_1_unsigned(A,int),A2)) ) ) ).

% word_lsb_def
tff(fact_6408_lsb__odd,axiom,
    ! [A: $tType] :
      ( least_6119777620449941438nt_lsb(A)
     => ! [X5: A] :
          ( aa(A,$o,least_8051144512741203767sb_lsb(A),X5)
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),X5) ) ) ).

% lsb_odd
tff(fact_6409_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_6410_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_6411_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 = aa(word(C),word(A),aa(word(B),fun(word(C),word(A)),word_cat(B,C,A),U),V) ) ) ) ) ).

% word_cat_split_size
tff(fact_6412_word__split__cat__alt,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [W: word(A),U: word(B),V: word(C)] :
          ( ( W = aa(word(C),word(A),aa(word(B),fun(word(C),word(A)),word_cat(B,C,A),U),V) )
         => ( aa(nat,$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_6413_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_6414_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_6415_int__lsb__numeral_I7_J,axiom,
    ! [W: num] : aa(int,$o,least_8051144512741203767sb_lsb(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W))) ).

% int_lsb_numeral(7)
tff(fact_6416_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_6417_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_6418_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),aa(num,num,bit1,W)))) ).

% int_lsb_numeral(9)
tff(fact_6419_bin__last__conv__lsb,axiom,
    aTP_Lamp_rr(int,$o) = least_8051144512741203767sb_lsb(int) ).

% bin_last_conv_lsb
tff(fact_6420_test__bit__cat,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [A2: word(B),B2: word(C),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(C),word(A),aa(word(B),fun(word(C),word(A)),word_cat(B,C,A),A2),B2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),aa(word(C),word(A),aa(word(B),fun(word(C),word(A)),word_cat(B,C,A),A2),B2)))
            & $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(C),nat,size_size(word(C)),B2)),aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),B2),Na),aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(word(C),nat,size_size(word(C)),B2)))) ) ) ) ).

% test_bit_cat
tff(fact_6421_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) )
           => ( aa(word(C),word(A),aa(word(B),fun(word(C),word(A)),word_cat(B,C,A),U),V) = W ) ) ) ) ).

% word_cat_split_alt
tff(fact_6422_cat__slices,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A)
        & type_len(C) )
     => ! [A2: word(A),Na: nat,C2: word(B),B2: word(C)] :
          ( ( A2 = slice2(B,A,Na,C2) )
         => ( ( B2 = slice2(B,C,zero_zero(nat),C2) )
           => ( ( Na = aa(word(C),nat,size_size(word(C)),B2) )
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(B),nat,size_size(word(B)),C2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,size_size(word(A)),A2)),aa(word(C),nat,size_size(word(C)),B2)))
               => ( aa(word(C),word(B),aa(word(A),fun(word(C),word(B)),word_cat(A,C,B),A2),B2) = C2 ) ) ) ) ) ) ).

% cat_slices
tff(fact_6423_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F3: set(A),I5: set(A),F2: fun(A,B),I: A] :
          ( finite_finite2(A,F3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_rs(set(A),fun(fun(A,B),fun(A,$o)),I5),F2))),F3)
           => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I),bot_bot(set(A))))) = $ite(member(A,I,I5),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F2,I5)),aa(A,B,F2,I)),groups1027152243600224163dd_sum(A,B,F2,I5)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_6424_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: fun(B,A)] : groups1027152243600224163dd_sum(B,A,P2,bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty'
tff(fact_6425_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),P2: fun(A,B),I: A] :
          ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_dn(set(A),fun(fun(A,B),fun(A,$o)),I5),P2)))
         => ( groups1027152243600224163dd_sum(A,B,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I),I5)) = $ite(member(A,I,I5),groups1027152243600224163dd_sum(A,B,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_6426_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_rt(fun(B,A),fun(set(B),fun(B,$o)),G),I5))) = groups1027152243600224163dd_sum(B,A,G,I5) ) ).

% sum.non_neutral'
tff(fact_6427_sum_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ru(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_6428_sum_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_dn(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_dn(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ru(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_6429_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: fun(B,A),I5: set(B)] :
          groups1027152243600224163dd_sum(B,A,P2,I5) = $ite(finite_finite2(B,collect(B,aa(set(B),fun(B,$o),aTP_Lamp_rt(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_rt(fun(B,A),fun(set(B),fun(B,$o)),P2),I5))),zero_zero(A)) ) ).

% sum.G_def
tff(fact_6430_slice__cat1,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(C)
        & type_len(B) )
     => ! [A2: word(A),B2: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,size_size(word(A)),A2)),aa(word(B),nat,size_size(word(B)),B2))),aa(word(C),nat,size_size(word(C)),aa(word(B),word(C),aa(word(A),fun(word(B),word(C)),word_cat(A,B,C),A2),B2)))
         => ( slice2(C,A,aa(word(B),nat,size_size(word(B)),B2),aa(word(B),word(C),aa(word(A),fun(word(B),word(C)),word_cat(A,B,C),A2),B2)) = A2 ) ) ) ).

% slice_cat1
tff(fact_6431_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F2: fun(A,B),I: A] :
          ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_rs(set(A),fun(fun(A,B),fun(A,$o)),I5),F2)))
         => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I),bot_bot(set(A))))) = $ite(member(A,I,I5),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F2,I5)),aa(A,B,F2,I)),groups1027152243600224163dd_sum(A,B,F2,I5)) ) ) ) ).

% sum_diff1'
tff(fact_6432_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list($o),Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(list($o),A,groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2))),Bs)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list($o),nat,size_size(list($o)),Bs))
            & aa(nat,$o,nth($o,Bs),Na) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_6433_time__array__of__list,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A),H: heap_ext(product_unit)] : time_time(array(A),array_of_list(A,Xs),H) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ) ).

% time_array_of_list
tff(fact_6434_of__list__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A)] : hoare_hoare_triple(array(A),one_one(assn),array_of_list(A,Xs),aTP_Lamp_rv(list(A),fun(array(A),assn),Xs)) ) ).

% of_list_rule
tff(fact_6435_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list($o)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(list($o),int,groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2))),Bs)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(list($o),nat,size_size(list($o)),Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_6436_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A,Xa: B,Xs: list(B)] : aa(list(B),A,groups4207007520872428315er_sum(B,A,F2,A2),aa(list(B),list(B),cons(B,Xa),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F2,Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(list(B),A,groups4207007520872428315er_sum(B,A,F2,A2),Xs))) ) ).

% horner_sum_simps(2)
tff(fact_6437_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B)] : aa(list(B),A,groups4207007520872428315er_sum(B,A,F2,A2),Xs) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_rw(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_6438_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B)] : aa(list(B),A,groups4207007520872428315er_sum(B,A,F2,A2),Xs) = foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_rx(fun(B,A),fun(A,fun(B,fun(A,A))),F2),A2),Xs,zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_6439_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] : aa(list($o),A,groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2))),map(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),upt(zero_zero(nat),Na))) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_6440_vebt__assn__def,axiom,
    ! [Na: nat,Sa: set(nat),Tia: vEBT_VEBTi] : aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Tia) = ex_assn(vEBT_VEBT,aa(vEBT_VEBTi,fun(vEBT_VEBT,assn),aa(set(nat),fun(vEBT_VEBTi,fun(vEBT_VEBT,assn)),aTP_Lamp_ry(nat,fun(set(nat),fun(vEBT_VEBTi,fun(vEBT_VEBT,assn))),Na),Sa),Tia)) ).

% vebt_assn_def
tff(fact_6441_ex__assn__const,axiom,
    ! [A: $tType,C2: assn] : ex_assn(A,aTP_Lamp_rz(assn,fun(A,assn),C2)) = C2 ).

% ex_assn_const
tff(fact_6442_norm__assertion__simps_I16_J,axiom,
    ! [A: $tType,Q: fun(A,assn),R2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),ex_assn(A,Q)),R2) = ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_cw(fun(A,assn),fun(assn,fun(A,assn)),Q),R2)) ).

% norm_assertion_simps(16)
tff(fact_6443_norm__assertion__simps_I17_J,axiom,
    ! [A: $tType,R2: assn,Q: fun(A,assn)] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R2),ex_assn(A,Q)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_sa(assn,fun(fun(A,assn),fun(A,assn)),R2),Q)) ).

% norm_assertion_simps(17)
tff(fact_6444_triv__exI,axiom,
    ! [A: $tType,Q: fun(A,assn),Xa: A] : entails(aa(A,assn,Q,Xa),ex_assn(A,Q)) ).

% triv_exI
tff(fact_6445_norm__assertion__simps_I21_J,axiom,
    ! [A: $tType,Q: assn,P: fun(A,assn)] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Q),ex_assn(A,P)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_sb(assn,fun(fun(A,assn),fun(A,assn)),Q),P)) ).

% norm_assertion_simps(21)
tff(fact_6446_norm__assertion__simps_I20_J,axiom,
    ! [A: $tType,Q: fun(A,assn),P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),ex_assn(A,Q)),P) = ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_sc(fun(A,assn),fun(assn,fun(A,assn)),Q),P)) ).

% norm_assertion_simps(20)
tff(fact_6447_norm__assertion__simps_I18_J,axiom,
    ! [A: $tType,Q: fun(A,assn),P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),ex_assn(A,Q)),P) = ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_sd(fun(A,assn),fun(assn,fun(A,assn)),Q),P)) ).

% norm_assertion_simps(18)
tff(fact_6448_norm__assertion__simps_I19_J,axiom,
    ! [A: $tType,Q: assn,P: fun(A,assn)] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Q),ex_assn(A,P)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_se(assn,fun(fun(A,assn),fun(A,assn)),Q),P)) ).

% norm_assertion_simps(19)
tff(fact_6449_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_6450_post__exI__rule,axiom,
    ! [B: $tType,A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,fun(B,assn)),Xa: B] :
      ( hoare_hoare_triple(A,P,C2,aa(B,fun(A,assn),aTP_Lamp_sf(fun(A,fun(B,assn)),fun(B,fun(A,assn)),Q),Xa))
     => hoare_hoare_triple(A,P,C2,aTP_Lamp_sg(fun(A,fun(B,assn)),fun(A,assn),Q)) ) ).

% post_exI_rule
tff(fact_6451_norm__pre__ex__rule,axiom,
    ! [A: $tType,B: $tType,P: fun(A,assn),F2: heap_Time_Heap(B),Q: fun(B,assn)] :
      ( ! [X: A] : hoare_hoare_triple(B,aa(A,assn,P,X),F2,Q)
     => hoare_hoare_triple(B,ex_assn(A,P),F2,Q) ) ).

% norm_pre_ex_rule
tff(fact_6452_enorm__exI_H,axiom,
    ! [A: $tType,Z7: fun(A,$o),P: assn,Q: fun(A,assn)] :
      ( ! [X: A] :
          ( aa(A,$o,Z7,X)
         => entails(P,aa(A,assn,Q,X)) )
     => ( ? [X_13: A] : aa(A,$o,Z7,X_13)
       => entails(P,ex_assn(A,Q)) ) ) ).

% enorm_exI'
tff(fact_6453_norm__post__ex__rule__htt,axiom,
    ! [B: $tType,A: $tType,P: assn,F2: heap_Time_Heap(A),Q: fun(B,fun(A,assn)),Xa: B,T: nat] :
      ( time_htt(A,P,F2,aa(B,fun(A,assn),Q,Xa),T)
     => time_htt(A,P,F2,aTP_Lamp_si(fun(B,fun(A,assn)),fun(A,assn),Q),T) ) ).

% norm_post_ex_rule_htt
tff(fact_6454_norm__pre__ex__rule__htt,axiom,
    ! [B: $tType,A: $tType,P: fun(A,assn),F2: heap_Time_Heap(B),Q: fun(B,assn),T: nat] :
      ( ! [X: A] : time_htt(B,aa(A,assn,P,X),F2,Q,T)
     => time_htt(B,ex_assn(A,P),F2,Q,T) ) ).

% norm_pre_ex_rule_htt
tff(fact_6455_map__add__upt,axiom,
    ! [Na: nat,M: nat] : map(nat,nat,aTP_Lamp_sj(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_6456_map__add__upt_H,axiom,
    ! [Ofs: nat,A2: nat,B2: nat] : map(nat,nat,aTP_Lamp_sj(nat,fun(nat,nat),Ofs),upt(A2,B2)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),Ofs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),Ofs)) ).

% map_add_upt'
tff(fact_6457_ex__distrib__star,axiom,
    ! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_cw(fun(A,assn),fun(assn,fun(A,assn)),P),Q)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),ex_assn(A,P)),Q) ).

% ex_distrib_star
tff(fact_6458_ent__ex__preI,axiom,
    ! [A: $tType,P: fun(A,assn),Q: assn] :
      ( ! [X: A] : entails(aa(A,assn,P,X),Q)
     => entails(ex_assn(A,P),Q) ) ).

% ent_ex_preI
tff(fact_6459_ent__ex__postI,axiom,
    ! [A: $tType,P: assn,Q: fun(A,assn),Xa: A] :
      ( entails(P,aa(A,assn,Q,Xa))
     => entails(P,ex_assn(A,Q)) ) ).

% ent_ex_postI
tff(fact_6460_ex__distrib__and,axiom,
    ! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_sd(fun(A,assn),fun(assn,fun(A,assn)),P),Q)) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),ex_assn(A,P)),Q) ).

% ex_distrib_and
tff(fact_6461_ex__join__or,axiom,
    ! [A: $tType,P: fun(A,assn),Q: fun(A,assn)] : ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_sk(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P),Q)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_sl(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P),Q)) ).

% ex_join_or
tff(fact_6462_ex__distrib__or,axiom,
    ! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_sc(fun(A,assn),fun(assn,fun(A,assn)),P),Q)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),ex_assn(A,P)),Q) ).

% ex_distrib_or
tff(fact_6463_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),cons(nat,I),upt(aa(nat,nat,suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_6464_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xa: nat,Xs: list(nat)] :
      ( ( upt(I,J) = aa(list(nat),list(nat),cons(nat,Xa),Xs) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
        & ( I = Xa )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_6465_map__decr__upt,axiom,
    ! [M: nat,Na: nat] : map(nat,nat,aTP_Lamp_sm(nat,nat),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = upt(M,Na) ).

% map_decr_upt
tff(fact_6466_map__replicate__trivial,axiom,
    ! [A: $tType,Xa: A,I: nat] : map(nat,A,aTP_Lamp_sn(A,fun(nat,A),Xa),upt(zero_zero(nat),I)) = replicate(A,I,Xa) ).

% map_replicate_trivial
tff(fact_6467_map__upt__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),Na: nat] : map(nat,A,F2,upt(zero_zero(nat),aa(nat,nat,suc,Na))) = aa(list(A),list(A),cons(A,aa(nat,A,F2,zero_zero(nat))),map(nat,A,aTP_Lamp_so(fun(nat,A),fun(nat,A),F2),upt(zero_zero(nat),Na))) ).

% map_upt_Suc
tff(fact_6468_map__nth,axiom,
    ! [A: $tType,Xs: list(A)] : map(nat,A,nth(A,Xs),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = Xs ).

% map_nth
tff(fact_6469_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))
     => ( aa(nat,A,nth(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_6470_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),Na: nat,M: nat,F2: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I3)) ) )
       => ( map(nat,A,F2,upt(M,Na)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_6471_make__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Na: nat,F2: fun(nat,A)] : hoare_hoare_triple(array(A),one_one(assn),array_make(A,Na,F2),aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_sp(nat,fun(fun(nat,A),fun(array(A),assn)),Na),F2)) ) ).

% make_rule
tff(fact_6472_sdiv__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),A2)),one_one(nat))))),aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),ring_1_signed(A,int,A2)),ring_1_signed(A,int,B2))) ) ).

% sdiv_word_min
tff(fact_6473_sdiv__int__numeral__numeral,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),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_6474_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),A2),B2) != zero_zero(int) )
     => ( sgn_sgn(int,aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),A2),B2)) = sgn_sgn(int,aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2)) ) ) ).

% sgn_sdiv_eq_sgn_mult
tff(fact_6475_signed__divide__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,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),sgn_sgn(int,K)),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_6476_time__array__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Na: nat,F2: fun(nat,A),H: heap_ext(product_unit)] : time_time(array(A),array_make(A,Na,F2),H) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)) ) ).

% time_array_make
tff(fact_6477_TBOUND__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Na: nat,F2: fun(nat,A)] : time_TBOUND(array(A),array_make(A,Na,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))) ) ).

% TBOUND_make
tff(fact_6478_sdiv__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),ring_1_signed(A,int,A2)),ring_1_signed(A,int,B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),A2)),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% sdiv_word_max
tff(fact_6479_array__of__list__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A)] : array_of_list(A,Xs) = array_make(A,aa(list(A),nat,size_size(list(A)),Xs),nth(A,Xs)) ) ).

% array_of_list_make
tff(fact_6480_length__corresp,axiom,
    ! [B: $tType,A: $tType] :
      ( heap(A)
     => ! [Tree_array: array(A),Tree_is: list(B)] :
          ( ( ex_assn(list(A),snga_assn(A,Tree_array)) = top_top(assn) )
         => ( heap_Time_return(nat,aa(list(B),nat,size_size(list(B)),Tree_is)) = array_len(A,Tree_array) ) ) ) ).

% length_corresp
tff(fact_6481_top__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( top(A)
     => ! [Xa: B] : aa(B,A,top_top(fun(B,A)),Xa) = top_top(A) ) ).

% top_apply
tff(fact_6482_inf__top_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),top_top(A)) = A2 ) ).

% inf_top.right_neutral
tff(fact_6483_inf__top_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A2: A,B2: A] :
          ( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) )
        <=> ( ( A2 = top_top(A) )
            & ( B2 = top_top(A) ) ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_6484_inf__top_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),A2) = A2 ) ).

% inf_top.left_neutral
tff(fact_6485_inf__top_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = top_top(A) )
        <=> ( ( A2 = top_top(A) )
            & ( B2 = top_top(A) ) ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_6486_top__eq__inf__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xa: A,Y: A] :
          ( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) )
        <=> ( ( Xa = top_top(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% top_eq_inf_iff
tff(fact_6487_inf__eq__top__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = top_top(A) )
        <=> ( ( Xa = top_top(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% inf_eq_top_iff
tff(fact_6488_inf__top__right,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),top_top(A)) = Xa ) ).

% inf_top_right
tff(fact_6489_inf__top__left,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),Xa) = Xa ) ).

% inf_top_left
tff(fact_6490_boolean__algebra_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),top_top(A)) = top_top(A) ) ).

% boolean_algebra.disj_one_right
tff(fact_6491_boolean__algebra_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),Xa) = top_top(A) ) ).

% boolean_algebra.disj_one_left
tff(fact_6492_sup__top__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),top_top(A)) = top_top(A) ) ).

% sup_top_right
tff(fact_6493_sup__top__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),Xa) = top_top(A) ) ).

% sup_top_left
tff(fact_6494_max__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),top_top(A)) = top_top(A) ) ).

% max_top2
tff(fact_6495_max__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),top_top(A)),Xa) = top_top(A) ) ).

% max_top
tff(fact_6496_assn__basic__inequalities_I5_J,axiom,
    top_top(assn) != bot_bot(assn) ).

% assn_basic_inequalities(5)
tff(fact_6497_boolean__algebra_Ocompl__one,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ( aa(A,A,uminus_uminus(A),top_top(A)) = bot_bot(A) ) ) ).

% boolean_algebra.compl_one
tff(fact_6498_boolean__algebra_Ocompl__zero,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ( aa(A,A,uminus_uminus(A),bot_bot(A)) = top_top(A) ) ) ).

% boolean_algebra.compl_zero
tff(fact_6499_sup__compl__top__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) = top_top(A) ) ).

% sup_compl_top_left1
tff(fact_6500_sup__compl__top__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xa)),Y)) = top_top(A) ) ).

% sup_compl_top_left2
tff(fact_6501_boolean__algebra_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xa)),Xa) = top_top(A) ) ).

% boolean_algebra.disj_cancel_left
tff(fact_6502_boolean__algebra_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,uminus_uminus(A),Xa)) = top_top(A) ) ).

% boolean_algebra.disj_cancel_right
tff(fact_6503_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( bounded_lattice(A)
     => ! [Xa: A,Y: A] :
          ( ( set_or1337092689740270186AtMost(A,Xa,Y) = top_top(set(A)) )
        <=> ( ( Xa = bot_bot(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_6504_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( ( A2 != top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),top_top(A)) ) ) ).

% top.not_eq_extremum
tff(fact_6505_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),A2) ) ).

% top.extremum_strict
tff(fact_6506_sup__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),A2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xa)),B2)) = top_top(A) ) ).

% sup_cancel_left1
tff(fact_6507_sup__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xa)),A2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),B2)) = top_top(A) ) ).

% sup_cancel_left2
tff(fact_6508_boolean__algebra_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),top_top(A)) = Xa ) ).

% boolean_algebra.conj_one_right
tff(fact_6509_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A2)
         => ( A2 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_6510_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A2)
        <=> ( A2 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_6511_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),top_top(A)) ) ).

% top_greatest
tff(fact_6512_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xa)),Y) ) ) ).

% sup_shunt
tff(fact_6513_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [A2: A,Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),Xa) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),Xa) = top_top(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),Y) = bot_bot(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),Y) = top_top(A) )
               => ( Xa = Y ) ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_6514_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = top_top(A) )
           => ( aa(A,A,uminus_uminus(A),Xa) = Y ) ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_6515_array__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Na: nat,Xa: A] : array_new(A,Na,Xa) = array_make(A,Na,aTP_Lamp_sq(A,fun(nat,A),Xa)) ) ).

% array_make
tff(fact_6516_Cauchy__iff2,axiom,
    ! [X7: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X7)
    <=> ! [J2: nat] :
        ? [M9: nat] :
        ! [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M3)
         => ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),N2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,X7,M3)),aa(nat,real,X7,N2)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J2)))) ) ) ) ).

% Cauchy_iff2
tff(fact_6517_is__singleton__the__elem,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
    <=> ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),the_elem(A,A3)),bot_bot(set(A))) ) ) ).

% is_singleton_the_elem
tff(fact_6518_UNIV__I,axiom,
    ! [A: $tType,Xa: A] : member(A,Xa,top_top(set(A))) ).

% UNIV_I
tff(fact_6519_Int__UNIV,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = top_top(set(A)) )
    <=> ( ( A3 = top_top(set(A)) )
        & ( B3 = top_top(set(A)) ) ) ) ).

% Int_UNIV
tff(fact_6520_range__mult,axiom,
    ! [A2: real] :
      aa(set(real),set(real),image(real,real,aa(real,fun(real,real),times_times(real),A2)),top_top(set(real))) = $ite(A2 = zero_zero(real),aa(set(real),set(real),aa(real,fun(set(real),set(real)),insert(real),zero_zero(real)),bot_bot(set(real))),top_top(set(real))) ).

% range_mult
tff(fact_6521_Collect__const,axiom,
    ! [A: $tType,P: $o] :
      collect(A,aTP_Lamp_sr($o,fun(A,$o),(P))) = $ite((P),top_top(set(A)),bot_bot(set(A))) ).

% Collect_const
tff(fact_6522_finite__Collect__not,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( finite_finite2(A,collect(A,P))
     => ( finite_finite2(A,collect(A,aTP_Lamp_az(fun(A,$o),fun(A,$o),P)))
      <=> finite_finite2(A,top_top(set(A))) ) ) ).

% finite_Collect_not
tff(fact_6523_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P: $o] :
      collect(product_prod(A,B),product_case_prod(A,B,$o,aTP_Lamp_ss($o,fun(A,fun(B,$o)),(P)))) = $ite((P),top_top(set(product_prod(A,B))),bot_bot(set(product_prod(A,B)))) ).

% Collect_const_case_prod
tff(fact_6524_surj__plus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_plus
tff(fact_6525_range__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% range_add
tff(fact_6526_Diff__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_6527_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_oe(A,fun(A,A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_diff_right
tff(fact_6528_is__singletonI,axiom,
    ! [A: $tType,Xa: A] : is_singleton(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) ).

% is_singletonI
tff(fact_6529_range__constant,axiom,
    ! [B: $tType,A: $tType,Xa: A] : aa(set(B),set(A),image(B,A,aTP_Lamp_mm(A,fun(B,A),Xa)),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ).

% range_constant
tff(fact_6530_Compl__partition2,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),A3) = top_top(set(A)) ).

% Compl_partition2
tff(fact_6531_Compl__partition,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3)) = top_top(set(A)) ).

% Compl_partition
tff(fact_6532_Compl__UNIV__eq,axiom,
    ! [A: $tType] : aa(set(A),set(A),uminus_uminus(set(A)),top_top(set(A))) = bot_bot(set(A)) ).

% Compl_UNIV_eq
tff(fact_6533_Compl__empty__eq,axiom,
    ! [A: $tType] : aa(set(A),set(A),uminus_uminus(set(A)),bot_bot(set(A))) = top_top(set(A)) ).

% Compl_empty_eq
tff(fact_6534_perfect__space__class_OUNIV__not__singleton,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [Xa: A] : top_top(set(A)) != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ) ).

% perfect_space_class.UNIV_not_singleton
tff(fact_6535_rangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A)] :
      ( member(A,B2,aa(set(B),set(A),image(B,A,F2),top_top(set(B))))
     => ~ ! [X: B] : B2 != aa(B,A,F2,X) ) ).

% rangeE
tff(fact_6536_rangeI,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xa: B] : member(A,aa(B,A,F2,Xa),aa(set(B),set(A),image(B,A,F2),top_top(set(B)))) ).

% rangeI
tff(fact_6537_range__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),Xa: B] :
      ( ( B2 = aa(B,A,F2,Xa) )
     => member(A,B2,aa(set(B),set(A),image(B,A,F2),top_top(set(B)))) ) ).

% range_eqI
tff(fact_6538_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_st(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_6539_insert__UNIV,axiom,
    ! [A: $tType,Xa: A] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),top_top(set(A))) = top_top(set(A)) ).

% insert_UNIV
tff(fact_6540_empty__not__UNIV,axiom,
    ! [A: $tType] : bot_bot(set(A)) != top_top(set(A)) ).

% empty_not_UNIV
tff(fact_6541_infinite__UNIV__int,axiom,
    ~ finite_finite2(int,top_top(set(int))) ).

% infinite_UNIV_int
tff(fact_6542_UNIV__witness,axiom,
    ! [A: $tType] :
    ? [X: A] : member(A,X,top_top(set(A))) ).

% UNIV_witness
tff(fact_6543_top__set__def,axiom,
    ! [A: $tType] : top_top(set(A)) = collect(A,top_top(fun(A,$o))) ).

% top_set_def
tff(fact_6544_UNIV__eq__I,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X: A] : member(A,X,A3)
     => ( top_top(set(A)) = A3 ) ) ).

% UNIV_eq_I
tff(fact_6545_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = collect(A,aTP_Lamp_rl(A,$o)) ).

% UNIV_def
tff(fact_6546_Int__UNIV__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),top_top(set(A))) = A3 ).

% Int_UNIV_right
tff(fact_6547_Int__UNIV__left,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),top_top(set(A))),B3) = B3 ).

% Int_UNIV_left
tff(fact_6548_Un__UNIV__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),top_top(set(A))) = top_top(set(A)) ).

% Un_UNIV_right
tff(fact_6549_Un__UNIV__left,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),top_top(set(A))),B3) = top_top(set(A)) ).

% Un_UNIV_left
tff(fact_6550_subset__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),top_top(set(A))) ).

% subset_UNIV
tff(fact_6551_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),B3: set(A),I: B] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F2),top_top(set(B)))),B3)
     => member(A,aa(B,A,F2,I),B3) ) ).

% range_subsetD
tff(fact_6552_Compl__eq__Diff__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),A3) ).

% Compl_eq_Diff_UNIV
tff(fact_6553_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X5: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X5),Xa2)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa2),top_top(set(product_prod(A,B)))) ) ).

% top_empty_eq2
tff(fact_6554_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),F2: fun(A,C)] :
      ( finite_finite2(A,aa(set(B),set(A),image(B,A,G),top_top(set(B))))
     => finite_finite2(C,aa(set(B),set(C),image(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_su(fun(B,A),fun(fun(A,C),fun(B,C)),G),F2)),top_top(set(B)))) ) ).

% finite_range_imageI
tff(fact_6555_is__singletonI_H,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ! [X: A,Y3: A] :
            ( member(A,X,A3)
           => ( member(A,Y3,A3)
             => ( X = Y3 ) ) )
       => is_singleton(A,A3) ) ) ).

% is_singletonI'
tff(fact_6556_range__eq__singletonD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A2: A,Xa: B] :
      ( ( aa(set(B),set(A),image(B,A,F2),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) )
     => ( aa(B,A,F2,Xa) = A2 ) ) ).

% range_eq_singletonD
tff(fact_6557_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_6558_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_6559_is__singletonE,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
     => ~ ! [X: A] : A3 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ).

% is_singletonE
tff(fact_6560_is__singleton__def,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
    <=> ? [X4: A] : A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),bot_bot(set(A))) ) ).

% is_singleton_def
tff(fact_6561_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A),E: real] :
          ( topolo3814608138187158403Cauchy(A,X7)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
           => ? [M8: nat] :
              ! [M2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M2)
               => ! [N9: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N9)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X7,M2)),aa(nat,A,X7,N9)))),E) ) ) ) ) ) ).

% CauchyD
tff(fact_6562_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [M10: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),M4)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X7,M4)),aa(nat,A,X7,N)))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X7) ) ) ).

% CauchyI
tff(fact_6563_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X7)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M9: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M3)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X7,M3)),aa(nat,A,X7,N2)))),E3) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_6564_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_sv(nat,fun(nat,nat),Na)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Na) ) ) ).

% range_mod
tff(fact_6565_UNIV__nat__eq,axiom,
    top_top(set(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),top_top(set(nat)))) ).

% UNIV_nat_eq
tff(fact_6566_root__def,axiom,
    ! [Na: nat,Xa: real] :
      aa(real,real,root(Na),Xa) = $ite(Na = zero_zero(nat),zero_zero(real),the_inv_into(real,real,top_top(set(real)),aTP_Lamp_sw(nat,fun(real,real),Na),Xa)) ).

% root_def
tff(fact_6567_top1I,axiom,
    ! [A: $tType,Xa: A] : aa(A,$o,top_top(fun(A,$o)),Xa) ).

% top1I
tff(fact_6568_top2I,axiom,
    ! [A: $tType,B: $tType,Xa: A,Y: B] : aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),Xa),Y) ).

% top2I
tff(fact_6569_the__inv__into__def,axiom,
    ! [B: $tType,A: $tType,A3: set(B),F2: fun(B,A),X5: A] : the_inv_into(B,A,A3,F2,X5) = the(B,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_sx(set(B),fun(fun(B,A),fun(A,fun(B,$o))),A3),F2),X5)) ).

% the_inv_into_def
tff(fact_6570_int__of__nat__def,axiom,
    code_T6385005292777649522of_nat = semiring_1_of_nat(int) ).

% int_of_nat_def
tff(fact_6571_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(A,B),B3: set(B)] :
      ( ! [X: A] :
          ( aa(A,$o,P,X)
         => member(B,aa(A,B,F2,X),B3) )
     => aa(set(B),$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))),B3) ) ).

% image_Collect_subsetI
tff(fact_6572_prop__restrict,axiom,
    ! [A: $tType,Xa: A,Z7: set(A),X7: set(A),P: fun(A,$o)] :
      ( member(A,Xa,Z7)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z7),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_al(set(A),fun(fun(A,$o),fun(A,$o)),X7),P)))
       => aa(A,$o,P,Xa) ) ) ).

% prop_restrict
tff(fact_6573_Collect__restrict,axiom,
    ! [A: $tType,X7: set(A),P: fun(A,$o)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_al(set(A),fun(fun(A,$o),fun(A,$o)),X7),P))),X7) ).

% Collect_restrict
tff(fact_6574_subset__emptyI,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X: A] : ~ member(A,X,A3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),bot_bot(set(A))) ) ).

% subset_emptyI
tff(fact_6575_insert__subsetI,axiom,
    ! [A: $tType,Xa: A,A3: set(A),X7: set(A)] :
      ( member(A,Xa,A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),A3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),X7)),A3) ) ) ).

% insert_subsetI
tff(fact_6576_length__product__lists,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),product_lists(A,Xss)) = foldr(nat,nat,times_times(nat),map(list(A),nat,size_size(list(A)),Xss),one_one(nat)) ).

% length_product_lists
tff(fact_6577_smod__int__range,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => member(int,aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2),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),B2))),one_one(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,abs_abs(int),B2)),one_one(int)))) ) ).

% smod_int_range
tff(fact_6578_smod__int__numeral__numeral,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),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_6579_smod__int__compares_I8_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2)) ) ) ).

% smod_int_compares(8)
tff(fact_6580_smod__int__compares_I7_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2)),zero_zero(int)) ) ) ).

% smod_int_compares(7)
tff(fact_6581_smod__int__compares_I6_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2)) ) ) ).

% smod_int_compares(6)
tff(fact_6582_smod__int__compares_I4_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2)),zero_zero(int)) ) ) ).

% smod_int_compares(4)
tff(fact_6583_smod__int__compares_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2)) ) ) ).

% smod_int_compares(2)
tff(fact_6584_smod__int__compares_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2)),B2) ) ) ).

% smod_int_compares(1)
tff(fact_6585_smod__mod__positive,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
       => ( aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) ) ) ) ).

% smod_mod_positive
tff(fact_6586_signed__modulo__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),K),L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),K),L)),L)) ).

% signed_modulo_int_def
tff(fact_6587_smod__int__compares_I5_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2)),aa(int,int,uminus_uminus(int),B2)) ) ) ).

% smod_int_compares(5)
tff(fact_6588_smod__int__compares_I3_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),B2)),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2)) ) ) ).

% smod_int_compares(3)
tff(fact_6589_smod__int__alt__def,axiom,
    ! [A2: int,B2: int] : aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2) = aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,A2)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,abs_abs(int),A2)),aa(int,int,abs_abs(int),B2))) ).

% smod_int_alt_def
tff(fact_6590_length__subseqs,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),subseqs(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_subseqs
tff(fact_6591_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),Na: nat] :
      ( ! [X: list(A)] :
          ( member(list(A),X,set2(list(A),Xs))
         => ( aa(list(A),nat,size_size(list(A)),X) = Na ) )
     => ( aa(list(A),nat,size_size(list(A)),concat(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(list(A)),nat,size_size(list(list(A))),Xs)),Na) ) ) ).

% length_mul_elem
tff(fact_6592_product__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] : product_lists(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs),Xss)) = concat(list(A),map(A,list(list(A)),aTP_Lamp_sy(list(list(A)),fun(A,list(list(A))),Xss),Xs)) ).

% product_lists.simps(2)
tff(fact_6593_product__concat__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B)] : product(A,B,Xs,Ys2) = concat(product_prod(A,B),map(A,list(product_prod(A,B)),aTP_Lamp_sz(list(B),fun(A,list(product_prod(A,B))),Ys2),Xs)) ).

% product_concat_map
tff(fact_6594_set__n__lists,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] : set2(list(A),n_lists(A,Na,Xs)) = collect(list(A),aa(list(A),fun(list(A),$o),aTP_Lamp_ta(nat,fun(list(A),fun(list(A),$o)),Na),Xs)) ).

% set_n_lists
tff(fact_6595_mod__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% mod_word_minus_1_minus_numeral
tff(fact_6596_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,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Bin)) ) ).

% uint_bintrunc
tff(fact_6597_word__exp__length__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A))) = zero_zero(word(A)) ) ) ).

% word_exp_length_eq_0
tff(fact_6598_less__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% less_eq_word_numeral_numeral
tff(fact_6599_less__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% less_word_numeral_numeral
tff(fact_6600_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,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Bin))) ) ).

% unat_bintrunc
tff(fact_6601_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_6602_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,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(B,type2(B))),aa(num,int,numeral_numeral(int),W))) ) ).

% ucast_bintr
tff(fact_6603_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,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),type_len0_len_of(B,type2(B))),aa(num,nat,numeral_numeral(nat),Na))) ) ).

% unsigned_numeral
tff(fact_6604_unat__lt2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_lt2p
tff(fact_6605_uint__lt2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_lt2p
tff(fact_6606_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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_bounded
tff(fact_6607_exp__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) = zero_zero(word(A)) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na) ) ) ).

% exp_eq_zero_iff
tff(fact_6608_of__nat__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) = zero_zero(word(A)) ) ) ).

% of_nat_2p
tff(fact_6609_signed__take__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_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,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,suc,Na)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% signed_take_bit_word_Suc_numeral
tff(fact_6610_signed__take__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_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,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(num,nat,numeral_numeral(nat),Na)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% signed_take_bit_word_numeral
tff(fact_6611_sint__sbintrunc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Bin)) = aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),Bin)) ) ).

% sint_sbintrunc
tff(fact_6612_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,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin))) ) ).

% uint_bintrunc_neg
tff(fact_6613_div__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% div_word_numeral_numeral
tff(fact_6614_mod__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% mod_word_numeral_numeral
tff(fact_6615_signed__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Na: num] : 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,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(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_6616_less__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% less_eq_word_minus_numeral_numeral
tff(fact_6617_less__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% less_eq_word_numeral_minus_numeral
tff(fact_6618_less__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% less_eq_word_minus_numeral_minus_numeral
tff(fact_6619_less__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% less_word_minus_numeral_minus_numeral
tff(fact_6620_less__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% less_word_numeral_minus_numeral
tff(fact_6621_less__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% less_word_minus_numeral_numeral
tff(fact_6622_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,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin)))) ) ).

% unat_bintrunc_neg
tff(fact_6623_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_6624_drop__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,suc,Na)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,Na)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% drop_bit_word_Suc_numeral
tff(fact_6625_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,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% drop_bit_word_numeral
tff(fact_6626_unat__power__lower,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ) ) ) ).

% unat_power_lower
tff(fact_6627_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,aa(nat,fun(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_6628_signed__take__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_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,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,suc,Na)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_take_bit_word_Suc_minus_numeral
tff(fact_6629_signed__take__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_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,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(num,nat,numeral_numeral(nat),Na)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_take_bit_word_minus_numeral
tff(fact_6630_sint__sbintrunc__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : 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,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin))) ) ).

% sint_sbintrunc_neg
tff(fact_6631_drop__bit__numeral__bit0__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,suc,zero_zero(nat))),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,zero_zero(nat))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% drop_bit_numeral_bit0_1
tff(fact_6632_div__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% div_word_minus_numeral_numeral
tff(fact_6633_div__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% div_word_numeral_minus_numeral
tff(fact_6634_div__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% div_word_minus_numeral_minus_numeral
tff(fact_6635_mod__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% mod_word_minus_numeral_numeral
tff(fact_6636_mod__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% mod_word_numeral_minus_numeral
tff(fact_6637_mod__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% mod_word_minus_numeral_minus_numeral
tff(fact_6638_word__less__sub__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A))))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% word_less_sub_le
tff(fact_6639_signed__neg__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Na: num] : 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,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(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_6640_drop__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,suc,Na)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,Na)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% drop_bit_word_Suc_minus_numeral
tff(fact_6641_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,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% drop_bit_word_minus_numeral
tff(fact_6642_less__word__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))) ) ) ).

% less_word_numeral_minus_1
tff(fact_6643_less__word__minus__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))) ) ) ).

% less_word_minus_numeral_minus_1
tff(fact_6644_div__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% div_word_minus_1_numeral
tff(fact_6645_mod__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% mod_word_minus_1_numeral
tff(fact_6646_div__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% div_word_minus_1_minus_numeral
tff(fact_6647_two__power__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
           => ( ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M) )
            <=> ( Na = M ) ) ) ) ) ).

% two_power_eq
tff(fact_6648_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,$o),bit_se5641148757651400278ts_bit(word(A)),slice2(B,A,Na,W)),M)
        <=> ( aa(nat,$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_6649_word__of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),M)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),type_len0_len_of(A,type2(A))),M)),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),type_len0_len_of(A,type2(A))),Na)) ) ) ).

% word_of_nat_less_iff
tff(fact_6650_ucast__ucast__mask,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(A)] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),type_len0_len_of(B,type2(B)))) ) ).

% ucast_ucast_mask
tff(fact_6651_ucast__less__ucast__weak,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ).

% ucast_less_ucast_weak
tff(fact_6652_ucast__ucast__id,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)) = Xa ) ) ) ).

% ucast_ucast_id
tff(fact_6653_word__and__full__mask__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),type_len0_len_of(A,type2(A)))) = Xa ) ).

% word_and_full_mask_simp
tff(fact_6654_word__of__int__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,L: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(int,word(A),ring_1_of_int(word(A)),K)),aa(int,word(A),ring_1_of_int(word(A)),L))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),K)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),L)) ) ) ).

% word_of_int_less_iff
tff(fact_6655_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,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),type_len0_len_of(A,type2(A))),M) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),type_len0_len_of(A,type2(A))),Na) ) ) ) ).

% word_of_nat_eq_iff
tff(fact_6656_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,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),type_len0_len_of(B,type2(B))),Na)) ) ).

% unsigned_of_nat
tff(fact_6657_signed__take__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [Na: nat,W: word(B)] :
          ring_1_signed(B,A,aa(word(B),word(B),aa(nat,fun(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,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),ring_1_signed(B,A,W)),ring_1_signed(B,A,W)) ) ).

% signed_take_bit_eq
tff(fact_6658_uint__word__arith__bintrs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = aa(int,int,aa(nat,fun(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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% uint_word_arith_bintrs(1)
tff(fact_6659_degenerate__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
         => ( ( Xa = zero_zero(word(A)) )
            | ( Xa = one_one(word(A)) ) ) ) ) ).

% degenerate_word
tff(fact_6660_max__word__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) = aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),type_len0_len_of(A,type2(A))) ) ) ).

% max_word_mask
tff(fact_6661_test__bit__1_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),one_one(word(A))),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),type_len0_len_of(A,type2(A)))
            & ( Na = zero_zero(nat) ) ) ) ) ).

% test_bit_1'
tff(fact_6662_test__bit__conj__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),M: nat] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),M)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) )
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),M) ) ) ).

% test_bit_conj_lt
tff(fact_6663_neg__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)),Na)
        <=> ( ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ) ).

% neg_test_bit
tff(fact_6664_word__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( Xa = Y )
        <=> ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),type_len0_len_of(A,type2(A)))
             => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),N2)
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),N2) ) ) ) ) ).

% word_eq_iff
tff(fact_6665_bit__word__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),type_len0_len_of(A,type2(A)))
             => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A2),N)
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),B2),N) ) )
         => ( A2 = B2 ) ) ) ).

% bit_word_eqI
tff(fact_6666_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_6667_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_6668_test__bit__wi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),ring_1_of_int(word(A)),Xa)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),Xa),Na) ) ) ) ).

% test_bit_wi
tff(fact_6669_max__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Na)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% max_test_bit
tff(fact_6670_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_6671_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_6672_bit__ucast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [A2: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),A2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),A2),Na) ) ) ) ).

% bit_ucast_iff
tff(fact_6673_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_6674_nth__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [W: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W)),Na)
        <=> ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_ucast
tff(fact_6675_test__bit__bin,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),Na) ) ) ) ).

% test_bit_bin
tff(fact_6676_num__abs__bintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num] : aa(num,word(A),numeral_numeral(word(A)),Xa) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Xa))) ) ).

% num_abs_bintr
tff(fact_6677_uint__word__arith__bintrs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)) = aa(int,int,aa(nat,fun(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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% uint_word_arith_bintrs(3)
tff(fact_6678_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_6679_num__of__bintr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2)) = aa(num,int,numeral_numeral(int),B2) )
         => ( aa(num,word(A),numeral_numeral(word(A)),A2) = aa(num,word(A),numeral_numeral(word(A)),B2) ) ) ) ).

% num_of_bintr'
tff(fact_6680_ucast__ucast__add,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(A),Y: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y)) ) ) ) ).

% ucast_ucast_add
tff(fact_6681_mask__over__length,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% mask_over_length
tff(fact_6682_ucast__mask__drop,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Na: nat,Xa: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),Xa),aa(nat,word(B),bit_se2239418461657761734s_mask(word(B)),Na))) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa) ) ) ) ).

% ucast_mask_drop
tff(fact_6683_up__ucast__inj,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(B),Y: word(B)] :
          ( ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( Xa = Y ) ) ) ) ).

% up_ucast_inj
tff(fact_6684_ucast__le__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y) ) ) ) ).

% ucast_le_ucast
tff(fact_6685_ucast__ucast__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [Xa: word(B),Y: word(C)] :
          ( ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(C),word(B),semiring_1_unsigned(C,word(B)),Y)) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(C,type2(C)))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(C,type2(C))),type_len0_len_of(A,type2(A)))
             => ( Xa = aa(word(C),word(B),semiring_1_unsigned(C,word(B)),Y) ) ) ) ) ) ).

% ucast_ucast_eq
tff(fact_6686_up__ucast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa) = aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y) )
          <=> ( Xa = Y ) ) ) ) ).

% up_ucast_inj_eq
tff(fact_6687_ucast__up__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_up_mono_le
tff(fact_6688_eq__ucast__ucast__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(B),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( ( Xa = aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y) )
           => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa) = Y ) ) ) ) ).

% eq_ucast_ucast_eq
tff(fact_6689_unat__ucast__up__simp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),nat,semiring_1_unsigned(B,nat),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)) = aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) ) ) ) ).

% unat_ucast_up_simp
tff(fact_6690_less__ucast__ucast__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(B),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),Xa),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa)),Y) ) ) ) ).

% less_ucast_ucast_less
tff(fact_6691_ucast__less__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ).

% ucast_less_ucast
tff(fact_6692_ucast__up__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y)
           => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_up_mono
tff(fact_6693_word__of__nat__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),M)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),type_len0_len_of(A,type2(A))),M)),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),type_len0_len_of(A,type2(A))),Na)) ) ) ).

% word_of_nat_less_eq_iff
tff(fact_6694_ucast__sub__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
           => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)) = aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),minus_minus(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ) ).

% ucast_sub_ucast
tff(fact_6695_bit__set__bit__aux,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,$o),Na: nat,W: word(A),M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),code_T2661198915054445665ts_aux(A,F2,Na,W)),M)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
            & $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na),aa(nat,$o,F2,M),aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na))) ) ) ) ).

% bit_set_bit_aux
tff(fact_6696_word__cat__inj,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [A2: word(A),B2: word(B),C2: word(A),D2: word(B)] :
          ( aa(nat,$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)))
         => ( ( aa(word(B),word(C),aa(word(A),fun(word(B),word(C)),word_cat(A,B,C),A2),B2) = aa(word(B),word(C),aa(word(A),fun(word(B),word(C)),word_cat(A,B,C),C2),D2) )
          <=> ( ( A2 = C2 )
              & ( B2 = D2 ) ) ) ) ) ).

% word_cat_inj
tff(fact_6697_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,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(C),word(A),aa(word(B),fun(word(C),word(A)),word_cat(B,C,A),V),W)),Na)
        <=> ( aa(nat,$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,aa(nat,fun(nat,nat),minus_minus(nat),Na),type_len0_len_of(C,type2(C))))) ) ) ) ).

% bit_word_cat_iff
tff(fact_6698_num__of__sbintr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( ( aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2)) = aa(num,int,numeral_numeral(int),B2) )
         => ( aa(num,word(A),numeral_numeral(word(A)),A2) = aa(num,word(A),numeral_numeral(word(A)),B2) ) ) ) ).

% num_of_sbintr'
tff(fact_6699_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,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(B,type2(B))),K))) ) ).

% unsigned_of_int
tff(fact_6700_word__sint__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( 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_6701_neg__mask__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))),M)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ) ).

% neg_mask_test_bit
tff(fact_6702_word__of__int__bin__cat__eq__iff,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(C)
        & type_len(B) )
     => ! [B2: word(B),A2: word(A),D2: word(B),C2: word(A)] :
          ( aa(nat,$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),B2)),aa(word(A),int,semiring_1_unsigned(A,int),A2))) = 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))) )
          <=> ( ( B2 = D2 )
              & ( A2 = C2 ) ) ) ) ) ).

% word_of_int_bin_cat_eq_iff
tff(fact_6703_mask__exceed,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) = zero_zero(word(A)) ) ) ) ).

% mask_exceed
tff(fact_6704_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,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),type_len0_len_of(B,type2(B)))) ) ).

% unsigned_less
tff(fact_6705_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_6706_word__nat__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ~ ! [N: nat] :
              ( ( Xa = aa(nat,word(A),semiring_1_of_nat(word(A)),N) )
             => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% word_nat_cases
tff(fact_6707_word__nchotomy,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W3: word(A)] :
        ? [N: nat] :
          ( ( W3 = aa(nat,word(A),semiring_1_of_nat(word(A)),N) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% word_nchotomy
tff(fact_6708_word__of__nat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Xa) = aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
             => ( Xa = Y ) ) ) ) ) ).

% word_of_nat_inj
tff(fact_6709_of__nat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Xa) = aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
            <=> ( Xa = Y ) ) ) ) ) ).

% of_nat_inj
tff(fact_6710_More__Word_Opower__not__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) != zero_zero(word(A)) ) ) ) ).

% More_Word.power_not_zero
tff(fact_6711_word__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Xa)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Y))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),type_len0_len_of(A,type2(A)))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y) ) ) ) ) ).

% word_power_increasing
tff(fact_6712_power__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) = zero_zero(word(A)) ) ) ) ).

% power_overflow
tff(fact_6713_nth__w2p__same,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),Na)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% nth_w2p_same
tff(fact_6714_nth__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),M)
        <=> ( ( M = Na )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_w2p
tff(fact_6715_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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) = aa(word(A),int,semiring_1_unsigned(A,int),W) ) ).

% uint_idem
tff(fact_6716_word__of__int__2p__len,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) = zero_zero(word(A)) ) ) ).

% word_of_int_2p_len
tff(fact_6717_of__nat__neq__iff__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) != aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) )
         => ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Xa) != aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
          <=> ( Xa != Y ) ) ) ) ).

% of_nat_neq_iff_word
tff(fact_6718_num__abs__sbintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num] : aa(num,word(A),numeral_numeral(word(A)),Xa) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),Xa))) ) ).

% num_abs_sbintr
tff(fact_6719_ucast__ucast__len,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(B,type2(B))))
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)) = Xa ) ) ) ).

% ucast_ucast_len
tff(fact_6720_ucast__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(B,type2(B))))
           => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_mono
tff(fact_6721_horner__sum__uint__exp__Cons__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Ws: list(word(A))] : aa(list(word(A)),int,groups4207007520872428315er_sum(word(A),int,semiring_1_unsigned(A,int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),aa(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)),aa(list(word(A)),int,groups4207007520872428315er_sum(word(A),int,semiring_1_unsigned(A,int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),Ws)) ) ).

% horner_sum_uint_exp_Cons_eq
tff(fact_6722_sint__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : ring_1_signed(A,int,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,aa(int,fun(int,int),plus_plus(int),ring_1_signed(A,int,A2)),ring_1_signed(A,int,B2))) ) ).

% sint_word_ariths(1)
tff(fact_6723_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),ring_1_signed(A,int,W)),Na)
        <=> $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Na),aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat)))) ) ) ).

% nth_sint
tff(fact_6724_sint__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : ring_1_signed(A,int,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)) = aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,aa(int,fun(int,int),times_times(int),ring_1_signed(A,int,A2)),ring_1_signed(A,int,B2))) ) ).

% sint_word_ariths(3)
tff(fact_6725_less__Suc__unat__less__bound,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% less_Suc_unat_less_bound
tff(fact_6726_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_6727_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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% lt2p_lem
tff(fact_6728_power__le__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ) ) ) ) ).

% power_le_mono
tff(fact_6729_two__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ).

% two_power_increasing
tff(fact_6730_unat__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(num,nat,numeral_numeral(nat),B2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_numeral
tff(fact_6731_of__nat__mono__maybe_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa)
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa)) ) ) ) ) ).

% of_nat_mono_maybe'
tff(fact_6732_of__nat__mono__maybe,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa)) ) ) ) ).

% of_nat_mono_maybe
tff(fact_6733_unat__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),Xa: word(A)] :
          ( aa(nat,$o,P,aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
        <=> ~ ? [N2: nat] :
                ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N2) = Xa )
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
                & ~ aa(nat,$o,P,N2) ) ) ) ).

% unat_split_asm
tff(fact_6734_of__nat__inverse,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [R: nat,A2: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),R) = A2 )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),nat,semiring_1_unsigned(A,nat),A2) = R ) ) ) ) ).

% of_nat_inverse
tff(fact_6735_unat__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),Xa: word(A)] :
          ( aa(nat,$o,P,aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
        <=> ! [N2: nat] :
              ( ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N2) = Xa )
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) )
             => aa(nat,$o,P,N2) ) ) ) ).

% unat_split
tff(fact_6736_unat__of__nat__len,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa)) = Xa ) ) ) ).

% unat_of_nat_len
tff(fact_6737_unat__eq__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) = Na )
          <=> ( Xa = aa(nat,word(A),semiring_1_of_nat(word(A)),Na) ) ) ) ) ).

% unat_eq_of_nat
tff(fact_6738_UNIV__word__eq__word__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( top_top(set(word(A))) = aa(set(nat),set(word(A)),image(nat,word(A),semiring_1_of_nat(word(A))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% UNIV_word_eq_word_of_nat
tff(fact_6739_x__less__2__0__1_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) != one_one(nat) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)))
           => ( ( Xa = zero_zero(word(A)) )
              | ( Xa = one_one(word(A)) ) ) ) ) ) ).

% x_less_2_0_1'
tff(fact_6740_test__bit__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))),M)
        <=> ( ( M = Na )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ) ).

% test_bit_2p
tff(fact_6741_Word_Oof__nat__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),M) = zero_zero(word(A)) )
        <=> ? [Q5: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q5),aa(nat,nat,aa(nat,fun(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_6742_word__1__le__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ).

% word_1_le_power
tff(fact_6743_ucast__of__nat__small,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa)) = aa(nat,word(B),semiring_1_of_nat(word(B)),Xa) ) ) ) ).

% ucast_of_nat_small
tff(fact_6744_uint__sub__lt2p,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(B)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(B),int,semiring_1_unsigned(B,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_sub_lt2p
tff(fact_6745_uint__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),B2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_numeral
tff(fact_6746_p2__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% p2_gt_0
tff(fact_6747_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,aa(nat,fun(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_6748_unat__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(B)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(word(B),nat,semiring_1_unsigned(B,nat),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_ucast
tff(fact_6749_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,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),I)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),I)) ) ).

% word_of_int_minus
tff(fact_6750_bit__last__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,W)),zero_zero(int)) ) ) ).

% bit_last_iff
tff(fact_6751_mask__lt__2pn,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ).

% mask_lt_2pn
tff(fact_6752_unat__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_of_nat
tff(fact_6753_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,aa(int,fun(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_6754_ucast__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(nat,word(B),aa(word(B),fun(nat,word(B)),power_power(word(B)),aa(num,word(B),numeral_numeral(word(B)),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% ucast_less
tff(fact_6755_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,aa(int,fun(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_6756_of__nat__n__less__equal__power__2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ).

% of_nat_n_less_equal_power_2
tff(fact_6757_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),aa(word(A),fun(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_6758_upper__trivial,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( Xa != aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(word(A))) )
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(word(A)))) ) ) ).

% upper_trivial
tff(fact_6759_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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% range_uint
tff(fact_6760_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),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(word(A))) ) ) ).

% minus_one_word
tff(fact_6761_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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% UNIV_eq
tff(fact_6762_word__power__less__diff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Q2: word(A),M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),Q2)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Q2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),Na)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Q2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na))) ) ) ) ).

% word_power_less_diff
tff(fact_6763_unat__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),A2),B2)) = 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),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(7)
tff(fact_6764_ucast__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(B,type2(B))))
           => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_mono_le
tff(fact_6765_take__bit__word__eq__self__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se2584673776208193580ke_bit(word(A)),Na),W) = W )
        <=> ( aa(nat,$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),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% take_bit_word_eq_self_iff
tff(fact_6766_msb0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),I: nat] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) ) ) ).

% msb0
tff(fact_6767_ucast__range__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
       => ( aa(set(word(A)),set(word(B)),image(word(A),word(B),semiring_1_unsigned(A,word(B))),top_top(set(word(A)))) = collect(word(B),aTP_Lamp_tb(word(B),$o)) ) ) ) ).

% ucast_range_less
tff(fact_6768_unat__add__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
        <=> ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_add_lem
tff(fact_6769_unat__add__lem_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_add_lem'
tff(fact_6770_Word_Oof__nat__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,word(A),semiring_1_of_nat(word(A)),K) != zero_zero(word(A)) ) ) ) ) ).

% Word.of_nat_neq_0
tff(fact_6771_More__Word_Oof__nat__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Na) = zero_zero(word(A)) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( Na = zero_zero(nat) ) ) ) ) ).

% More_Word.of_nat_0
tff(fact_6772_of__nat__mono__maybe__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xa)
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa)) ) ) ) ) ).

% of_nat_mono_maybe_le
tff(fact_6773_unat__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),zero_zero(word(A))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% unat_word_ariths(4)
tff(fact_6774_unat__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = 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),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(1)
tff(fact_6775_bool__mask_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),one_one(word(A))))
          <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),one_one(word(A))) = one_one(word(A)) ) ) ) ) ).

% bool_mask'
tff(fact_6776_uint__range_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),Xa))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% uint_range'
tff(fact_6777_sint__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,Xa)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat)))) ) ).

% sint_lt
tff(fact_6778_ucast__mono__le_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(B,type2(B))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
             => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ) ).

% ucast_mono_le'
tff(fact_6779_of__nat__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Na) = W )
        <=> ? [Q5: nat] : Na = aa(nat,nat,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),Q5),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% of_nat_eq
tff(fact_6780_word__int__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ~ ! [N: int] :
              ( ( Xa = aa(int,word(A),ring_1_of_int(word(A)),N) )
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
               => ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),N),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ) ).

% word_int_cases
tff(fact_6781_word__of__int__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Y: int] :
          ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) )
         => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) )
           => ( ( aa(int,word(A),ring_1_of_int(word(A)),Xa) = aa(int,word(A),ring_1_of_int(word(A)),Y) )
            <=> ( Xa = Y ) ) ) ) ) ).

% word_of_int_inj
tff(fact_6782_unat__mult__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
        <=> ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_mult_lem
tff(fact_6783_unat__ucast__no__overflow__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [B2: word(A),F2: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(B,type2(B))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),F2)),B2)
            <=> aa(nat,$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),B2)) ) ) ) ) ).

% unat_ucast_no_overflow_le
tff(fact_6784_uint__m2p__not__non__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% uint_m2p_not_non_neg
tff(fact_6785_uint__m2p__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),zero_zero(int)) ) ).

% uint_m2p_neg
tff(fact_6786_unat__ucast__less__no__overflow__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,F2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),F2)),Na)
          <=> aa(word(A),$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_6787_unat__ucast__less__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,F2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),F2)),Na)
           => aa(word(A),$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_6788_uint__power__lower,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na) ) ) ) ).

% uint_power_lower
tff(fact_6789_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),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% upper_bits_unset_is_l2p
tff(fact_6790_nth__bounded,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ) ).

% nth_bounded
tff(fact_6791_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),$o),ord_less(word(A)),P2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & ! [N11: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N11)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N11),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P2),N11) ) ) ) ) ) ).

% less_2p_is_upper_bits_unset
tff(fact_6792_uint__add__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_add_lem
tff(fact_6793_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,aa(int,fun(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_6794_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,aa(int,fun(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_6795_wi__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: int,M: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(int,word(A),ring_1_of_int(word(A)),Na)),aa(int,word(A),ring_1_of_int(word(A)),M))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% wi_le
tff(fact_6796_uint__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = 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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(1)
tff(fact_6797_word__2p__mult__inc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ).

% word_2p_mult_inc
tff(fact_6798_unat__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)) = 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),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(2)
tff(fact_6799_wi__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: int,M: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(int,word(A),ring_1_of_int(word(A)),Na)),aa(int,word(A),ring_1_of_int(word(A)),M))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% wi_less
tff(fact_6800_unat__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),B2)) = 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),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(6)
tff(fact_6801_power__2__ge__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% power_2_ge_iff
tff(fact_6802_word__power__less__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Sz),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)) ) ) ).

% word_power_less_1
tff(fact_6803_uint__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),uminus_uminus(word(A)),A2)) = 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),A2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(4)
tff(fact_6804_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),$o),ord_less_eq(word(A)),W),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% le_mask_iff_lt_2n
tff(fact_6805_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),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ).

% and_mask_less'
tff(fact_6806_eq__mask__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( ( W = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% eq_mask_less
tff(fact_6807_sint__1__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
           => ( ( A2 = zero_zero(word(A)) )
             => ( ring_1_signed(A,int,A2) != zero_zero(int) ) ) )
         => ( ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
             => ( ( A2 = one_one(word(A)) )
               => ( ring_1_signed(A,int,one_one(word(A))) != aa(int,int,uminus_uminus(int),one_one(int)) ) ) )
           => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),type_len0_len_of(A,type2(A)))
               => ( ring_1_signed(A,int,one_one(word(A))) != one_one(int) ) ) ) ) ) ).

% sint_1_cases
tff(fact_6808_uint__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),A2),B2)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(2)
tff(fact_6809_uint__mult__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_mult_lem
tff(fact_6810_uint__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)) = 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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(3)
tff(fact_6811_signed__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Na: nat] : 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,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),aa(nat,int,semiring_1_of_nat(int),Na))) ) ).

% signed_of_nat
tff(fact_6812_word__power__mod__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))) ) ) ) ) ).

% word_power_mod_div
tff(fact_6813_msb1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% msb1
tff(fact_6814_unat__plus__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% unat_plus_if'
tff(fact_6815_unat__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),one_one(word(A))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% unat_word_ariths(5)
tff(fact_6816_unat__sub__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))) ) ).

% unat_sub_if'
tff(fact_6817_no__olen__add__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% no_olen_add_nat
tff(fact_6818_word__add__le__mono2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),I)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),J)) ) ) ) ).

% word_add_le_mono2
tff(fact_6819_word__add__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K)) ) ) ) ).

% word_add_le_mono1
tff(fact_6820_word__add__le__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ).

% word_add_le_dest
tff(fact_6821_word__add__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ).

% word_add_le_iff
tff(fact_6822_sint__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% sint_less
tff(fact_6823_word__add__less__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K)) ) ) ) ).

% word_add_less_mono1
tff(fact_6824_word__add__less__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J) ) ) ) ) ).

% word_add_less_dest
tff(fact_6825_word__add__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J) ) ) ) ) ).

% word_add_less_iff
tff(fact_6826_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,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(nat)) ) ) ).

% unat_minus_one_word
tff(fact_6827_sint__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))),ring_1_signed(A,int,Xa)) ) ).

% sint_ge
tff(fact_6828_unat__less__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat,K: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Sz),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),K),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),K)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Sz)) ) ) ) ).

% unat_less_power
tff(fact_6829_uint__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(int,$o),Xa: word(A)] :
          ( aa(int,$o,P,aa(word(A),int,semiring_1_unsigned(A,int),Xa))
        <=> ! [I2: int] :
              ( ( ( aa(int,word(A),ring_1_of_int(word(A)),I2) = Xa )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I2)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) )
             => aa(int,$o,P,I2) ) ) ) ).

% uint_split
tff(fact_6830_uint__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(int,$o),Xa: word(A)] :
          ( aa(int,$o,P,aa(word(A),int,semiring_1_unsigned(A,int),Xa))
        <=> ~ ? [I2: int] :
                ( ( aa(int,word(A),ring_1_of_int(word(A)),I2) = Xa )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I2)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))
                & ~ aa(int,$o,P,I2) ) ) ) ).

% uint_split_asm
tff(fact_6831_word__of__int__inverse,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [R: int,A2: word(A)] :
          ( ( aa(int,word(A),ring_1_of_int(word(A)),R) = A2 )
         => ( aa(int,$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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(word(A),int,semiring_1_unsigned(A,int),A2) = R ) ) ) ) ) ).

% word_of_int_inverse
tff(fact_6832_word__mult__less__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J) ) ) ) ) ).

% word_mult_less_dest
tff(fact_6833_div__lt_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% div_lt'
tff(fact_6834_div__lt_H_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% div_lt''
tff(fact_6835_double__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A2) = zero_zero(word(A)) )
        <=> ( ( A2 = zero_zero(word(A)) )
            | ( A2 = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ) ).

% double_eq_zero_iff
tff(fact_6836_no__olen__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% no_olen_add
tff(fact_6837_no__olen__add_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Xa))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),Xa))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% no_olen_add'
tff(fact_6838_word__le__exists_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
         => ? [Z3: word(A)] :
              ( ( Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z3) )
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Z3))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ) ).

% word_le_exists'
tff(fact_6839_More__Word_Oof__nat__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: nat,Xa: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xa))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),P2)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Xa)) ) ) ) ).

% More_Word.of_nat_power
tff(fact_6840_uint__plus__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% uint_plus_if'
tff(fact_6841_word__less__power__trans2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: nat,K: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Na),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Na),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),K))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ) ).

% word_less_power_trans2
tff(fact_6842_word__less__power__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: nat,K: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Na),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),K)),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ) ).

% word_less_power_trans
tff(fact_6843_uint__sub__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),A2),B2)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),B2)),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% uint_sub_if'
tff(fact_6844_word__less__two__pow__divI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))) ) ) ) ) ).

% word_less_two_pow_divI
tff(fact_6845_uint__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2))) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_neg_numeral
tff(fact_6846_word__power__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),Na)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
           => ( ( Xa != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) != zero_zero(word(A)) ) ) ) ) ) ).

% word_power_nonzero
tff(fact_6847_mult__pow2__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat,Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))
             => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) )
               => ( Xa = Y ) ) ) ) ) ) ).

% mult_pow2_inj
tff(fact_6848_div__lt__uint_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),I)),aa(word(A),int,semiring_1_unsigned(A,int),Xa))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% div_lt_uint'
tff(fact_6849_div__lt__uint_H_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),I)),aa(word(A),int,semiring_1_unsigned(A,int),Xa))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% div_lt_uint''
tff(fact_6850_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),$o),ord_less(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),type_len0_len_of(A,type2(A)))
           => ( ( 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_6851_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] : n_lists(A,aa(nat,nat,suc,Na),Xs) = concat(list(A),map(list(A),list(list(A)),aTP_Lamp_td(list(A),fun(list(A),list(list(A))),Xs),n_lists(A,Na,Xs))) ).

% n_lists.simps(2)
tff(fact_6852_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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))),ring_1_signed(A,int,W)) ) ).

% sint_greater_eq
tff(fact_6853_int__eq__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))
         => ( ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),Xa)) = aa(nat,int,semiring_1_of_nat(int),Xa) ) ) ) ).

% int_eq_sint
tff(fact_6854_word__mult__less__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),I: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))
              <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J) ) ) ) ) ) ).

% word_mult_less_cancel
tff(fact_6855_word__mult__less__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K)) ) ) ) ) ).

% word_mult_less_mono1
tff(fact_6856_smod__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),ring_1_signed(A,int,A2)),ring_1_signed(A,int,B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% smod_word_max
tff(fact_6857_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),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
           => ! [N4: nat] :
                ( aa(nat,$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_6858_le2p__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A))))
         => ! [N4: nat] :
              ( aa(nat,$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_6859_word__add__offset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Na: nat,Xa: word(A),M: nat,Sz: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Sz),type_len0_len_of(A,type2(A)))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),Na)))
               => ( ( Sz = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) )
                 => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),Y)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)) ) ) ) ) ) ) ).

% word_add_offset_less
tff(fact_6860_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,$o),bit_se5641148757651400278ts_bit(int),aa(list(word(A)),int,groups4207007520872428315er_sum(word(A),int,semiring_1_unsigned(A,int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),Ws)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),type_len0_len_of(A,type2(A)))),aa(list(word(A)),nat,size_size(list(word(A))),Ws))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),nth(word(A),Ws),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),type_len0_len_of(A,type2(A))))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),type_len0_len_of(A,type2(A)))) ) ) ) ).

% bit_horner_sum_uint_exp_iff
tff(fact_6861_div__power__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Y)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Y)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Xa)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y),Xa))),one_one(word(A))) ) ) ) ) ).

% div_power_helper
tff(fact_6862_even__mult__exp__div__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),M: nat,Na: nat] :
          ( aa(word(A),$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)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)))
        <=> ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
              & ~ aa(word(A),$o,dvd_dvd(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)))) ) ) ) ).

% even_mult_exp_div_word_iff
tff(fact_6863_Suc__2p__unat__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),K)),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ) ) ).

% Suc_2p_unat_mask
tff(fact_6864_sint__of__nat__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: nat,A2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),A2))),ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),B2))) ) ) ) ).

% sint_of_nat_le
tff(fact_6865_sint__of__nat__ge__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),Xa))) ) ) ).

% sint_of_nat_ge_zero
tff(fact_6866_sint__int__max__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) = aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% sint_int_max_plus_1
tff(fact_6867_sint__of__int__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))),Xa)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))
           => ( ring_1_signed(A,int,aa(int,word(A),ring_1_of_int(word(A)),Xa)) = Xa ) ) ) ) ).

% sint_of_int_eq
tff(fact_6868_word__mult__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K)) ) ) ) ) ).

% word_mult_le_mono1
tff(fact_6869_word__mult__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),I: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))
              <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ) ).

% word_mult_le_iff
tff(fact_6870_sint__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,int,aa(int,fun(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),B2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat)))) ) ).

% sint_numeral
tff(fact_6871_smod__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),ring_1_signed(A,int,A2)),ring_1_signed(A,int,B2))) ) ).

% smod_word_min
tff(fact_6872_int__word__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] : ring_1_signed(A,int,aa(int,word(A),ring_1_of_int(word(A)),Xa)) = aa(int,int,aa(int,fun(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),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat)))) ) ).

% int_word_sint
tff(fact_6873_Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Y)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),Y)),Y) = Xa ) ) ) ) ).

% Word.word_div_mult
tff(fact_6874_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_te(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_tf(nat,fun(nat,fun(nat,$o)),Na),M))) ) ) ) ).

% of_nat_less_two_pow_div_set
tff(fact_6875_sint__int__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,aa(word(A),word(A),uminus_uminus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))) = aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% sint_int_min
tff(fact_6876_word__less__power__trans__ofnat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),K))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ) ).

% word_less_power_trans_ofnat
tff(fact_6877_word__bit__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),A2: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [A5: word(A)] :
                ( aa(word(A),$o,P,A5)
               => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),A5)
                 => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A5),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
                   => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A5)) ) ) )
           => ( ! [A5: word(A)] :
                  ( aa(word(A),$o,P,A5)
                 => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A5),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
                   => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A5))) ) )
             => aa(word(A),$o,P,A2) ) ) ) ) ).

% word_bit_induct
tff(fact_6878_unat__mult__power__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,Sz: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),Sz)))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)),aa(nat,word(A),semiring_1_of_nat(word(A)),K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Sz)),K) ) ) ) ).

% unat_mult_power_lem
tff(fact_6879_bit__word__half__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: $o] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa($o,word(A),zero_neq_one_of_bool(word(A)),(B2))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))))),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) = A2 ) ) ) ).

% bit_word_half_eq
tff(fact_6880_length__n__lists,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),n_lists(A,Na,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(list(A),nat,size_size(list(A)),Xs)),Na) ).

% length_n_lists
tff(fact_6881_word__of__int__via__signed,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Mask: int,Shift: int,Index: nat,Overflow: int,Least: int,I: int,Arbitrary1: fun(int,word(A)),Arbitrary2: fun(int,word(A))] :
          ( ( Mask = aa(nat,int,bit_se2239418461657761734s_mask(int),type_len0_len_of(A,type2(A))) )
         => ( ( Shift = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),type_len0_len_of(A,type2(A))),one_one(int)) )
           => ( ( Index = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat)) )
             => ( ( Overflow = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),one_one(int)) )
               => ( ( Least = aa(int,int,uminus_uminus(int),Overflow) )
                 => ( aa(int,word(A),ring_1_of_int(word(A)),I) = $let(
                        i: int,
                        i:= aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),Mask),
                        $ite(
                          aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),i),Index),
                          $ite(
                            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),i),Shift)),Least)
                            | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Overflow),aa(int,int,aa(int,fun(int,int),minus_minus(int),i),Shift)) ),
                            aa(int,word(A),Arbitrary1,i),
                            aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),minus_minus(int),i),Shift)) ),
                          $ite(
                            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),i),Least)
                            | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Overflow),i) ),
                            aa(int,word(A),Arbitrary2,i),
                            aa(int,word(A),ring_1_of_int(word(A)),i) ) ) ) ) ) ) ) ) ) ) ).

% word_of_int_via_signed
tff(fact_6882_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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Sz),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),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Us)))) ) ) ) ) ).

% Suc_div_unat_helper
tff(fact_6883_alignUp__div__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,Na: nat,Xa: word(A),A2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),Na)))
         => ( ( Xa = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),semiring_1_of_nat(word(A)),K)) )
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),Xa)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
               => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) != zero_zero(word(A)) )
                 => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),aa(nat,word(A),semiring_1_of_nat(word(A)),K)) ) ) ) ) ) ) ).

% alignUp_div_helper
tff(fact_6884_less__eq__decr__length__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% less_eq_decr_length_iff
tff(fact_6885_decr__length__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Na)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na) ) ) ).

% decr_length_less_iff
tff(fact_6886_len__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),type_len0_len_of(A,type2(A))) ) ).

% len_gt_0
tff(fact_6887_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_6888_scast__sbintr,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: num] : 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,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),aa(num,int,numeral_numeral(int),W))) ) ).

% scast_sbintr
tff(fact_6889_up__scast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( ( ring_1_signed(A,word(B),Xa) = ring_1_signed(A,word(B),Y) )
          <=> ( Xa = Y ) ) ) ) ).

% up_scast_inj_eq
tff(fact_6890_up__scast__inj,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(B),Y: word(B)] :
          ( ( ring_1_signed(B,word(A),Xa) = ring_1_signed(B,word(A),Y) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(B),nat,size_size(word(B)),Xa)),type_len0_len_of(A,type2(A)))
           => ( Xa = Y ) ) ) ) ).

% up_scast_inj
tff(fact_6891_scast__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( 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_6892_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,$o),bit_se5641148757651400278ts_bit(word(A)),ring_1_signed(B,word(A),W)),Na)
        <=> ( aa(nat,$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,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ) ) ).

% bit_word_scast_iff
tff(fact_6893_scast__1_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( ring_1_signed(B,word(A),one_one(word(B))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),one_one(int))) ) ) ).

% scast_1'
tff(fact_6894_len__of__finite__3__def,axiom,
    ! [Xa: itself(finite_3)] : type_len0_len_of(finite_3,Xa) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% len_of_finite_3_def
tff(fact_6895_len__of__finite__2__def,axiom,
    ! [Xa: itself(finite_2)] : type_len0_len_of(finite_2,Xa) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% len_of_finite_2_def
tff(fact_6896_len__bit0,axiom,
    ! [A: $tType] :
      ( type_len0(A)
     => ! [Uu: itself(numeral_bit0(A))] : type_len0_len_of(numeral_bit0(A),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))) ) ).

% len_bit0
tff(fact_6897_two__less__eq__exp__length,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & linordered_idom(A) )
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),type_len0_len_of(B,type2(B)))) ) ).

% two_less_eq_exp_length
tff(fact_6898_len__bit1,axiom,
    ! [A: $tType] :
      ( type_len0(A)
     => ! [Uu: itself(numeral_bit1(A))] : type_len0_len_of(numeral_bit1(A),Uu) = 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_6899_divmod__via__sdivmod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Xa),Y)) = $ite(
                aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),one_one(word(A)))),Y),
                $ite(aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),zero_zero(word(A))),Xa),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y))),
                $let(
                  q: word(A),
                  q:= aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),one_one(nat)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed7115095781618012415divide(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),one_one(nat)),Xa)),Y)),
                  $let(
                    r: word(A),
                    r:= aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),q),Y)),
                    $ite(aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),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),aa(word(A),fun(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_6900_product__code,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B)] : product_product(A,B,set2(A,Xs),set2(B,Ys2)) = set2(product_prod(A,B),concat(product_prod(A,B),map(A,list(product_prod(A,B)),aTP_Lamp_sz(list(B),fun(A,list(product_prod(A,B))),Ys2),Xs))) ).

% product_code
tff(fact_6901_sdiv__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed7115095781618012415divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% sdiv_word_numeral_numeral
tff(fact_6902_sdiv__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed7115095781618012415divide(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% sdiv_word_minus_numeral_numeral
tff(fact_6903_sdiv__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed7115095781618012415divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% sdiv_word_numeral_minus_numeral
tff(fact_6904_sdiv__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed7115095781618012415divide(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% sdiv_word_minus_numeral_minus_numeral
tff(fact_6905_word__sdiv__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num,Y: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed7115095781618012415divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),Xa)),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Xa))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_sdiv_numerals(1)
tff(fact_6906_word__sdiv__numerals__lhs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num,W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed7115095781618012415divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),Xa)),W) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Xa))),ring_1_signed(A,int,W))) ) ).

% word_sdiv_numerals_lhs(1)
tff(fact_6907_word__sdiv__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed7115095781618012415divide(word(A)),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),ring_1_signed(A,int,one_one(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_sdiv_numerals(3)
tff(fact_6908_word__sdiv__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed7115095781618012415divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),Xa)),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Xa))),ring_1_signed(A,int,one_one(word(A))))) ) ).

% word_sdiv_numerals(7)
tff(fact_6909_word__sdiv__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed7115095781618012415divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),Xa)),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Xa))),ring_1_signed(A,int,zero_zero(word(A))))) ) ).

% word_sdiv_numerals(4)
tff(fact_6910_word__sdiv__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed7115095781618012415divide(word(A)),zero_zero(word(A))),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),ring_1_signed(A,int,zero_zero(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_sdiv_numerals(2)
tff(fact_6911_signed__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)),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,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_drop_bit_word_minus_numeral
tff(fact_6912_signed__drop__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),aa(nat,nat,suc,Na)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,Na)),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_drop_bit_word_Suc_minus_numeral
tff(fact_6913_signed__drop__bit__signed__drop__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat,W: word(A)] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),M),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),Na),W)) = aa(word(A),word(A),aa(nat,fun(word(A),word(A)),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_6914_signed__drop__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),aa(nat,nat,suc,Na)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,Na)),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),K)))) ) ).

% signed_drop_bit_word_Suc_numeral
tff(fact_6915_signed__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)),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,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),K)))) ) ).

% signed_drop_bit_word_numeral
tff(fact_6916_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,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),M),W)),Na)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),
              $ite(
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),M)),Na)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ),
                aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat)),
                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) )) ) ) ).

% bit_signed_drop_bit_iff
tff(fact_6917_word__int__split__asm,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P: fun(A,$o),F2: fun(int,A),Xa: word(B)] :
          ( aa(A,$o,P,aa(word(B),A,aa(fun(int,A),fun(word(B),A),word_int_case(A,B),F2),Xa))
        <=> ~ ? [N2: int] :
                ( ( Xa = aa(int,word(B),ring_1_of_int(word(B)),N2) )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N2)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),N2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(B,type2(B))))
                & ~ aa(A,$o,P,aa(int,A,F2,N2)) ) ) ) ).

% word_int_split_asm
tff(fact_6918_word__int__split,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P: fun(A,$o),F2: fun(int,A),Xa: word(B)] :
          ( aa(A,$o,P,aa(word(B),A,aa(fun(int,A),fun(word(B),A),word_int_case(A,B),F2),Xa))
        <=> ! [I2: int] :
              ( ( ( Xa = aa(int,word(B),ring_1_of_int(word(B)),I2) )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I2)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(B,type2(B)))) )
             => aa(A,$o,P,aa(int,A,F2,I2)) ) ) ) ).

% word_int_split
tff(fact_6919_word__int__case__wi,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [F2: fun(int,A),I: int] : aa(word(B),A,aa(fun(int,A),fun(word(B),A),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,aa(int,fun(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_6920_smod__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed6721504322012087516modulo(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% smod_word_minus_numeral_minus_numeral
tff(fact_6921_smod__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed6721504322012087516modulo(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% smod_word_numeral_minus_numeral
tff(fact_6922_smod__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed6721504322012087516modulo(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% smod_word_numeral_numeral
tff(fact_6923_smod__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed6721504322012087516modulo(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% smod_word_minus_numeral_numeral
tff(fact_6924_word__smod__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num,Y: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed6721504322012087516modulo(word(A)),aa(num,word(A),numeral_numeral(word(A)),Xa)),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Xa))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_smod_numerals(1)
tff(fact_6925_word__smod__numerals__lhs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num,W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed6721504322012087516modulo(word(A)),aa(num,word(A),numeral_numeral(word(A)),Xa)),W) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Xa))),ring_1_signed(A,int,W))) ) ).

% word_smod_numerals_lhs(1)
tff(fact_6926_word__smod__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed6721504322012087516modulo(word(A)),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),ring_1_signed(A,int,one_one(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_smod_numerals(3)
tff(fact_6927_word__smod__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed6721504322012087516modulo(word(A)),aa(num,word(A),numeral_numeral(word(A)),Xa)),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Xa))),ring_1_signed(A,int,one_one(word(A))))) ) ).

% word_smod_numerals(7)
tff(fact_6928_word__smod__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed6721504322012087516modulo(word(A)),aa(num,word(A),numeral_numeral(word(A)),Xa)),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Xa))),ring_1_signed(A,int,zero_zero(word(A))))) ) ).

% word_smod_numerals(4)
tff(fact_6929_word__smod__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed6721504322012087516modulo(word(A)),zero_zero(word(A))),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),ring_1_signed(A,int,zero_zero(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_smod_numerals(2)
tff(fact_6930_uint__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),word_pred(A),A2)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),one_one(int))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(6)
tff(fact_6931_slice1__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Na: nat,W: word(B)] :
          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,aa(nat,fun(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,aa(nat,fun(nat,nat),minus_minus(nat),Na),type_len0_len_of(B,type2(B)))),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W))) ) ).

% slice1_def
tff(fact_6932_succ__pred__no_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] : aa(word(A),word(A),word_pred(A),aa(num,word(A),numeral_numeral(word(A)),W)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W)),one_one(word(A))) ) ).

% succ_pred_no(2)
tff(fact_6933_succ__pred__no_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] : aa(word(A),word(A),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),aa(word(A),fun(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_6934_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)),aa(word(A),word(A),word_pred(A),zero_zero(word(A)))),Y) ) ).

% word_not_simps(2)
tff(fact_6935_uint__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),word_succ(A),A2)) = 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),A2)),one_one(int))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(5)
tff(fact_6936_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,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(int,fun(word(A),word(A)),word_roti(A),K),W)),Na)
        <=> ( aa(nat,$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_6937_succ__pred__no_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] : aa(word(A),word(A),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_6938_succ__pred__no_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] : aa(word(A),word(A),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_6939_word__roti__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: int,Na: int,W: word(A)] : aa(word(A),word(A),aa(int,fun(word(A),word(A)),word_roti(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),M),Na)),W) = aa(word(A),word(A),aa(int,fun(word(A),word(A)),word_roti(A),M),aa(word(A),word(A),aa(int,fun(word(A),word(A)),word_roti(A),Na),W)) ) ).

% word_roti_add
tff(fact_6940_Abs__fnat__hom__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: nat] : aa(word(A),word(A),word_succ(A),aa(nat,word(A),semiring_1_of_nat(word(A)),A2)) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,suc,A2)) ) ).

% Abs_fnat_hom_Suc
tff(fact_6941_wi__hom__succ,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int] : aa(word(A),word(A),word_succ(A),aa(int,word(A),ring_1_of_int(word(A)),A2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),one_one(int))) ) ).

% wi_hom_succ
tff(fact_6942_word__arith__nat__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),word(A),word_succ(A),A2) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),A2))) ) ).

% word_arith_nat_Suc
tff(fact_6943_word__roti__conv__mod_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: int,W: word(A)] : aa(word(A),word(A),aa(int,fun(word(A),word(A)),word_roti(A),Na),W) = aa(word(A),word(A),aa(int,fun(word(A),word(A)),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_6944_word__succ__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),word(A),word_succ(A),A2) = 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),A2)),one_one(int))) ) ).

% word_succ_alt
tff(fact_6945_word__roti__conv__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: int,W: word(A)] : aa(word(A),word(A),aa(int,fun(word(A),word(A)),word_roti(A),Na),W) = aa(word(A),word(A),aa(int,fun(word(A),word(A)),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_6946_uint__word__arith__bintrs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),word_succ(A),A2)) = aa(int,int,aa(nat,fun(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),A2)),one_one(int))) ) ).

% uint_word_arith_bintrs(5)
tff(fact_6947_sint__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ring_1_signed(A,int,aa(word(A),word(A),word_succ(A),A2)) = aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,aa(int,fun(int,int),plus_plus(int),ring_1_signed(A,int,A2)),one_one(int))) ) ).

% sint_word_ariths(5)
tff(fact_6948_unat__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),word_succ(A),A2)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),A2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(3)
tff(fact_6949_sless__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% sless_eq_word_minus_numeral_minus_numeral
tff(fact_6950_sless__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% sless_eq_word_numeral_minus_numeral
tff(fact_6951_word__sle__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),A2))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),B2))) ) ) ).

% word_sle_no
tff(fact_6952_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),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Na))),ring_1_signed(A,int,one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(6)
tff(fact_6953_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),ring_1_signed(A,int,one_one(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Na))) ) ) ).

% extra_sle_sless_unfolds(4)
tff(fact_6954_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),ring_1_signed(A,int,zero_zero(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Na))) ) ) ).

% extra_sle_sless_unfolds(2)
tff(fact_6955_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),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Na))),ring_1_signed(A,int,zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(5)
tff(fact_6956_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),ring_1_signed(A,int,zero_zero(word(A)))),ring_1_signed(A,int,one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(1)
tff(fact_6957_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),ring_1_signed(A,int,one_one(word(A)))),ring_1_signed(A,int,zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(3)
tff(fact_6958_sless__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% sless_eq_word_numeral_numeral
tff(fact_6959_sless__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% sless_eq_word_minus_numeral_numeral
tff(fact_6960_signed_Ofinite__has__minimal,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] :
          ( finite_finite2(word(A),A3)
         => ( ( A3 != bot_bot(set(word(A))) )
           => ? [X: word(A)] :
                ( member(word(A),X,A3)
                & ! [Xa2: word(A)] :
                    ( member(word(A),Xa2,A3)
                   => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa2),X)
                     => ( X = Xa2 ) ) ) ) ) ) ) ).

% signed.finite_has_minimal
tff(fact_6961_signed_Ofinite__has__maximal,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] :
          ( finite_finite2(word(A),A3)
         => ( ( A3 != bot_bot(set(word(A))) )
           => ? [X: word(A)] :
                ( member(word(A),X,A3)
                & ! [Xa2: word(A)] :
                    ( member(word(A),Xa2,A3)
                   => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Xa2)
                     => ( X = Xa2 ) ) ) ) ) ) ) ).

% signed.finite_has_maximal
tff(fact_6962_signed_Ofinite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [S: set(A),P: fun(set(A),$o),F2: fun(A,word(B))] :
          ( finite_finite2(A,S)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [X: A,S4: set(A)] :
                  ( finite_finite2(A,S4)
                 => ( ! [Y2: A] :
                        ( member(A,Y2,S4)
                       => aa(word(B),$o,aa(word(B),fun(word(B),$o),word_sle(B),aa(A,word(B),F2,Y2)),aa(A,word(B),F2,X)) )
                   => ( aa(set(A),$o,P,S4)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S4)) ) ) )
             => aa(set(A),$o,P,S) ) ) ) ) ).

% signed.finite_ranking_induct
tff(fact_6963_word__0__sle__from__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),zero_zero(word(A))),Xa) ) ) ).

% word_0_sle_from_less
tff(fact_6964_sless__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% sless_word_minus_numeral_numeral
tff(fact_6965_sless__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% sless_word_numeral_minus_numeral
tff(fact_6966_word__sless__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
            & ( aa(num,word(A),numeral_numeral(word(A)),A2) != aa(num,word(A),numeral_numeral(word(A)),B2) ) ) ) ) ).

% word_sless_no
tff(fact_6967_extra__sle__sless__unfolds_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(num,word(A),numeral_numeral(word(A)),Na)),one_one(word(A)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Na))),ring_1_signed(A,int,one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(12)
tff(fact_6968_extra__sle__sless__unfolds_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,one_one(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Na))) ) ) ).

% extra_sle_sless_unfolds(10)
tff(fact_6969_extra__sle__sless__unfolds_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),zero_zero(word(A))),aa(num,word(A),numeral_numeral(word(A)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,zero_zero(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Na))) ) ) ).

% extra_sle_sless_unfolds(8)
tff(fact_6970_extra__sle__sless__unfolds_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(num,word(A),numeral_numeral(word(A)),Na)),zero_zero(word(A)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Na))),ring_1_signed(A,int,zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(11)
tff(fact_6971_extra__sle__sless__unfolds_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),one_one(word(A))),zero_zero(word(A)))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,one_one(word(A)))),ring_1_signed(A,int,zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(9)
tff(fact_6972_extra__sle__sless__unfolds_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),zero_zero(word(A))),one_one(word(A)))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,zero_zero(word(A)))),ring_1_signed(A,int,one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(7)
tff(fact_6973_sless__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% sless_word_numeral_numeral
tff(fact_6974_sless__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% sless_word_minus_numeral_minus_numeral
tff(fact_6975_word__sless__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,A2)),ring_1_signed(A,int,B2)) ) ) ).

% word_sless_alt
tff(fact_6976_signed_Ofinite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),P: fun(set(word(A)),$o)] :
          ( finite_finite2(word(A),A3)
         => ( aa(set(word(A)),$o,P,bot_bot(set(word(A))))
           => ( ! [B6: word(A),A7: set(word(A))] :
                  ( finite_finite2(word(A),A7)
                 => ( ! [X5: word(A)] :
                        ( member(word(A),X5,A7)
                       => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B6),X5) )
                   => ( aa(set(word(A)),$o,P,A7)
                     => aa(set(word(A)),$o,P,aa(set(word(A)),set(word(A)),aa(word(A),fun(set(word(A)),set(word(A))),insert(word(A)),B6),A7)) ) ) )
             => aa(set(word(A)),$o,P,A3) ) ) ) ) ).

% signed.finite_linorder_min_induct
tff(fact_6977_signed_Ofinite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A)),P: fun(set(word(A)),$o)] :
          ( finite_finite2(word(A),A3)
         => ( aa(set(word(A)),$o,P,bot_bot(set(word(A))))
           => ( ! [B6: word(A),A7: set(word(A))] :
                  ( finite_finite2(word(A),A7)
                 => ( ! [X5: word(A)] :
                        ( member(word(A),X5,A7)
                       => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X5),B6) )
                   => ( aa(set(word(A)),$o,P,A7)
                     => aa(set(word(A)),$o,P,aa(set(word(A)),set(word(A)),aa(word(A),fun(set(word(A)),set(word(A))),insert(word(A)),B6),A7)) ) ) )
             => aa(set(word(A)),$o,P,A3) ) ) ) ) ).

% signed.finite_linorder_max_induct
tff(fact_6978_signed_Oinfinite__growing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X7: set(word(A))] :
          ( ( X7 != bot_bot(set(word(A))) )
         => ( ! [X: word(A)] :
                ( member(word(A),X,X7)
               => ? [Xa2: word(A)] :
                    ( member(word(A),Xa2,X7)
                    & aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Xa2) ) )
           => ~ finite_finite2(word(A),X7) ) ) ) ).

% signed.infinite_growing
tff(fact_6979_finite__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: set(word(A))] : finite_finite2(word(A),A3) ) ).

% finite_word
tff(fact_6980_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_6981_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_6982_word__sless__sint__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,Xa)),aa(int,int,aa(int,fun(int,int),minus_minus(int),ring_1_signed(A,int,Y)),one_one(int))) ) ) ).

% word_sless_sint_le
tff(fact_6983_take__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_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,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),aa(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_6984_take__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se2584673776208193580ke_bit(word(A)),aa(nat,nat,suc,Na)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,Na))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))) ) ).

% take_bit_word_Suc_minus_numeral
tff(fact_6985_min_Oright__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ).

% min.right_idem
tff(fact_6986_min_Oleft__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ).

% min.left_idem
tff(fact_6987_min_Oidem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A2),A2) = A2 ) ).

% min.idem
tff(fact_6988_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% min.bounded_iff
tff(fact_6989_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_6990_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb1
tff(fact_6991_min__arg__not__ge_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),M),Na)),M)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),Na) = M ) ) ) ).

% min_arg_not_ge(1)
tff(fact_6992_min__arg__not__ge_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),M),Na)),Na)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),Na) = Na ) ) ) ).

% min_arg_not_ge(2)
tff(fact_6993_min__less__self__conv_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% min_less_self_conv(1)
tff(fact_6994_min__less__self__conv_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% min_less_self_conv(2)
tff(fact_6995_min__simps_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min_simps(1)
tff(fact_6996_min__simps_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min_simps(2)
tff(fact_6997_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Y) ) ) ) ).

% min_less_iff_conj
tff(fact_6998_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb4
tff(fact_6999_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb3
tff(fact_7000_min__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),bot_bot(A)) = bot_bot(A) ) ).

% min_bot2
tff(fact_7001_min__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),bot_bot(A)),Xa) = bot_bot(A) ) ).

% min_bot
tff(fact_7002_min__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),top_top(A)) = Xa ) ).

% min_top2
tff(fact_7003_min__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),top_top(A)),Xa) = Xa ) ).

% min_top
tff(fact_7004_min__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na)) ).

% min_Suc_Suc
tff(fact_7005_min__0L,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),Na) = zero_zero(nat) ).

% min_0L
tff(fact_7006_min__0R,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Na),zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_7007_max__min__same_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)) = Xa ) ).

% max_min_same(1)
tff(fact_7008_max__min__same_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),Xa) = Xa ) ).

% max_min_same(2)
tff(fact_7009_max__min__same_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),Y) = Y ) ).

% max_min_same(3)
tff(fact_7010_max__min__same_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xa: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Y),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)) = Y ) ).

% max_min_same(4)
tff(fact_7011_min__minus,axiom,
    ! [M: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K) ).

% min_minus
tff(fact_7012_min__minus_H,axiom,
    ! [M: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)),M) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K) ).

% min_minus'
tff(fact_7013_take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na)),A2) ) ).

% take_bit_take_bit
tff(fact_7014_signed__take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),Na),A2)) = aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na)),A2) ) ).

% signed_take_bit_signed_take_bit
tff(fact_7015_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Xa)) = zero_zero(A) ) ).

% min_0_1(3)
tff(fact_7016_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),Xa)),zero_zero(A)) = zero_zero(A) ) ).

% min_0_1(4)
tff(fact_7017_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),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_7018_min__0__1_I2_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),zero_zero(A)) = zero_zero(A) ) ) ).

% min_0_1(2)
tff(fact_7019_min__0__1_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),one_one(A)) = zero_zero(A) ) ) ).

% min_0_1(1)
tff(fact_7020_min__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),aa(num,A,numeral_numeral(A),Xa)) = one_one(A) ) ).

% min_0_1(5)
tff(fact_7021_min__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),Xa)),one_one(A)) = one_one(A) ) ).

% min_0_1(6)
tff(fact_7022_min__Suc__gt_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),B2),aa(nat,nat,suc,A2)) = aa(nat,nat,suc,A2) ) ) ).

% min_Suc_gt(2)
tff(fact_7023_min__Suc__gt_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,A2)),B2) = aa(nat,nat,suc,A2) ) ) ).

% min_Suc_gt(1)
tff(fact_7024_min__pm,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),A2),B2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) = A2 ).

% min_pm
tff(fact_7025_min__pm1,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),A2),B2)) = A2 ).

% min_pm1
tff(fact_7026_rev__min__pm,axiom,
    ! [B2: nat,A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),B2),A2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) = A2 ).

% rev_min_pm
tff(fact_7027_rev__min__pm1,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),B2),A2)) = A2 ).

% rev_min_pm1
tff(fact_7028_take__bit__of__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(nat,A,bit_se2239418461657761734s_mask(A),Na)) = aa(nat,A,bit_se2239418461657761734s_mask(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na)) ) ).

% take_bit_of_mask
tff(fact_7029_min__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_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_7030_min__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_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_7031_min__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_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_7032_min__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(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_min(nat),pred_numeral(K)),Na)) ).

% min_numeral_Suc
tff(fact_7033_min__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(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_min(nat),Na),pred_numeral(K))) ).

% min_Suc_numeral
tff(fact_7034_Min__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),lattic643756798350308766er_Min(A,A3)) ) ) ) ) ).

% Min_insert
tff(fact_7035_take__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se2584673776208193580ke_bit(word(A)),aa(nat,nat,suc,Na)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,Na))),aa(num,int,numeral_numeral(int),K))) ) ).

% take_bit_word_Suc_numeral
tff(fact_7036_take__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_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,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),aa(num,nat,numeral_numeral(nat),Na))),aa(num,int,numeral_numeral(int),K))) ) ).

% take_bit_word_numeral
tff(fact_7037_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(aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na),K),aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na),L),R)) ).

% concat_bit_assoc_sym
tff(fact_7038_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z)) ) ).

% min_diff_distrib_left
tff(fact_7039_min__diff,axiom,
    ! [M: nat,I: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),I)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),I)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na)),I) ).

% min_diff
tff(fact_7040_of__int__min,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: int,Y: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_min(int),Xa),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(int,A,ring_1_of_int(A),Xa)),aa(int,A,ring_1_of_int(A),Y)) ) ).

% of_int_min
tff(fact_7041_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z) ) ) ) ).

% min_less_iff_disj
tff(fact_7042_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% min.strict_boundedE
tff(fact_7043_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% min.strict_order_iff
tff(fact_7044_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2) ) ) ).

% min.strict_coboundedI1
tff(fact_7045_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2) ) ) ).

% min.strict_coboundedI2
tff(fact_7046_mask__and__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: nat,B2: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),A2)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),B2)) = aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),A2),B2)) ) ).

% mask_and_mask
tff(fact_7047_mask__twice,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat,M: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na))) ) ).

% mask_twice
tff(fact_7048_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Xa),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_min
tff(fact_7049_min_Oleft__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),B2),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ).

% min.left_commute
tff(fact_7050_min_Ocommute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = aa(A,A,aa(A,fun(A,A),ord_min(A),B2),A2) ) ).

% min.commute
tff(fact_7051_min_Oassoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ).

% min.assoc
tff(fact_7052_min__max__distrib2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A2),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),C2)) ) ).

% min_max_distrib2
tff(fact_7053_min__max__distrib1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),A2)),aa(A,A,aa(A,fun(A,A),ord_min(A),C2),A2)) ) ).

% min_max_distrib1
tff(fact_7054_max__min__distrib2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C2)) ) ).

% max_min_distrib2
tff(fact_7055_max__min__distrib1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),A2)),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),A2)) ) ).

% max_min_distrib1
tff(fact_7056_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xa: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_max_eq_min
tff(fact_7057_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xa: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_min_eq_max
tff(fact_7058_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% min_add_distrib_left
tff(fact_7059_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),ord_min(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z)) ) ).

% min_add_distrib_right
tff(fact_7060_nat__mult__min__left,axiom,
    ! [M: nat,Na: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q2)) ).

% nat_mult_min_left
tff(fact_7061_nat__mult__min__right,axiom,
    ! [M: nat,Na: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Na),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)) ).

% nat_mult_min_right
tff(fact_7062_inf__min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( inf_inf(A) = ord_min(A) ) ) ).

% inf_min
tff(fact_7063_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_7064_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X5: A,Xa2: A] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),X5),Xa2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Xa2),X5,Xa2) ) ).

% min_def_raw
tff(fact_7065_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% min_le_iff_disj
tff(fact_7066_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2) ) ) ).

% min.coboundedI2
tff(fact_7067_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2) ) ) ).

% min.coboundedI1
tff(fact_7068_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb_iff2
tff(fact_7069_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb_iff1
tff(fact_7070_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),B2) ) ).

% min.cobounded2
tff(fact_7071_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),A2) ) ).

% min.cobounded1
tff(fact_7072_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ) ) ).

% min.order_iff
tff(fact_7073_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ) ) ).

% min.boundedI
tff(fact_7074_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% min.boundedE
tff(fact_7075_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% min.orderI
tff(fact_7076_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ) ) ).

% min.orderE
tff(fact_7077_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),C2),D2)) ) ) ) ).

% min.mono
tff(fact_7078_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y) = Y ) ) ) ).

% min_absorb2
tff(fact_7079_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y) = Xa ) ) ) ).

% min_absorb1
tff(fact_7080_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2),A2,B2) ) ).

% min_def
tff(fact_7081_take__bit__concat__bit__eq,axiom,
    ! [M: nat,Na: nat,K: int,L: int] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),M),aa(int,int,bit_concat_bit(Na,K),L)) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na),K),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),L)) ).

% take_bit_concat_bit_eq
tff(fact_7082_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: 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_min(A),Xa),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_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),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),Xa),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2))) ) ).

% min_mult_distrib_right
tff(fact_7083_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: 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),Xa),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),Xa),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2))) ) ).

% max_mult_distrib_right
tff(fact_7084_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,Xa: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Xa)),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),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y))) ) ).

% min_mult_distrib_left
tff(fact_7085_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,Xa: 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),Xa),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),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y))) ) ).

% max_mult_distrib_left
tff(fact_7086_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: 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),Xa),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),Xa),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2))) ) ).

% max_divide_distrib_right
tff(fact_7087_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: 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_min(A),Xa),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_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),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),Xa),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2))) ) ).

% min_divide_distrib_right
tff(fact_7088_hom__Min__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,H,X)),aa(A,A,H,Y3))
         => ( finite_finite2(A,N3)
           => ( ( N3 != bot_bot(set(A)) )
             => ( aa(A,A,H,lattic643756798350308766er_Min(A,N3)) = lattic643756798350308766er_Min(A,aa(set(A),set(A),image(A,A,H),N3)) ) ) ) ) ) ).

% hom_Min_commute
tff(fact_7089_Min_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( B3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
             => ( aa(A,A,aa(A,fun(A,A),ord_min(A),lattic643756798350308766er_Min(A,B3)),lattic643756798350308766er_Min(A,A3)) = lattic643756798350308766er_Min(A,A3) ) ) ) ) ) ).

% Min.subset
tff(fact_7090_Min_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X: A,Y3: A] : member(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A)))))
             => member(A,lattic643756798350308766er_Min(A,A3),A3) ) ) ) ) ).

% Min.closed
tff(fact_7091_Min_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( ~ member(A,Xa,A3)
           => ( ( A3 != bot_bot(set(A)) )
             => ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),lattic643756798350308766er_Min(A,A3)) ) ) ) ) ) ).

% Min.insert_not_elem
tff(fact_7092_Min_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( finite_finite2(A,B3)
             => ( ( B3 != bot_bot(set(A)) )
               => ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),lattic643756798350308766er_Min(A,A3)),lattic643756798350308766er_Min(A,B3)) ) ) ) ) ) ) ).

% Min.union
tff(fact_7093_Min_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) = finite_fold(A,A,ord_min(A),Xa,A3) ) ) ) ).

% Min.eq_fold
tff(fact_7094_mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: 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),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na))) ) ).

% mod_exp_eq
tff(fact_7095_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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na))) ).

% mod_mod_power
tff(fact_7096_Min_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A3)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))))) ) ) ) ).

% Min.insert_remove
tff(fact_7097_Min_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xa: A] :
          ( finite_finite2(A,A3)
         => ( member(A,Xa,A3)
           => ( lattic643756798350308766er_Min(A,A3) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))))) ) ) ) ) ).

% Min.remove
tff(fact_7098_Word_Obit__mask__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M)),Na)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),M)) ) ) ).

% Word.bit_mask_iff
tff(fact_7099_unat__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = aa(nat,nat,bit_se2239418461657761734s_mask(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),Na)) ) ).

% unat_mask_eq
tff(fact_7100_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,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na))),one_one(A)) ) ).

% mask_mod_exp
tff(fact_7101_Min_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : lattic643756798350308766er_Min(A,A3) = the2(A,finite_fold(A,option(A),aTP_Lamp_tg(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Min.eq_fold'
tff(fact_7102_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,$o),bit_se5641148757651400278ts_bit(word(A)),slice2(B,A,M,W)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),M)))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),type_len0_len_of(B,type2(B)))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),M))) ) ) ) ).

% bit_slice_iff
tff(fact_7103_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,$o),bit_se5641148757651400278ts_bit(word(A)),slice1(B,A,M,W)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),type_len0_len_of(B,type2(B)))),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),M))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),M))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),type_len0_len_of(B,type2(B))))) ) ) ) ).

% bit_slice1_iff
tff(fact_7104_unat__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Na),type_len0_len_of(A,type2(A))))),one_one(nat)) ) ).

% unat_mask
tff(fact_7105_bit__horner__sum__bit__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bs: list($o),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(list($o),word(A),groups4207007520872428315er_sum($o,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Bs)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),aa(list($o),nat,size_size(list($o)),Bs)))
            & aa(nat,$o,nth($o,Bs),Na) ) ) ) ).

% bit_horner_sum_bit_word_iff
tff(fact_7106_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] :
          aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
                & bit_se6407376104438227557le_bit(A,type2(A),Na) ))),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))) ) ).

% drop_bit_exp_eq
tff(fact_7107_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_7108_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_min(extended_enat),Q2),zero_zero(extended_enat)) = zero_zero(extended_enat) ).

% min_enat_simps(2)
tff(fact_7109_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_min(extended_enat),zero_zero(extended_enat)),Q2) = zero_zero(extended_enat) ).

% min_enat_simps(3)
tff(fact_7110_possible__bit__min,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I),J))
        <=> ( bit_se6407376104438227557le_bit(A,Tyrep,I)
            | bit_se6407376104438227557le_bit(A,Tyrep,J) ) ) ) ).

% possible_bit_min
tff(fact_7111_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_7112_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_7113_inf__int__def,axiom,
    inf_inf(int) = ord_min(int) ).

% inf_int_def
tff(fact_7114_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_7115_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_7116_bit__imp__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
         => bit_se6407376104438227557le_bit(A,type2(A),Na) ) ) ).

% bit_imp_possible_bit
tff(fact_7117_impossible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat,A2: A] :
          ( ~ bit_se6407376104438227557le_bit(A,type2(A),Na)
         => ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na) ) ) ).

% impossible_bit
tff(fact_7118_bit__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ! [N2: nat] :
              ( bit_se6407376104438227557le_bit(A,type2(A),N2)
             => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),N2)
              <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B2),N2) ) ) ) ) ).

% bit_eq_iff
tff(fact_7119_bit__eqI,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] :
          ( ! [N: nat] :
              ( bit_se6407376104438227557le_bit(A,type2(A),N)
             => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),N)
              <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B2),N) ) )
         => ( A2 = B2 ) ) ) ).

% bit_eqI
tff(fact_7120_possible__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] : bit_se6407376104438227557le_bit(A,type2(A),Na) ) ).

% possible_bit
tff(fact_7121_bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na) ) ) ) ).

% bit_not_iff
tff(fact_7122_semiring__bit__operations__class_Obit__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)),Na)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
            | ( ( M = Na )
              & bit_se6407376104438227557le_bit(A,type2(A),Na) ) ) ) ) ).

% semiring_bit_operations_class.bit_set_bit_iff
tff(fact_7123_bit__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),M),A2)),Na)
        <=> ( ( ( M = Na )
            <=> ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na) )
            & bit_se6407376104438227557le_bit(A,type2(A),Na) ) ) ) ).

% bit_flip_bit_iff
tff(fact_7124_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(nat,A,bit_se2239418461657761734s_mask(A),M)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ).

% bit_mask_iff
tff(fact_7125_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_7126_bit__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_ri4674362597316999326ke_bit(A),M),A2)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na)) ) ) ) ).

% bit_signed_take_bit_iff
tff(fact_7127_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_7128_possible__bit__def,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),Na: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,Na)
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) ) ) ) ).

% possible_bit_def
tff(fact_7129_bit__minus__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),A2)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),Na) ) ) ) ).

% bit_minus_iff
tff(fact_7130_bit__twiddle__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Y),
            aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y)),
              $ite(aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),zero_zero(word(A))))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),ord_min(word(A)),Xa),Y) ) ).

% bit_twiddle_min
tff(fact_7131_bit__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),A2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
            & bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) ) ) ) ).

% bit_push_bit_iff
tff(fact_7132_fold__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
        <=> ~ bit_se6407376104438227557le_bit(A,type2(A),Na) ) ) ).

% fold_possible_bit
tff(fact_7133_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,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & ( M = Na ) ) ) ) ).

% bit_exp_iff
tff(fact_7134_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_7135_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,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & ( Na != M ) ) ) ) ).

% bit_not_exp_iff
tff(fact_7136_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% bit_minus_exp_iff
tff(fact_7137_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ).

% bit_mask_sub_iff
tff(fact_7138_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),Na)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))
            & ( Na != zero_zero(nat) )
            & bit_se6407376104438227557le_bit(A,type2(A),Na) ) ) ) ).

% bit_double_iff
tff(fact_7139_word__set__bits__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o)] : bit_bi4170147762399595738t_bits(word(A),P) = aa(list($o),word(A),groups4207007520872428315er_sum($o,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),map(nat,$o,P,upt(zero_zero(nat),type_len0_len_of(A,type2(A))))) ) ).

% word_set_bits_def
tff(fact_7140_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),aa(nat,fun(word(A),word(A)),word_rotr(A),Na),W)) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),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),aa(nat,fun(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_7141_set__bits__False__eq,axiom,
    ! [A: $tType] :
      ( bit_bi6583157726757044596ension(A)
     => ( bit_bi4170147762399595738t_bits(A,aTP_Lamp_th(nat,$o)) = zero_zero(A) ) ) ).

% set_bits_False_eq
tff(fact_7142_set__bits__K__False,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( bit_bi4170147762399595738t_bits(word(A),aTP_Lamp_th(nat,$o)) = zero_zero(word(A)) ) ) ).

% set_bits_K_False
tff(fact_7143_inf__enat__def,axiom,
    inf_inf(extended_enat) = ord_min(extended_enat) ).

% inf_enat_def
tff(fact_7144_word__rotr__word__rotr__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat,W: word(A)] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),word_rotr(A),M),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),word_rotr(A),Na),W)) = aa(word(A),word(A),aa(nat,fun(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_7145_bit__set__bits__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_bi4170147762399595738t_bits(word(A),P)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,P,Na) ) ) ) ).

% bit_set_bits_word_iff
tff(fact_7146_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_7147_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_7148_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,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),word_rotr(A),M),W)),Na)
        <=> ( aa(nat,$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_7149_word__roti__eq__word__rotr__word__rotl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int,W: word(A)] :
          aa(word(A),word(A),aa(int,fun(word(A),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),aa(nat,fun(word(A),word(A)),word_rotr(A),aa(int,nat,nat2,I)),W),aa(word(A),word(A),aa(nat,fun(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_7150_word__roti_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : aa(word(A),word(A),aa(int,fun(word(A),word(A)),word_roti(A),Xa),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(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),Xa),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),Xa),aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Xb))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xa),aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))),Xb))) ) ).

% word_roti.abs_eq
tff(fact_7151_int__set__bits__K__False,axiom,
    bit_bi4170147762399595738t_bits(int,aTP_Lamp_th(nat,$o)) = zero_zero(int) ).

% int_set_bits_K_False
tff(fact_7152_int__set__bits__K__True,axiom,
    bit_bi4170147762399595738t_bits(int,aTP_Lamp_ti(nat,$o)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% int_set_bits_K_True
tff(fact_7153_times__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),times_times(int),Xa),Xb)) ) ).

% times_word.abs_eq
tff(fact_7154_plus__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Xb)) ) ).

% plus_word.abs_eq
tff(fact_7155_word__succ_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] : aa(word(A),word(A),word_succ(A),aa(int,word(A),word2(A),Xa)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),one_one(int))) ) ).

% word_succ.abs_eq
tff(fact_7156_word__rotl_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Xb: int] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),word_rotl(A),Xa),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),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),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),type_len0_len_of(A,type2(A))))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Xb))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),type_len0_len_of(A,type2(A))))),Xb))) ) ).

% word_rotl.abs_eq
tff(fact_7157_less__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Xa)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Xb)) ) ) ).

% less_word.abs_eq
tff(fact_7158_bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,X5: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),word2(A),Xa)),X5)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X5),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),Xa),X5) ) ) ) ).

% bit_word.abs_eq
tff(fact_7159_divide__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Xa)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Xb))) ) ).

% divide_word.abs_eq
tff(fact_7160_modulo__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Xa)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Xb))) ) ).

% modulo_word.abs_eq
tff(fact_7161_unsigned_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [Xa: int] : aa(word(B),A,semiring_1_unsigned(B,A),aa(int,word(B),word2(B),Xa)) = aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(B,type2(B))),Xa))) ) ).

% unsigned.abs_eq
tff(fact_7162_word__rotl__eq__word__rotr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa(nat,fun(word(A),word(A)),word_rotl(A),Na) = aa(nat,fun(word(A),word(A)),word_rotr(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),type_len0_len_of(A,type2(A))))) ) ).

% word_rotl_eq_word_rotr
tff(fact_7163_word__sless_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(int,word(A),word2(A),Xa)),aa(int,word(A),word2(A),Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xa)),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xb)) ) ) ).

% word_sless.abs_eq
tff(fact_7164_word__rotr_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Xb: int] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),word_rotr(A),Xa),aa(int,word(A),word2(A),Xb)) = aa(int,word(A),word2(A),aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),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),Xa),type_len0_len_of(A,type2(A)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Xb))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),type_len0_len_of(A,type2(A)))),Xb))) ) ).

% word_rotr.abs_eq
tff(fact_7165_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,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),word_rotl(A),M),W)),Na)
        <=> ( aa(nat,$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,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),type_len0_len_of(A,type2(A)))))),type_len0_len_of(A,type2(A)))) ) ) ) ).

% bit_word_rotl_iff
tff(fact_7166_of__nat_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat] : aa(nat,word(A),of_nat(A),Xa) = aa(int,word(A),word2(A),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),type_len0_len_of(A,type2(A))),Xa))) ) ).

% of_nat.abs_eq
tff(fact_7167_the__nat_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] : aa(word(A),nat,the_nat(A),aa(int,word(A),word2(A),Xa)) = aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Xa)) ) ).

% the_nat.abs_eq
tff(fact_7168_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_7169_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_7170_wf__set__bits__int__Suc,axiom,
    ! [F2: fun(nat,$o)] :
      ( bit_wf_set_bits_int(aTP_Lamp_tj(fun(nat,$o),fun(nat,$o),F2))
    <=> bit_wf_set_bits_int(F2) ) ).

% wf_set_bits_int_Suc
tff(fact_7171_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_7172_wf__set__bits__int__const,axiom,
    ! [B2: $o] : bit_wf_set_bits_int(aTP_Lamp_tk($o,fun(nat,$o),(B2))) ).

% wf_set_bits_int_const
tff(fact_7173_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_7174_CHAR__eqI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
         => ( ! [X: nat] :
                ( ( aa(nat,A,semiring_1_of_nat(A),X) = zero_zero(A) )
               => aa(nat,$o,dvd_dvd(nat,C2),X) )
           => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ).

% CHAR_eqI
tff(fact_7175_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_7176_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
         => ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
           => ( ! [X: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),X)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),C2)
                   => ( aa(nat,A,semiring_1_of_nat(A),X) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_7177_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_7178_bit__sshiftr__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: nat,Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),bit_Sh8784991116023147202shiftr(A),W),M)),Na)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),
              $ite(
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),M)),Na)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ),
                aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat)),
                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) )) ) ) ).

% bit_sshiftr_iff
tff(fact_7179_sshiftr__numeral__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Na: nat] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),bit_Sh8784991116023147202shiftr(A),aa(num,word(A),numeral_numeral(word(A)),M)),aa(nat,nat,suc,Na)) = aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),aa(nat,nat,suc,Na)),aa(num,word(A),numeral_numeral(word(A)),M)) ) ).

% sshiftr_numeral_Suc
tff(fact_7180_sshiftr__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Na: num] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),bit_Sh8784991116023147202shiftr(A),aa(num,word(A),numeral_numeral(word(A)),M)),aa(num,nat,numeral_numeral(nat),Na)) = aa(word(A),word(A),aa(nat,fun(word(A),word(A)),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_7181_sshiftr__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Na: nat] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),bit_Sh8784991116023147202shiftr(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M))),aa(nat,nat,suc,Na)) = aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),aa(nat,nat,suc,Na)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M))) ) ).

% sshiftr_minus_numeral_Suc
tff(fact_7182_sshiftr__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Na: num] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),bit_Sh8784991116023147202shiftr(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M))),aa(num,nat,numeral_numeral(nat),Na)) = aa(word(A),word(A),aa(nat,fun(word(A),word(A)),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_7183_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_7184_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),A3: set(C)] :
          ( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
         => ( ! [X: B,Y3: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X)),aa(B,A,H,Y3))
           => ( aa(set(C),A,groups7311177749621191930dd_sum(C,A,comp(B,A,C,H,G)),A3) = aa(B,A,H,aa(set(C),B,groups7311177749621191930dd_sum(C,B,G),A3)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_7185_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A2)) = comp(nat,$o,nat,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,fun(nat,nat),plus_plus(nat),Na)) ) ).

% bit_drop_bit_eq
tff(fact_7186_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_7187_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_7188_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_7189_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_7190_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_tl(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_inverse_divide
tff(fact_7191_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_7192_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_7193_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_7194_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_7195_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_7196_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_7197_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_sm(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_7198_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_sm(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_7199_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_sm(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_7200_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_sm(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_7201_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_7202_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,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_7203_shiftl__Suc__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_Sh4282982442137083160shiftl(nat),aa(nat,nat,suc,zero_zero(nat))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ).

% shiftl_Suc_0
tff(fact_7204_shiftl__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: num] : aa(nat,A,aa(A,fun(nat,A),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),aa(nat,A,bit_se2239418461657761734s_mask(A),aa(num,nat,numeral_numeral(nat),Na))) ) ).

% shiftl_minus_1_numeral
tff(fact_7205_shiftl__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,Na: nat] : aa(nat,A,aa(A,fun(nat,A),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_7206_shiftl__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,Na: num] : aa(nat,A,aa(A,fun(nat,A),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_7207_shiftl__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,Na: nat] : aa(nat,A,aa(A,fun(nat,A),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_7208_shiftl__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,Na: num] : aa(nat,A,aa(A,fun(nat,A),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_7209_shiftl__of__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),bit_Sh4282982442137083160shiftl(A),A2),aa(nat,nat,suc,Na)) = aa(nat,A,aa(A,fun(nat,A),bit_Sh4282982442137083160shiftl(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),Na) ) ).

% shiftl_of_Suc
tff(fact_7210_shiftl__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(nat,A,aa(A,fun(nat,A),bit_Sh4282982442137083160shiftl(A),one_one(A)),Na) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ).

% shiftl_1
tff(fact_7211_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) = finite_fold(B,A,comp(A,fun(A,A),B,plus_plus(A),G),zero_zero(A),A3) ) ).

% sum.eq_fold
tff(fact_7212_prod_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A3: set(B)] : groups7121269368397514597t_prod(B,A,G,A3) = finite_fold(B,A,comp(A,fun(A,A),B,times_times(A),G),one_one(A),A3) ) ).

% prod.eq_fold
tff(fact_7213_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S: set(int)] :
      ( ~ finite_finite2(int,S)
    <=> ~ finite_finite2(nat,aa(set(int),set(nat),image(int,nat,comp(int,nat,int,nat2,abs_abs(int))),S)) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_7214_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_7215_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_7216_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_7217_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_7218_shiftl__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Xa: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),bit_Sh4282982442137083160shiftl(A),Xa),Na) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% shiftl_eq_mult
tff(fact_7219_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_7220_shiftr__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,Na: num] : aa(nat,A,aa(A,fun(nat,A),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_7221_shiftr__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,Na: nat] : aa(nat,A,aa(A,fun(nat,A),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_7222_shiftr__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,Na: num] : aa(nat,A,aa(A,fun(nat,A),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_7223_shiftr__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,Na: nat] : aa(nat,A,aa(A,fun(nat,A),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_7224_bit__shiftr__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] : aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(nat,A,aa(A,fun(nat,A),bit_Sh4282982442137083166shiftr(A),A2),Na)) = comp(nat,$o,nat,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,fun(nat,nat),plus_plus(nat),Na)) ) ).

% bit_shiftr_eq
tff(fact_7225_shiftr__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Xa: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),bit_Sh4282982442137083166shiftr(A),Xa),Na) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% shiftr_eq_div
tff(fact_7226_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,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A))))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))) ) ) ) ).

% bit_revcast_iff
tff(fact_7227_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,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))) ) ) ).

% word_msb_neg_numeral
tff(fact_7228_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,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))) ) ) ).

% word_msb_numeral
tff(fact_7229_word__msb__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),ring_1_signed(A,int,W)),zero_zero(int)) ) ) ).

% word_msb_sint
tff(fact_7230_word__sless__msb__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
        <=> ( ( most_s684356279273892711sb_msb(word(A),Y)
             => most_s684356279273892711sb_msb(word(A),Xa) )
            & ( ( most_s684356279273892711sb_msb(word(A),Xa)
                & ~ most_s684356279273892711sb_msb(word(A),Y) )
              | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ) ).

% word_sless_msb_less
tff(fact_7231_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),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))
         => ~ most_s684356279273892711sb_msb(word(A),V) ) ) ).

% not_msb_from_less
tff(fact_7232_word__sint__msb__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ring_1_signed(A,int,Xa) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),
            $ite(most_s684356279273892711sb_msb(word(A),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa)),zero_zero(int))) ) ).

% word_sint_msb_eq
tff(fact_7233_msb__big,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),A2)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))),A2) ) ) ).

% msb_big
tff(fact_7234_signed_Orsp,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => aa(fun(int,A),$o,aa(fun(int,A),fun(fun(int,A),$o),bNF_rel_fun(int,int,A,A,aTP_Lamp_tm(int,fun(int,$o)),fequal(A)),comp(int,A,int,ring_1_of_int(A),aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))))),comp(int,A,int,ring_1_of_int(A),aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% signed.rsp
tff(fact_7235_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,aa(int,fun(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_7236_inj__on__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : inj_on(A,B,F2,bot_bot(set(A))) ).

% inj_on_empty
tff(fact_7237_inj__mult__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A] :
          ( inj_on(A,A,aa(A,fun(A,A),times_times(A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_mult_left
tff(fact_7238_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( inj_on(A,A,aTP_Lamp_tn(A,fun(A,A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_7239_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A2: A,A3: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3))
    <=> ( inj_on(A,B,F2,A3)
        & ~ member(B,aa(A,B,F2,A2),aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))) ) ) ).

% inj_on_insert
tff(fact_7240_inj__on__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xs: list(A),Ys2: list(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set2(A,Xs)),set2(A,Ys2)))
     => ( ( map(A,B,F2,Xs) = map(A,B,F2,Ys2) )
      <=> ( Xs = Ys2 ) ) ) ).

% inj_on_map_eq_map
tff(fact_7241_map__inj__on,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),Ys2: list(B)] :
      ( ( map(B,A,F2,Xs) = map(B,A,F2,Ys2) )
     => ( inj_on(B,A,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set2(B,Xs)),set2(B,Ys2)))
       => ( Xs = Ys2 ) ) ) ).

% map_inj_on
tff(fact_7242_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(B,A),D4: set(B)] :
      ( finite_finite2(A,A3)
     => ( inj_on(B,A,F2,D4)
       => finite_finite2(B,collect(B,aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_to(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),A3),F2),D4))) ) ) ).

% finite_inverse_image_gen
tff(fact_7243_finite__Collect,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(B,A)] :
      ( finite_finite2(A,S)
     => ( inj_on(B,A,F2,top_top(set(B)))
       => finite_finite2(B,collect(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_tp(set(A),fun(fun(B,A),fun(B,$o)),S),F2))) ) ) ).

% finite_Collect
tff(fact_7244_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(B,A)] :
      ( finite_finite2(A,A3)
     => ( inj_on(B,A,F2,top_top(set(B)))
       => finite_finite2(B,collect(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_tp(set(A),fun(fun(B,A),fun(B,$o)),A3),F2))) ) ) ).

% finite_inverse_image
tff(fact_7245_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : inj_on(A,A,aTP_Lamp_oe(A,fun(A,A),A2),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_7246_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F2: fun(A,B)] :
          ( ! [X: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
             => ( aa(A,B,F2,X) != aa(A,B,F2,Y3) ) )
         => inj_on(A,B,F2,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_7247_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_tq(fun(A,B),fun(A,fun(C,B)),F2),top_top(set(A))) ) ).

% inj_fun
tff(fact_7248_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_tr(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_7249_inj__add__left,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A2),top_top(set(A))) ) ).

% inj_add_left
tff(fact_7250_power__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & power(A) )
     => ! [R2: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R2,one_one(A)),one_one(B))
         => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R2,bNF_rel_fun(A,B,A,B,R2,R2)),times_times(A)),times_times(B))
           => aa(fun(B,fun(nat,B)),$o,aa(fun(A,fun(nat,A)),fun(fun(B,fun(nat,B)),$o),bNF_rel_fun(A,B,fun(nat,A),fun(nat,B),R2,bNF_rel_fun(nat,nat,A,B,fequal(nat),R2)),power_power(A)),power_power(B)) ) ) ) ).

% power_transfer
tff(fact_7251_less__integer_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),fequal(int),bNF_rel_fun(int,int,$o,$o,fequal(int),fequal($o))),ord_less(int)),ord_less(int)) ).

% less_integer.rsp
tff(fact_7252_inj__on__id2,axiom,
    ! [A: $tType,A3: set(A)] : inj_on(A,A,aTP_Lamp_ma(A,A),A3) ).

% inj_on_id2
tff(fact_7253_natural__of__integer_Orsp,axiom,
    aa(fun(int,nat),$o,aa(fun(int,nat),fun(fun(int,nat),$o),bNF_rel_fun(int,int,nat,nat,fequal(int),fequal(nat)),nat2),nat2) ).

% natural_of_integer.rsp
tff(fact_7254_modulo__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),modulo_modulo(int)),modulo_modulo(int)) ).

% modulo_integer.rsp
tff(fact_7255_inj__on__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A)] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A2),A3) ) ).

% inj_on_add
tff(fact_7256_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A)] : inj_on(A,A,aTP_Lamp_ts(A,fun(A,A),A2),A3) ) ).

% inj_on_add'
tff(fact_7257_plus__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),plus_plus(int)),plus_plus(int)) ).

% plus_integer.rsp
tff(fact_7258_dup_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),aTP_Lamp_tt(int,int)),aTP_Lamp_tt(int,int)) ).

% dup.rsp
tff(fact_7259_transfer__rule__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ring_1(B)
        & ring_1(A) )
     => ! [R2: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R2,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R2,one_one(A)),one_one(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R2,bNF_rel_fun(A,B,A,B,R2,R2)),plus_plus(A)),plus_plus(B))
             => ( aa(fun(B,B),$o,aa(fun(A,A),fun(fun(B,B),$o),bNF_rel_fun(A,B,A,B,R2,R2),uminus_uminus(A)),uminus_uminus(B))
               => aa(fun(int,B),$o,aa(fun(int,A),fun(fun(int,B),$o),bNF_rel_fun(int,int,A,B,fequal(int),R2),ring_1_of_int(A)),ring_1_of_int(B)) ) ) ) ) ) ).

% transfer_rule_of_int
tff(fact_7260_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,A3: set(A)] :
          ( ( A2 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A2),A3) ) ) ).

% inj_on_mult
tff(fact_7261_times__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),times_times(int)),times_times(int)) ).

% times_integer.rsp
tff(fact_7262_transfer__rule__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & semiring_numeral(B)
        & monoid_add(A)
        & semiring_numeral(A) )
     => ! [R2: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R2,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R2,one_one(A)),one_one(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R2,bNF_rel_fun(A,B,A,B,R2,R2)),plus_plus(A)),plus_plus(B))
             => aa(fun(num,B),$o,aa(fun(num,A),fun(fun(num,B),$o),bNF_rel_fun(num,num,A,B,fequal(num),R2),numeral_numeral(A)),numeral_numeral(B)) ) ) ) ) ).

% transfer_rule_numeral
tff(fact_7263_divide__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),divide_divide(int)),divide_divide(int)) ).

% divide_integer.rsp
tff(fact_7264_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
             => ( member(A,X,A3)
               => ( member(A,Y3,A3)
                 => ( aa(A,B,F2,X) != aa(A,B,F2,Y3) ) ) ) )
         => ( ! [X: A,Y3: A] :
                ( member(A,X,A3)
               => ( member(A,Y3,A3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                    | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X) ) ) )
           => inj_on(A,B,F2,A3) ) ) ) ).

% linorder_inj_onI
tff(fact_7265_word__cat_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,int,int,aTP_Lamp_tm(int,fun(int,$o)),aTP_Lamp_tv(int,fun(int,$o)))),aTP_Lamp_tw(int,fun(int,int))),aTP_Lamp_tw(int,fun(int,int))) ).

% word_cat.rsp
tff(fact_7266_minus__word_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),minus_minus(int)),minus_minus(int)) ).

% minus_word.rsp
tff(fact_7267_xor__word_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),bit_se5824344971392196577ns_xor(int)),bit_se5824344971392196577ns_xor(int)) ).

% xor_word.rsp
tff(fact_7268_less__word_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,$o,$o,aTP_Lamp_tu(int,fun(int,$o)),fequal($o))),aTP_Lamp_tx(int,fun(int,$o))),aTP_Lamp_tx(int,fun(int,$o))) ).

% less_word.rsp
tff(fact_7269_or__word_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),bit_se1065995026697491101ons_or(int)),bit_se1065995026697491101ons_or(int)) ).

% or_word.rsp
tff(fact_7270_equal__word_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,$o,$o,aTP_Lamp_tu(int,fun(int,$o)),fequal($o))),aTP_Lamp_tu(int,fun(int,$o))),aTP_Lamp_tu(int,fun(int,$o))) ).

% equal_word.rsp
tff(fact_7271_udvd_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,$o,$o,aTP_Lamp_tu(int,fun(int,$o)),fequal($o))),aTP_Lamp_ty(int,fun(int,$o))),aTP_Lamp_ty(int,fun(int,$o))) ).

% udvd.rsp
tff(fact_7272_modulo__word_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),aTP_Lamp_tz(int,fun(int,int))),aTP_Lamp_tz(int,fun(int,int))) ).

% modulo_word.rsp
tff(fact_7273_and__word_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),bit_se5824344872417868541ns_and(int)),bit_se5824344872417868541ns_and(int)) ).

% and_word.rsp
tff(fact_7274_plus__word_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),plus_plus(int)),plus_plus(int)) ).

% plus_word.rsp
tff(fact_7275_times__word_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),times_times(int)),times_times(int)) ).

% times_word.rsp
tff(fact_7276_divide__word_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),aTP_Lamp_ua(int,fun(int,int))),aTP_Lamp_ua(int,fun(int,int))) ).

% divide_word.rsp
tff(fact_7277_less__eq__word_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,$o,$o,aTP_Lamp_tu(int,fun(int,$o)),fequal($o))),aTP_Lamp_ub(int,fun(int,$o))),aTP_Lamp_ub(int,fun(int,$o))) ).

% less_eq_word.rsp
tff(fact_7278_inj__on__Un__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
     => ( ( aa(set(A),set(B),image(A,B,F2),A3) = aa(set(A),set(B),image(A,B,F2),B3) )
      <=> ( A3 = B3 ) ) ) ).

% inj_on_Un_image_eq_iff
tff(fact_7279_prod_Oimage__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(A,B),A3: set(A)] :
          ( inj_on(A,B,G,A3)
         => ( groups7121269368397514597t_prod(B,B,aTP_Lamp_uc(B,B),aa(set(A),set(B),image(A,B,G),A3)) = groups7121269368397514597t_prod(A,B,G,A3) ) ) ) ).

% prod.image_eq
tff(fact_7280_inj__on__strict__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B3: set(A),A3: set(A)] :
      ( inj_on(A,B,F2,B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),aa(set(A),set(B),image(A,B,F2),B3)) ) ) ).

% inj_on_strict_subset
tff(fact_7281_inj__img__insertE,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),Xa: B,B3: set(B)] :
      ( inj_on(A,B,F2,A3)
     => ( ~ member(B,Xa,B3)
       => ( ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xa),B3) = aa(set(A),set(B),image(A,B,F2),A3) )
         => ~ ! [X8: A,A6: set(A)] :
                ( ~ member(A,X8,A6)
               => ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X8),A6) )
                 => ( ( Xa = aa(A,B,F2,X8) )
                   => ( B3 != aa(set(A),set(B),image(A,B,F2),A6) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_7282_sum_Oimage__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(A,B),A3: set(A)] :
          ( inj_on(A,B,G,A3)
         => ( aa(set(B),B,groups7311177749621191930dd_sum(B,B,aTP_Lamp_ud(B,B)),aa(set(A),set(B),image(A,B,G),A3)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) ) ) ) ).

% sum.image_eq
tff(fact_7283_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A3: set(A),A11: set(B)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ? [F6: fun(A,B)] :
            ( inj_on(A,B,F6,A3)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F6),A3)),A11) )
      <=> ? [G3: fun(B,A)] : aa(set(B),set(A),image(B,A,G3),A11) = A3 ) ) ).

% inj_on_iff_surj
tff(fact_7284_num__of__integer_Orsp,axiom,
    aa(fun(int,num),$o,aa(fun(int,num),fun(fun(int,num),$o),bNF_rel_fun(int,int,num,num,fequal(int),fequal(num)),comp(nat,num,int,num_of_nat,nat2)),comp(nat,num,int,num_of_nat,nat2)) ).

% num_of_integer.rsp
tff(fact_7285_size__word_Orsp,axiom,
    aa(fun(int,nat),$o,aa(fun(int,nat),fun(fun(int,nat),$o),bNF_rel_fun(int,int,nat,nat,aTP_Lamp_tu(int,fun(int,$o)),fequal(nat)),aTP_Lamp_ue(int,nat)),aTP_Lamp_ue(int,nat)) ).

% size_word.rsp
tff(fact_7286_the__int_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),fequal(int)),aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A)))),aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A)))) ) ).

% the_int.rsp
tff(fact_7287_Word_Oof__int_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),aTP_Lamp_tu(int,fun(int,$o))),aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A)))),aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A)))) ) ).

% Word.of_int.rsp
tff(fact_7288_cast_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tm(int,fun(int,$o))),aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A)))),aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A)))) ) ).

% cast.rsp
tff(fact_7289_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_7290_word__sle_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,$o,$o,aTP_Lamp_tu(int,fun(int,$o)),fequal($o))),aTP_Lamp_uf(int,fun(int,$o))),aTP_Lamp_uf(int,fun(int,$o))) ).

% word_sle.rsp
tff(fact_7291_word__sless_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,$o,$o,aTP_Lamp_tu(int,fun(int,$o)),fequal($o))),aTP_Lamp_ug(int,fun(int,$o))),aTP_Lamp_ug(int,fun(int,$o))) ).

% word_sless.rsp
tff(fact_7292_signed__modulo__word_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),aTP_Lamp_uh(int,fun(int,int))),aTP_Lamp_uh(int,fun(int,int))) ).

% signed_modulo_word.rsp
tff(fact_7293_signed__divide__word_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),aTP_Lamp_tu(int,fun(int,$o)),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),aTP_Lamp_ui(int,fun(int,int))),aTP_Lamp_ui(int,fun(int,int))) ).

% signed_divide_word.rsp
tff(fact_7294_bin__rest__integer_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),aTP_Lamp_uj(int,int)),aTP_Lamp_uj(int,int)) ).

% bin_rest_integer.rsp
tff(fact_7295_uminus__word_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o))),uminus_uminus(int)),uminus_uminus(int)) ).

% uminus_word.rsp
tff(fact_7296_bin__last__integer_Orsp,axiom,
    aa(fun(int,$o),$o,aa(fun(int,$o),fun(fun(int,$o),$o),bNF_rel_fun(int,int,$o,$o,fequal(int),fequal($o)),aTP_Lamp_rr(int,$o)),aTP_Lamp_rr(int,$o)) ).

% bin_last_integer.rsp
tff(fact_7297_inj__on__disjoint__Un,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),G: fun(A,B),B3: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( inj_on(A,B,G,B3)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),aa(set(A),set(B),image(A,B,G),B3)) = bot_bot(set(B)) )
         => inj_on(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_oo(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),A3),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_7298_not__word_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o))),bit_ri4277139882892585799ns_not(int)),bit_ri4277139882892585799ns_not(int)) ).

% not_word.rsp
tff(fact_7299_word__roti_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),aTP_Lamp_uk(int,fun(int,int))),aTP_Lamp_uk(int,fun(int,int))) ).

% word_roti.rsp
tff(fact_7300_inj__on__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
    <=> ( inj_on(A,B,F2,A3)
        & inj_on(A,B,F2,B3)
        & ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3))) = bot_bot(set(B)) ) ) ) ).

% inj_on_Un
tff(fact_7301_bit__word_Orsp,axiom,
    aa(fun(int,fun(nat,$o)),$o,aa(fun(int,fun(nat,$o)),fun(fun(int,fun(nat,$o)),$o),bNF_rel_fun(int,int,fun(nat,$o),fun(nat,$o),aTP_Lamp_tu(int,fun(int,$o)),fequal(fun(nat,$o))),aTP_Lamp_ul(int,fun(nat,$o))),aTP_Lamp_ul(int,fun(nat,$o))) ).

% bit_word.rsp
tff(fact_7302_word__succ_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o))),aTP_Lamp_um(int,int)),aTP_Lamp_um(int,int)) ).

% word_succ.rsp
tff(fact_7303_word__pred_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o))),aTP_Lamp_un(int,int)),aTP_Lamp_un(int,int)) ).

% word_pred.rsp
tff(fact_7304_the__nat_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(int,nat),$o,aa(fun(int,nat),fun(fun(int,nat),$o),bNF_rel_fun(int,int,nat,nat,aTP_Lamp_tu(int,fun(int,$o)),fequal(nat)),comp(int,nat,int,nat2,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))))),comp(int,nat,int,nat2,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))))) ) ).

% the_nat.rsp
tff(fact_7305_unsigned_Orsp,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => aa(fun(int,A),$o,aa(fun(int,A),fun(fun(int,A),$o),bNF_rel_fun(int,int,A,A,aTP_Lamp_tm(int,fun(int,$o)),fequal(A)),comp(int,A,int,comp(nat,A,int,semiring_1_of_nat(A),nat2),aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(B,type2(B))))),comp(int,A,int,comp(nat,A,int,semiring_1_of_nat(A),nat2),aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(B,type2(B))))) ) ).

% unsigned.rsp
tff(fact_7306_signed__cast_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tm(int,fun(int,$o))),aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))),aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% signed_cast.rsp
tff(fact_7307_the__signed__int_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),fequal(int)),aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))),aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% the_signed_int.rsp
tff(fact_7308_transfer__rule__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semiring_1(B)
        & semiring_1(A) )
     => ! [R2: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R2,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R2,one_one(A)),one_one(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R2,bNF_rel_fun(A,B,A,B,R2,R2)),plus_plus(A)),plus_plus(B))
             => aa(fun(nat,B),$o,aa(fun(nat,A),fun(fun(nat,B),$o),bNF_rel_fun(nat,nat,A,B,fequal(nat),R2),semiring_1_of_nat(A)),semiring_1_of_nat(B)) ) ) ) ) ).

% transfer_rule_of_nat
tff(fact_7309_If__the__inv__into__in__Func,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),C3: set(A),B3: set(A),Xa: A] :
      ( inj_on(A,B,G,C3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))
       => member(fun(B,A),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_uo(fun(A,B),fun(set(A),fun(A,fun(B,A))),G),C3),Xa),bNF_Wellorder_Func(B,A,top_top(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_7310_inj__split__Cons,axiom,
    ! [A: $tType,X7: 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_tc(list(A),fun(A,list(A)))),X7) ).

% inj_split_Cons
tff(fact_7311_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_7312_signed_Osorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => inj_on(word(A),word(A),aTP_Lamp_up(word(A),word(A)),top_top(set(word(A)))) ) ).

% signed.sorted_list_of_set.inj_on
tff(fact_7313_inj__graph,axiom,
    ! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_ur(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).

% inj_graph
tff(fact_7314_unat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => inj_on(word(A),nat,semiring_1_unsigned(A,nat),top_top(set(word(A)))) ) ).

% unat_inj
tff(fact_7315_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),X7: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_us(fun(A,B),fun(A,product_prod(A,B)),F2),X7) ).

% inj_on_convol_ident
tff(fact_7316_sub_Orsp,axiom,
    aa(fun(num,fun(num,int)),$o,aa(fun(num,fun(num,int)),fun(fun(num,fun(num,int)),$o),bNF_rel_fun(num,num,fun(num,int),fun(num,int),fequal(num),bNF_rel_fun(num,num,int,int,fequal(num),fequal(int))),aTP_Lamp_ut(num,fun(num,int))),aTP_Lamp_ut(num,fun(num,int))) ).

% sub.rsp
tff(fact_7317_less__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,$o)),$o,aa(fun(nat,fun(nat,$o)),fun(fun(nat,fun(nat,$o)),$o),bNF_rel_fun(nat,nat,fun(nat,$o),fun(nat,$o),fequal(nat),bNF_rel_fun(nat,nat,$o,$o,fequal(nat),fequal($o))),ord_less(nat)),ord_less(nat)) ).

% less_natural.rsp
tff(fact_7318_integer__of__natural_Orsp,axiom,
    aa(fun(nat,int),$o,aa(fun(nat,int),fun(fun(nat,int),$o),bNF_rel_fun(nat,nat,int,int,fequal(nat),fequal(int)),semiring_1_of_nat(int)),semiring_1_of_nat(int)) ).

% integer_of_natural.rsp
tff(fact_7319_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_7320_inj__Suc,axiom,
    ! [N3: set(nat)] : inj_on(nat,nat,suc,N3) ).

% inj_Suc
tff(fact_7321_modulo__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),modulo_modulo(nat)),modulo_modulo(nat)) ).

% modulo_natural.rsp
tff(fact_7322_transfer__rule__of__bool,axiom,
    ! [A: $tType,B: $tType] :
      ( ( zero_neq_one(B)
        & zero_neq_one(A) )
     => ! [R2: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R2,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R2,one_one(A)),one_one(B))
           => aa(fun($o,B),$o,aa(fun($o,A),fun(fun($o,B),$o),bNF_rel_fun($o,$o,A,B,fequal($o),R2),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B)) ) ) ) ).

% transfer_rule_of_bool
tff(fact_7323_plus__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),plus_plus(nat)),plus_plus(nat)) ).

% plus_natural.rsp
tff(fact_7324_times__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),times_times(nat)),times_times(nat)) ).

% times_natural.rsp
tff(fact_7325_divide__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),divide_divide(nat)),divide_divide(nat)) ).

% divide_natural.rsp
tff(fact_7326_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_ql(nat,fun(nat,nat),K),N3) ) ).

% inj_on_diff_nat
tff(fact_7327_swap__inj__on,axiom,
    ! [B: $tType,A: $tType,A3: 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_uu(A,fun(B,product_prod(B,A)))),A3) ).

% swap_inj_on
tff(fact_7328_inj__Pair_I1_J,axiom,
    ! [B: $tType,A: $tType,C2: fun(A,B),S: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_us(fun(A,B),fun(A,product_prod(A,B)),C2),S) ).

% inj_Pair(1)
tff(fact_7329_inj__Pair_I2_J,axiom,
    ! [B: $tType,A: $tType,C2: fun(A,B),S: set(A)] : inj_on(A,product_prod(B,A),aTP_Lamp_uv(fun(A,B),fun(A,product_prod(B,A)),C2),S) ).

% inj_Pair(2)
tff(fact_7330_inj__singleton,axiom,
    ! [A: $tType,A3: set(A)] : inj_on(A,set(A),aTP_Lamp_uw(A,set(A)),A3) ).

% inj_singleton
tff(fact_7331_target__size_Orsp,axiom,
    ! [A: $tType] : aa(fun(fun(A,int),nat),$o,aa(fun(fun(A,int),nat),fun(fun(fun(A,int),nat),$o),bNF_rel_fun(fun(A,int),fun(A,int),nat,nat,bNF_rel_fun(A,A,int,int,fequal(A),aTP_Lamp_tm(int,fun(int,$o))),fequal(nat)),aTP_Lamp_ux(fun(A,int),nat)),aTP_Lamp_ux(fun(A,int),nat)) ).

% target_size.rsp
tff(fact_7332_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => ? [N: nat,F8: fun(nat,A)] :
          ( ( A3 = aa(set(nat),set(A),image(nat,A,F8),collect(nat,aTP_Lamp_cy(nat,fun(nat,$o),N))) )
          & inj_on(nat,A,F8,collect(nat,aTP_Lamp_cy(nat,fun(nat,$o),N))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_7333_finite__imp__inj__to__nat__seg_H,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => ~ ! [F8: fun(A,nat)] :
            ( ? [N: nat] : aa(set(A),set(nat),image(A,nat,F8),A3) = collect(nat,aTP_Lamp_cy(nat,fun(nat,$o),N))
           => ~ inj_on(A,nat,F8,A3) ) ) ).

% finite_imp_inj_to_nat_seg'
tff(fact_7334_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => ? [F8: fun(A,nat),N: nat] :
          ( ( aa(set(A),set(nat),image(A,nat,F8),A3) = collect(nat,aTP_Lamp_cy(nat,fun(nat,$o),N)) )
          & inj_on(A,nat,F8,A3) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_7335_drop__bit__word_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),aTP_Lamp_uy(nat,fun(int,int))),aTP_Lamp_uy(nat,fun(int,int))) ).

% drop_bit_word.rsp
tff(fact_7336_take__bit__word_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),aTP_Lamp_uz(nat,fun(int,int))),aTP_Lamp_uz(nat,fun(int,int))) ).

% take_bit_word.rsp
tff(fact_7337_push__bit__word_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),bit_se4730199178511100633sh_bit(int)),bit_se4730199178511100633sh_bit(int)) ).

% push_bit_word.rsp
tff(fact_7338_unset__bit__word_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),bit_se2638667681897837118et_bit(int)),bit_se2638667681897837118et_bit(int)) ).

% unset_bit_word.rsp
tff(fact_7339_set__bit__word_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),bit_se5668285175392031749et_bit(int)),bit_se5668285175392031749et_bit(int)) ).

% set_bit_word.rsp
tff(fact_7340_flip__bit__word_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),bit_se8732182000553998342ip_bit(int)),bit_se8732182000553998342ip_bit(int)) ).

% flip_bit_word.rsp
tff(fact_7341_source__size_Orsp,axiom,
    ! [A: $tType] : aa(fun(fun(int,A),nat),$o,aa(fun(fun(int,A),nat),fun(fun(fun(int,A),nat),$o),bNF_rel_fun(fun(int,A),fun(int,A),nat,nat,bNF_rel_fun(int,int,A,A,aTP_Lamp_tm(int,fun(int,$o)),fequal(A)),fequal(nat)),aTP_Lamp_va(fun(int,A),nat)),aTP_Lamp_va(fun(int,A),nat)) ).

% source_size.rsp
tff(fact_7342_is__up_Orsp,axiom,
    aa(fun(fun(int,int),$o),$o,aa(fun(fun(int,int),$o),fun(fun(fun(int,int),$o),$o),bNF_rel_fun(fun(int,int),fun(int,int),$o,$o,bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tm(int,fun(int,$o))),fequal($o)),aTP_Lamp_vb(fun(int,int),$o)),aTP_Lamp_vb(fun(int,int),$o)) ).

% is_up.rsp
tff(fact_7343_is__down_Orsp,axiom,
    aa(fun(fun(int,int),$o),$o,aa(fun(fun(int,int),$o),fun(fun(fun(int,int),$o),$o),bNF_rel_fun(fun(int,int),fun(int,int),$o,$o,bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tm(int,fun(int,$o))),fequal($o)),aTP_Lamp_vc(fun(int,int),$o)),aTP_Lamp_vc(fun(int,int),$o)) ).

% is_down.rsp
tff(fact_7344_mask__word_Orsp,axiom,
    aa(fun(nat,int),$o,aa(fun(nat,int),fun(fun(nat,int),$o),bNF_rel_fun(nat,nat,int,int,fequal(nat),aTP_Lamp_tu(int,fun(int,$o))),bit_se2239418461657761734s_mask(int)),bit_se2239418461657761734s_mask(int)) ).

% mask_word.rsp
tff(fact_7345_of__nat_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(nat,int),$o,aa(fun(nat,int),fun(fun(nat,int),$o),bNF_rel_fun(nat,nat,int,int,fequal(nat),aTP_Lamp_tu(int,fun(int,$o))),comp(nat,int,nat,semiring_1_of_nat(int),aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),type_len0_len_of(A,type2(A))))),comp(nat,int,nat,semiring_1_of_nat(int),aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),type_len0_len_of(A,type2(A))))) ) ).

% of_nat.rsp
tff(fact_7346_word__int__case_Orsp,axiom,
    ! [A: $tType] : aa(fun(fun(int,A),fun(int,A)),$o,aa(fun(fun(int,A),fun(int,A)),fun(fun(fun(int,A),fun(int,A)),$o),bNF_rel_fun(fun(int,A),fun(int,A),fun(int,A),fun(int,A),fequal(fun(int,A)),bNF_rel_fun(int,int,A,A,aTP_Lamp_tm(int,fun(int,$o)),fequal(A))),aTP_Lamp_vd(fun(int,A),fun(int,A))),aTP_Lamp_vd(fun(int,A),fun(int,A))) ).

% word_int_case.rsp
tff(fact_7347_Bit__integer_Orsp,axiom,
    aa(fun(int,fun($o,int)),$o,aa(fun(int,fun($o,int)),fun(fun(int,fun($o,int)),$o),bNF_rel_fun(int,int,fun($o,int),fun($o,int),fequal(int),bNF_rel_fun($o,$o,int,int,fequal($o),fequal(int))),aTP_Lamp_ve(int,fun($o,int))),aTP_Lamp_ve(int,fun($o,int))) ).

% Bit_integer.rsp
tff(fact_7348_word__rotr_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),aTP_Lamp_vf(nat,fun(int,int))),aTP_Lamp_vf(nat,fun(int,int))) ).

% word_rotr.rsp
tff(fact_7349_word__rotl_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),aTP_Lamp_vg(nat,fun(int,int))),aTP_Lamp_vg(nat,fun(int,int))) ).

% word_rotl.rsp
tff(fact_7350_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_sw(nat,fun(real,real),Na),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_7351_signed__drop__bit_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,aTP_Lamp_tu(int,fun(int,$o)),aTP_Lamp_tu(int,fun(int,$o)))),aTP_Lamp_vh(nat,fun(int,int))),aTP_Lamp_vh(nat,fun(int,int))) ).

% signed_drop_bit.rsp
tff(fact_7352_inj__on__word__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => inj_on(nat,word(A),semiring_1_of_nat(word(A)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% inj_on_word_of_nat
tff(fact_7353_word__roti_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(int,fun(word(A),word(A))),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(word(A),word(A))),$o),bNF_rel_fun(int,int,fun(int,int),fun(word(A),word(A)),fequal(int),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A))),aTP_Lamp_uk(int,fun(int,int))),word_roti(A)) ) ).

% word_roti.transfer
tff(fact_7354_fun_Orel__map_I2_J,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,Sa2: fun(B,fun(C,$o)),Xa: fun(A,B),G: fun(D,C),Y: fun(A,D)] :
      ( aa(fun(A,C),$o,aa(fun(A,B),fun(fun(A,C),$o),bNF_rel_fun(A,A,B,C,fequal(A),Sa2),Xa),comp(D,C,A,G,Y))
    <=> aa(fun(A,D),$o,aa(fun(A,B),fun(fun(A,D),$o),bNF_rel_fun(A,A,B,D,fequal(A),aa(fun(D,C),fun(B,fun(D,$o)),aTP_Lamp_vi(fun(B,fun(C,$o)),fun(fun(D,C),fun(B,fun(D,$o))),Sa2),G)),Xa),Y) ) ).

% fun.rel_map(2)
tff(fact_7355_shiftl__word__transfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(nat,word(A))),$o,aa(fun(int,fun(nat,int)),fun(fun(word(A),fun(nat,word(A))),$o),bNF_rel_fun(int,word(A),fun(nat,int),fun(nat,word(A)),pcr_word(A),bNF_rel_fun(nat,nat,int,word(A),fequal(nat),pcr_word(A))),aTP_Lamp_vj(int,fun(nat,int))),bit_Sh4282982442137083160shiftl(word(A))) ) ).

% shiftl_word_transfer
tff(fact_7356_word_Oid__abs__transfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(int,word(A)),$o,aa(fun(int,int),fun(fun(int,word(A)),$o),bNF_rel_fun(int,int,int,word(A),fequal(int),pcr_word(A)),aTP_Lamp_gt(int,int)),word2(A)) ) ).

% word.id_abs_transfer
tff(fact_7357_power__transfer__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(nat,word(A))),$o,aa(fun(int,fun(nat,int)),fun(fun(word(A),fun(nat,word(A))),$o),bNF_rel_fun(int,word(A),fun(nat,int),fun(nat,word(A)),pcr_word(A),bNF_rel_fun(nat,nat,int,word(A),fequal(nat),pcr_word(A))),power_power(int)),power_power(word(A))) ) ).

% power_transfer_word
tff(fact_7358_plus__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),word(A))),$o,aa(fun(int,fun(int,int)),fun(fun(word(A),fun(word(A),word(A))),$o),bNF_rel_fun(int,word(A),fun(int,int),fun(word(A),word(A)),pcr_word(A),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A))),plus_plus(int)),plus_plus(word(A))) ) ).

% plus_word.transfer
tff(fact_7359_times__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),word(A))),$o,aa(fun(int,fun(int,int)),fun(fun(word(A),fun(word(A),word(A))),$o),bNF_rel_fun(int,word(A),fun(int,int),fun(word(A),word(A)),pcr_word(A),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A))),times_times(int)),times_times(word(A))) ) ).

% times_word.transfer
tff(fact_7360_word_Orel__eq__transfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(word(A),fun(word(A),$o)),$o),bNF_rel_fun(int,word(A),fun(int,$o),fun(word(A),$o),pcr_word(A),bNF_rel_fun(int,word(A),$o,$o,pcr_word(A),fequal($o))),aTP_Lamp_tu(int,fun(int,$o))),fequal(word(A))) ) ).

% word.rel_eq_transfer
tff(fact_7361_size__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),nat),$o,aa(fun(int,nat),fun(fun(word(A),nat),$o),bNF_rel_fun(int,word(A),nat,nat,pcr_word(A),fequal(nat)),aTP_Lamp_ue(int,nat)),size_size(word(A))) ) ).

% size_word.transfer
tff(fact_7362_word__succ_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),word(A)),$o,aa(fun(int,int),fun(fun(word(A),word(A)),$o),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A)),aTP_Lamp_um(int,int)),word_succ(A)) ) ).

% word_succ.transfer
tff(fact_7363_word__pred_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),word(A)),$o,aa(fun(int,int),fun(fun(word(A),word(A)),$o),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A)),aTP_Lamp_un(int,int)),word_pred(A)) ) ).

% word_pred.transfer
tff(fact_7364_shiftr__word__transfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(nat,word(A))),$o,aa(fun(int,fun(nat,int)),fun(fun(word(A),fun(nat,word(A))),$o),bNF_rel_fun(int,word(A),fun(nat,int),fun(nat,word(A)),pcr_word(A),bNF_rel_fun(nat,nat,int,word(A),fequal(nat),pcr_word(A))),aTP_Lamp_vk(int,fun(nat,int))),bit_Sh4282982442137083166shiftr(word(A))) ) ).

% shiftr_word_transfer
tff(fact_7365_fun_Omap__ident,axiom,
    ! [B: $tType,A: $tType,T: fun(A,B)] : comp(B,B,A,aTP_Lamp_vl(B,B),T) = T ).

% fun.map_ident
tff(fact_7366_less__eq__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(word(A),fun(word(A),$o)),$o),bNF_rel_fun(int,word(A),fun(int,$o),fun(word(A),$o),pcr_word(A),bNF_rel_fun(int,word(A),$o,$o,pcr_word(A),fequal($o))),aTP_Lamp_ub(int,fun(int,$o))),ord_less_eq(word(A))) ) ).

% less_eq_word.transfer
tff(fact_7367_less__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(word(A),fun(word(A),$o)),$o),bNF_rel_fun(int,word(A),fun(int,$o),fun(word(A),$o),pcr_word(A),bNF_rel_fun(int,word(A),$o,$o,pcr_word(A),fequal($o))),aTP_Lamp_tx(int,fun(int,$o))),ord_less(word(A))) ) ).

% less_word.transfer
tff(fact_7368_bit__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(nat,$o)),$o,aa(fun(int,fun(nat,$o)),fun(fun(word(A),fun(nat,$o)),$o),bNF_rel_fun(int,word(A),fun(nat,$o),fun(nat,$o),pcr_word(A),fequal(fun(nat,$o))),aTP_Lamp_ul(int,fun(nat,$o))),bit_se5641148757651400278ts_bit(word(A))) ) ).

% bit_word.transfer
tff(fact_7369_word__test__bit__transfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(nat,$o)),$o,aa(fun(int,fun(nat,$o)),fun(fun(word(A),fun(nat,$o)),$o),bNF_rel_fun(int,word(A),fun(nat,$o),fun(nat,$o),pcr_word(A),bNF_rel_fun(nat,nat,$o,$o,fequal(nat),fequal($o))),aTP_Lamp_ul(int,fun(nat,$o))),bit_se5641148757651400278ts_bit(word(A))) ) ).

% word_test_bit_transfer
tff(fact_7370_divide__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),word(A))),$o,aa(fun(int,fun(int,int)),fun(fun(word(A),fun(word(A),word(A))),$o),bNF_rel_fun(int,word(A),fun(int,int),fun(word(A),word(A)),pcr_word(A),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A))),aTP_Lamp_ua(int,fun(int,int))),divide_divide(word(A))) ) ).

% divide_word.transfer
tff(fact_7371_modulo__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),word(A))),$o,aa(fun(int,fun(int,int)),fun(fun(word(A),fun(word(A),word(A))),$o),bNF_rel_fun(int,word(A),fun(int,int),fun(word(A),word(A)),pcr_word(A),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A))),aTP_Lamp_tz(int,fun(int,int))),modulo_modulo(word(A))) ) ).

% modulo_word.transfer
tff(fact_7372_take__bit__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(nat,fun(word(A),word(A))),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(word(A),word(A))),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(word(A),word(A)),fequal(nat),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A))),aTP_Lamp_uz(nat,fun(int,int))),bit_se2584673776208193580ke_bit(word(A))) ) ).

% take_bit_word.transfer
tff(fact_7373_drop__bit__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(nat,fun(word(A),word(A))),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(word(A),word(A))),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(word(A),word(A)),fequal(nat),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A))),aTP_Lamp_uy(nat,fun(int,int))),bit_se4197421643247451524op_bit(word(A))) ) ).

% drop_bit_word.transfer
tff(fact_7374_word__cat_Otransfer,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B)
        & type_len(C) )
     => aa(fun(word(A),fun(word(B),word(C))),$o,aa(fun(int,fun(int,int)),fun(fun(word(A),fun(word(B),word(C))),$o),bNF_rel_fun(int,word(A),fun(int,int),fun(word(B),word(C)),pcr_word(A),bNF_rel_fun(int,word(B),int,word(C),pcr_word(B),pcr_word(C))),aTP_Lamp_tw(int,fun(int,int))),word_cat(A,B,C)) ) ).

% word_cat.transfer
tff(fact_7375_word__int__case_Otransfer,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => aa(fun(fun(int,A),fun(word(B),A)),$o,aa(fun(fun(int,A),fun(int,A)),fun(fun(fun(int,A),fun(word(B),A)),$o),bNF_rel_fun(fun(int,A),fun(int,A),fun(int,A),fun(word(B),A),fequal(fun(int,A)),bNF_rel_fun(int,word(B),A,A,pcr_word(B),fequal(A))),aTP_Lamp_vd(fun(int,A),fun(int,A))),word_int_case(A,B)) ) ).

% word_int_case.transfer
tff(fact_7376_signed__take__bit__word__transfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(nat,fun(word(A),word(A))),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(word(A),word(A))),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(word(A),word(A)),fequal(nat),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A))),aTP_Lamp_vm(nat,fun(int,int))),bit_ri4674362597316999326ke_bit(word(A))) ) ).

% signed_take_bit_word_transfer
tff(fact_7377_of__nat_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(nat,word(A)),$o,aa(fun(nat,int),fun(fun(nat,word(A)),$o),bNF_rel_fun(nat,nat,int,word(A),fequal(nat),pcr_word(A)),comp(nat,int,nat,semiring_1_of_nat(int),aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),type_len0_len_of(A,type2(A))))),of_nat(A)) ) ).

% of_nat.transfer
tff(fact_7378_the__nat_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),nat),$o,aa(fun(int,nat),fun(fun(word(A),nat),$o),bNF_rel_fun(int,word(A),nat,nat,pcr_word(A),fequal(nat)),comp(int,nat,int,nat2,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))))),the_nat(A)) ) ).

% the_nat.transfer
tff(fact_7379_word__sle_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(word(A),fun(word(A),$o)),$o),bNF_rel_fun(int,word(A),fun(int,$o),fun(word(A),$o),pcr_word(A),bNF_rel_fun(int,word(A),$o,$o,pcr_word(A),fequal($o))),aTP_Lamp_uf(int,fun(int,$o))),word_sle(A)) ) ).

% word_sle.transfer
tff(fact_7380_word__sless_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(word(A),fun(word(A),$o)),$o),bNF_rel_fun(int,word(A),fun(int,$o),fun(word(A),$o),pcr_word(A),bNF_rel_fun(int,word(A),$o,$o,pcr_word(A),fequal($o))),aTP_Lamp_ug(int,fun(int,$o))),word_sless(A)) ) ).

% word_sless.transfer
tff(fact_7381_signed__drop__bit_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(nat,fun(word(A),word(A))),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(word(A),word(A))),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(word(A),word(A)),fequal(nat),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A))),aTP_Lamp_vh(nat,fun(int,int))),signed_drop_bit(A)) ) ).

% signed_drop_bit.transfer
tff(fact_7382_unsigned_Otransfer,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & semiring_1(B) )
     => aa(fun(word(A),B),$o,aa(fun(int,B),fun(fun(word(A),B),$o),bNF_rel_fun(int,word(A),B,B,pcr_word(A),fequal(B)),comp(int,B,int,comp(nat,B,int,semiring_1_of_nat(B),nat2),aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))))),semiring_1_unsigned(A,B)) ) ).

% unsigned.transfer
tff(fact_7383_signed__modulo__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),word(A))),$o,aa(fun(int,fun(int,int)),fun(fun(word(A),fun(word(A),word(A))),$o),bNF_rel_fun(int,word(A),fun(int,int),fun(word(A),word(A)),pcr_word(A),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A))),aTP_Lamp_uh(int,fun(int,int))),signed6721504322012087516modulo(word(A))) ) ).

% signed_modulo_word.transfer
tff(fact_7384_signed__divide__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),word(A))),$o,aa(fun(int,fun(int,int)),fun(fun(word(A),fun(word(A),word(A))),$o),bNF_rel_fun(int,word(A),fun(int,int),fun(word(A),word(A)),pcr_word(A),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A))),aTP_Lamp_ui(int,fun(int,int))),signed7115095781618012415divide(word(A))) ) ).

% signed_divide_word.transfer
tff(fact_7385_sshiftr__transfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(nat,word(A))),$o,aa(fun(int,fun(nat,int)),fun(fun(word(A),fun(nat,word(A))),$o),bNF_rel_fun(int,word(A),fun(nat,int),fun(nat,word(A)),pcr_word(A),bNF_rel_fun(nat,nat,int,word(A),fequal(nat),pcr_word(A))),aTP_Lamp_vn(int,fun(nat,int))),bit_Sh8784991116023147202shiftr(A)) ) ).

% sshiftr_transfer
tff(fact_7386_word__rotr_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(nat,fun(word(A),word(A))),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(word(A),word(A))),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(word(A),word(A)),fequal(nat),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A))),aTP_Lamp_vf(nat,fun(int,int))),word_rotr(A)) ) ).

% word_rotr.transfer
tff(fact_7387_word__rotl_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(nat,fun(word(A),word(A))),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(word(A),word(A))),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(word(A),word(A)),fequal(nat),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A))),aTP_Lamp_vg(nat,fun(int,int))),word_rotl(A)) ) ).

% word_rotl.transfer
tff(fact_7388_fun_Orel__map_I1_J,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,Sb: fun(B,fun(C,$o)),I: fun(D,B),Xa: fun(A,D),Y: fun(A,C)] :
      ( aa(fun(A,C),$o,aa(fun(A,B),fun(fun(A,C),$o),bNF_rel_fun(A,A,B,C,fequal(A),Sb),comp(D,B,A,I,Xa)),Y)
    <=> aa(fun(A,C),$o,aa(fun(A,D),fun(fun(A,C),$o),bNF_rel_fun(A,A,D,C,fequal(A),aa(fun(D,B),fun(D,fun(C,$o)),aTP_Lamp_vo(fun(B,fun(C,$o)),fun(fun(D,B),fun(D,fun(C,$o))),Sb),I)),Xa),Y) ) ).

% fun.rel_map(1)
tff(fact_7389_target__size_Otransfer,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => aa(fun(fun(A,word(B)),nat),$o,aa(fun(fun(A,int),nat),fun(fun(fun(A,word(B)),nat),$o),bNF_rel_fun(fun(A,int),fun(A,word(B)),nat,nat,bNF_rel_fun(A,A,int,word(B),fequal(A),pcr_word(B)),fequal(nat)),aTP_Lamp_ux(fun(A,int),nat)),target_size(A,B)) ) ).

% target_size.transfer
tff(fact_7390_source__size_Otransfer,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => aa(fun(fun(word(B),A),nat),$o,aa(fun(fun(int,A),nat),fun(fun(fun(word(B),A),nat),$o),bNF_rel_fun(fun(int,A),fun(word(B),A),nat,nat,bNF_rel_fun(int,word(B),A,A,pcr_word(B),fequal(A)),fequal(nat)),aTP_Lamp_va(fun(int,A),nat)),source_size(B,A)) ) ).

% source_size.transfer
tff(fact_7391_revcast__down__us,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Na: nat,W: word(A)] :
          ( ( aa(fun(word(A),word(B)),nat,source_size(A,word(B)),revcast(A,B)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(fun(word(A),word(B)),nat,target_size(word(A),B),revcast(A,B))),Na) )
         => ( aa(word(A),word(B),revcast(A,B),W) = aa(word(A),word(B),semiring_1_unsigned(A,word(B)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),Na),W)) ) ) ) ).

% revcast_down_us
tff(fact_7392_revcast__down__uu,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Na: nat,W: word(A)] :
          ( ( aa(fun(word(A),word(B)),nat,source_size(A,word(B)),revcast(A,B)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(fun(word(A),word(B)),nat,target_size(word(A),B),revcast(A,B))),Na) )
         => ( aa(word(A),word(B),revcast(A,B),W) = 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)) ) ) ) ).

% revcast_down_uu
tff(fact_7393_cast__down__rev,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Na: nat,W: word(A)] :
          ( ( aa(fun(word(A),word(B)),nat,source_size(A,word(B)),semiring_1_unsigned(A,word(B))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(fun(word(A),word(B)),nat,target_size(word(A),B),semiring_1_unsigned(A,word(B)))),Na) )
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),W) = aa(word(A),word(B),revcast(A,B),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),Na),W)) ) ) ) ).

% cast_down_rev
tff(fact_7394_revcast__up,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Na: nat,W: word(A)] :
          ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(fun(word(A),word(B)),nat,source_size(A,word(B)),revcast(A,B))),Na) = aa(fun(word(A),word(B)),nat,target_size(word(A),B),revcast(A,B)) )
         => ( aa(word(A),word(B),revcast(A,B),W) = aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se4730199178511100633sh_bit(word(B)),Na),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),W)) ) ) ) ).

% revcast_up
tff(fact_7395_revcast__down__ss,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Na: nat,W: word(A)] :
          ( ( aa(fun(word(A),word(B)),nat,source_size(A,word(B)),revcast(A,B)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(fun(word(A),word(B)),nat,target_size(word(A),B),revcast(A,B))),Na) )
         => ( aa(word(A),word(B),revcast(A,B),W) = ring_1_signed(A,word(B),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),Na),W)) ) ) ) ).

% revcast_down_ss
tff(fact_7396_revcast__down__su,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Na: nat,W: word(A)] :
          ( ( aa(fun(word(A),word(B)),nat,source_size(A,word(B)),revcast(A,B)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(fun(word(A),word(B)),nat,target_size(word(A),B),revcast(A,B))),Na) )
         => ( aa(word(A),word(B),revcast(A,B),W) = ring_1_signed(A,word(B),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),Na),W)) ) ) ) ).

% revcast_down_su
tff(fact_7397_udvd_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(word(A),fun(word(A),$o)),$o),bNF_rel_fun(int,word(A),fun(int,$o),fun(word(A),$o),pcr_word(A),bNF_rel_fun(int,word(A),$o,$o,pcr_word(A),fequal($o))),aTP_Lamp_ty(int,fun(int,$o))),udvd(A)) ) ).

% udvd.transfer
tff(fact_7398_num_Orec__transfer,axiom,
    ! [A: $tType,B: $tType,S: fun(A,fun(B,$o))] : aa(fun(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)))),$o,aa(fun(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)))),fun(fun(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)))),$o),bNF_rel_fun(A,B,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B))),S,bNF_rel_fun(fun(num,fun(A,A)),fun(num,fun(B,B)),fun(fun(num,fun(A,A)),fun(num,A)),fun(fun(num,fun(B,B)),fun(num,B)),bNF_rel_fun(num,num,fun(A,A),fun(B,B),fequal(num),bNF_rel_fun(A,B,A,B,S,S)),bNF_rel_fun(fun(num,fun(A,A)),fun(num,fun(B,B)),fun(num,A),fun(num,B),bNF_rel_fun(num,num,fun(A,A),fun(B,B),fequal(num),bNF_rel_fun(A,B,A,B,S,S)),bNF_rel_fun(num,num,A,B,fequal(num),S)))),rec_num(A)),rec_num(B)) ).

% num.rec_transfer
tff(fact_7399_udvdI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [W: word(A),V: word(A),U: word(B)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),W) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),V)),aa(word(B),nat,semiring_1_unsigned(B,nat),U)) )
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),udvd(A),V),W) ) ) ).

% udvdI
tff(fact_7400_verit__eq__simplify_I19_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,fun(A,A)),F32: fun(num,fun(A,A))] : aa(num,A,aa(fun(num,fun(A,A)),fun(num,A),aa(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)),aa(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),rec_num(A),F1),F22),F32),one2) = F1 ).

% verit_eq_simplify(19)
tff(fact_7401_verit__eq__simplify_I20_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,fun(A,A)),F32: fun(num,fun(A,A)),X22: num] : aa(num,A,aa(fun(num,fun(A,A)),fun(num,A),aa(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)),aa(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),rec_num(A),F1),F22),F32),bit0(X22)) = aa(A,A,aa(num,fun(A,A),F22,X22),aa(num,A,aa(fun(num,fun(A,A)),fun(num,A),aa(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)),aa(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),rec_num(A),F1),F22),F32),X22)) ).

% verit_eq_simplify(20)
tff(fact_7402_verit__eq__simplify_I21_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,fun(A,A)),F32: fun(num,fun(A,A)),X32: num] : aa(num,A,aa(fun(num,fun(A,A)),fun(num,A),aa(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)),aa(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),rec_num(A),F1),F22),F32),aa(num,num,bit1,X32)) = aa(A,A,aa(num,fun(A,A),F32,X32),aa(num,A,aa(fun(num,fun(A,A)),fun(num,A),aa(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)),aa(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),rec_num(A),F1),F22),F32),X32)) ).

% verit_eq_simplify(21)
tff(fact_7403_udvdE,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),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_7404_udvd__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),udvd(A),A2),B2)
        <=> ? [N2: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),B2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)) ) ) ).

% udvd_nat_alt
tff(fact_7405_mod__eq__0__imp__udvd,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),W),V) = zero_zero(word(A)) )
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),udvd(A),V),W) ) ) ).

% mod_eq_0_imp_udvd
tff(fact_7406_udvd__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),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_7407_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),$o),ord_less(word(A)),Xy),K)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),udvd(A),Z),Xy)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),udvd(A),Z),K)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xy),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),K),Z)) ) ) ) ) ).

% udvd_minus_le'
tff(fact_7408_udvd__unfold__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),udvd(A),A2),B2)
        <=> ? [N2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N2)
              & ( aa(word(A),int,semiring_1_unsigned(A,int),B2) = aa(int,int,aa(int,fun(int,int),times_times(int),N2),aa(word(A),int,semiring_1_unsigned(A,int),A2)) ) ) ) ) ).

% udvd_unfold_int
tff(fact_7409_udvd__incr2__K,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),A2: word(A),Sa: word(A),K6: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),Sa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),Sa))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),udvd(A),K6),Sa)
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),udvd(A),K6),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),P2),A2))
               => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),P2)
                 => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K6)
                   => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),K6))
                      & aa(word(A),$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),K6)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),Sa)) ) ) ) ) ) ) ) ) ).

% udvd_incr2_K
tff(fact_7410_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),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),W)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),udvd(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),W) ) ) ).

% exp_dvd_iff_exp_udvd
tff(fact_7411_num_Ocase__transfer,axiom,
    ! [A: $tType,B: $tType,S: fun(A,fun(B,$o))] : aa(fun(B,fun(fun(num,B),fun(fun(num,B),fun(num,B)))),$o,aa(fun(A,fun(fun(num,A),fun(fun(num,A),fun(num,A)))),fun(fun(B,fun(fun(num,B),fun(fun(num,B),fun(num,B)))),$o),bNF_rel_fun(A,B,fun(fun(num,A),fun(fun(num,A),fun(num,A))),fun(fun(num,B),fun(fun(num,B),fun(num,B))),S,bNF_rel_fun(fun(num,A),fun(num,B),fun(fun(num,A),fun(num,A)),fun(fun(num,B),fun(num,B)),bNF_rel_fun(num,num,A,B,fequal(num),S),bNF_rel_fun(fun(num,A),fun(num,B),fun(num,A),fun(num,B),bNF_rel_fun(num,num,A,B,fequal(num),S),bNF_rel_fun(num,num,A,B,fequal(num),S)))),case_num(A)),case_num(B)) ).

% num.case_transfer
tff(fact_7412_rc1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Na: nat,W: word(A)] :
          ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(fun(word(A),word(B)),nat,source_size(A,word(B)),revcast(A,B))),Na) = aa(fun(word(A),word(B)),nat,target_size(word(A),B),revcast(A,B)) )
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),W) = word_reverse(B,aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se4730199178511100633sh_bit(word(B)),Na),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),word_reverse(A,W)))) ) ) ) ).

% rc1
tff(fact_7413_num_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: B,F22: fun(num,B),F32: fun(num,B),Num: num] : aa(B,A,H,aa(num,B,aa(fun(num,B),fun(num,B),aa(fun(num,B),fun(fun(num,B),fun(num,B)),aa(B,fun(fun(num,B),fun(fun(num,B),fun(num,B))),case_num(B),F1),F22),F32),Num)) = aa(num,A,aa(fun(num,A),fun(num,A),aa(fun(num,A),fun(fun(num,A),fun(num,A)),aa(A,fun(fun(num,A),fun(fun(num,A),fun(num,A))),case_num(A),aa(B,A,H,F1)),aa(fun(num,B),fun(num,A),aTP_Lamp_vp(fun(B,A),fun(fun(num,B),fun(num,A)),H),F22)),aa(fun(num,B),fun(num,A),aTP_Lamp_vp(fun(B,A),fun(fun(num,B),fun(num,A)),H),F32)),Num) ).

% num.case_distrib
tff(fact_7414_verit__eq__simplify_I18_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X32: num] : aa(num,A,aa(fun(num,A),fun(num,A),aa(fun(num,A),fun(fun(num,A),fun(num,A)),aa(A,fun(fun(num,A),fun(fun(num,A),fun(num,A))),case_num(A),F1),F22),F32),aa(num,num,bit1,X32)) = aa(num,A,F32,X32) ).

% verit_eq_simplify(18)
tff(fact_7415_verit__eq__simplify_I17_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X22: num] : aa(num,A,aa(fun(num,A),fun(num,A),aa(fun(num,A),fun(fun(num,A),fun(num,A)),aa(A,fun(fun(num,A),fun(fun(num,A),fun(num,A))),case_num(A),F1),F22),F32),bit0(X22)) = aa(num,A,F22,X22) ).

% verit_eq_simplify(17)
tff(fact_7416_verit__eq__simplify_I16_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A)] : aa(num,A,aa(fun(num,A),fun(num,A),aa(fun(num,A),fun(fun(num,A),fun(num,A)),aa(A,fun(fun(num,A),fun(fun(num,A),fun(num,A))),case_num(A),F1),F22),F32),one2) = F1 ).

% verit_eq_simplify(16)
tff(fact_7417_rev__slice,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Na: nat,K: nat,W: word(B)] :
          ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K)),type_len0_len_of(A,type2(A))) = type_len0_len_of(B,type2(B)) )
         => ( slice2(B,A,Na,word_reverse(B,W)) = word_reverse(A,slice2(B,A,K,W)) ) ) ) ).

% rev_slice
tff(fact_7418_rev__slice1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Na: nat,K: nat,W: word(B)] :
          ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K) = 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))) )
         => ( slice1(B,A,Na,word_reverse(B,W)) = word_reverse(A,slice1(B,A,K,W)) ) ) ) ).

% rev_slice1
tff(fact_7419_bit__word__reverse__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_reverse(A,W)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,Na))) ) ) ) ).

% bit_word_reverse_iff
tff(fact_7420_rc2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Na: nat,W: word(A)] :
          ( ( aa(fun(word(A),word(B)),nat,source_size(A,word(B)),revcast(A,B)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(fun(word(A),word(B)),nat,target_size(word(A),B),revcast(A,B))),Na) )
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),W) = word_reverse(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),word_reverse(A,W)))) ) ) ) ).

% rc2
tff(fact_7421_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(Na,M) = aa(product_prod(nat,num),option(num),product_case_prod(nat,num,option(num),aTP_Lamp_vt(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),aa(nat,fun(num,product_prod(nat,num)),product_Pair(nat,num),Na),M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_7422_word__reverse__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : word_reverse(A,W) = aa(list($o),word(A),groups4207007520872428315er_sum($o,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(list($o),list($o),rev($o),map(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),upt(zero_zero(nat),type_len0_len_of(A,type2(A)))))) ) ).

% word_reverse_def
tff(fact_7423_case__nat__numeral,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,A),V: num] : case_nat(A,A2,F2,aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,F2,pred_numeral(V)) ).

% case_nat_numeral
tff(fact_7424_case__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,A),V: num,Na: nat] : case_nat(A,A2,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),Na)) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),Na)) ).

% case_nat_add_eq_if
tff(fact_7425_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A),X22: nat] : case_nat(A,F1,F22,aa(nat,nat,suc,X22)) = aa(nat,A,F22,X22) ).

% old.nat.simps(5)
tff(fact_7426_old_Onat_Osimps_I4_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A)] : case_nat(A,F1,F22,zero_zero(nat)) = F1 ).

% old.nat.simps(4)
tff(fact_7427_nat_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: B,F22: fun(nat,B),Nat: nat] : aa(B,A,H,case_nat(B,F1,F22,Nat)) = case_nat(A,aa(B,A,H,F1),aa(fun(nat,B),fun(nat,A),aTP_Lamp_vu(fun(B,A),fun(fun(nat,B),fun(nat,A)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_7428_nth__Cons,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Na: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),Na) = case_nat(A,Xa,nth(A,Xs),Na) ).

% nth_Cons
tff(fact_7429_rev__nth,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rev(A),Xs)),Na) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,suc,Na))) ) ) ).

% rev_nth
tff(fact_7430_rev__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),list(A),rev(A),list_update(A,Xs,K,Y)) = list_update(A,aa(list(A),list(A),rev(A),Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),K)),one_one(nat)),Y) ) ) ).

% rev_update
tff(fact_7431_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(Na,aa(num,num,bit1,M)) = case_nat(option(num),none(num),aTP_Lamp_vv(num,fun(nat,option(num)),M),Na) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_7432_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(Na,bit0(M)) = case_nat(option(num),none(num),aTP_Lamp_vw(num,fun(nat,option(num)),M),Na) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_7433_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> case_nat($o,$false,aTP_Lamp_ti(nat,$o),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_7434_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> case_nat($o,$true,aTP_Lamp_th(nat,$o),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_7435_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na)
    <=> case_nat($o,$false,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% less_eq_nat.simps(2)
tff(fact_7436_max__Suc2,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),aa(nat,nat,suc,Na)) = case_nat(nat,aa(nat,nat,suc,Na),aTP_Lamp_vx(nat,fun(nat,nat),Na),M) ).

% max_Suc2
tff(fact_7437_max__Suc1,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Na)),M) = case_nat(nat,aa(nat,nat,suc,Na),aTP_Lamp_vy(nat,fun(nat,nat),Na),M) ).

% max_Suc1
tff(fact_7438_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),I: nat,V: A] : list_update(A,aa(list(A),list(A),cons(A,Xa),Xs),I,V) = case_nat(list(A),aa(list(A),list(A),cons(A,V),Xs),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_vz(A,fun(list(A),fun(A,fun(nat,list(A)))),Xa),Xs),V),I) ).

% list_update.simps(2)
tff(fact_7439_diff__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,Na)) = case_nat(nat,zero_zero(nat),aTP_Lamp_fj(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)) ).

% diff_Suc
tff(fact_7440_bit__numeral__rec_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),bit0(W))),Na)
        <=> case_nat($o,$false,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),W)),Na) ) ) ).

% bit_numeral_rec(1)
tff(fact_7441_bit__numeral__rec_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,W))),Na)
        <=> case_nat($o,$true,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),W)),Na) ) ) ).

% bit_numeral_rec(2)
tff(fact_7442_min__Suc1,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,Na)),M) = case_nat(nat,zero_zero(nat),aTP_Lamp_wa(nat,fun(nat,nat),Na),M) ).

% min_Suc1
tff(fact_7443_min__Suc2,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),aa(nat,nat,suc,Na)) = case_nat(nat,zero_zero(nat),aTP_Lamp_wb(nat,fun(nat,nat),Na),M) ).

% min_Suc2
tff(fact_7444_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [Na: nat] : bit_take_bit_num(Na,one2) = case_nat(option(num),none(num),aTP_Lamp_wc(nat,option(num)),Na) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_7445_word__rcat__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( word_rcat(A,B) = comp(list(word(A)),word(B),list(word(A)),comp(int,word(B),list(word(A)),ring_1_of_int(word(B)),groups4207007520872428315er_sum(word(A),int,semiring_1_unsigned(A,int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),rev(word(A))) ) ) ).

% word_rcat_def
tff(fact_7446_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_wd(C,set(B)),F2) = comp(set(D),set(B),A,image(D,B,G),aTP_Lamp_we(A,set(D))) ).

% empty_natural
tff(fact_7447_conj__subset__def,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),collect(A,P))
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),collect(A,Q)) ) ) ).

% conj_subset_def
tff(fact_7448_conj__comp__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(B,$o),Q: fun(B,$o),G: fun(A,B),X5: A] :
      ( aa(A,$o,comp(B,$o,A,aa(fun(B,$o),fun(B,$o),aTP_Lamp_wf(fun(B,$o),fun(fun(B,$o),fun(B,$o)),P),Q),G),X5)
    <=> ( aa(A,$o,comp(B,$o,A,P,G),X5)
        & aa(A,$o,comp(B,$o,A,Q,G),X5) ) ) ).

% conj_comp_iff
tff(fact_7449_test__bit__rcat,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Sw: nat,Wl: list(word(A)),Rc: word(B),Na: nat] :
          ( ( Sw = aa(word(A),nat,size_size(word(A)),hd(word(A),Wl)) )
         => ( ( Rc = aa(list(word(A)),word(B),word_rcat(A,B),Wl) )
           => ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),Rc),Na)
            <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(B),nat,size_size(word(B)),Rc))
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),Sw)),aa(list(word(A)),nat,size_size(list(word(A))),Wl))
                & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),nth(word(A),aa(list(word(A)),list(word(A)),rev(word(A)),Wl)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),Sw))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),Sw)) ) ) ) ) ) ).

% test_bit_rcat
tff(fact_7450_DERIV__real__root__generic,axiom,
    ! [Na: nat,Xa: real,D4: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( Xa != 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,$o),ord_less(real),zero_zero(real)),Xa)
             => ( D4 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))) ) ) )
         => ( ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
               => ( D4 = aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat))))))) ) ) )
           => ( ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
               => ( D4 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(Na),D4,topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_7451_at__within__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A] : topolo174197925503356063within(A,A2,bot_bot(set(A))) = bot_bot(filter(A)) ) ).

% at_within_empty
tff(fact_7452_DERIV__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),Xa: A,Db: A,Sa: set(A)] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,Xa),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xa,Sa))
           => 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,Xa,Sa)) ) ) ) ).

% DERIV_chain
tff(fact_7453_DERIV__image__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),Xa: A,Sa: set(A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,Xa),aa(set(A),set(A),image(A,A,G),Sa)))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xa,Sa))
           => 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,Xa,Sa)) ) ) ) ).

% DERIV_image_chain
tff(fact_7454_DERIV__at__within__shift__lemma,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,Xa: A,S: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Xa),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Z)),S)))
         => has_field_derivative(A,comp(A,A,A,F2,aa(A,fun(A,A),plus_plus(A),Z)),Y,topolo174197925503356063within(A,Xa,S)) ) ) ).

% DERIV_at_within_shift_lemma
tff(fact_7455_DERIV__pow,axiom,
    ! [Na: nat,Xa: real,Sa: set(real)] : has_field_derivative(real,aTP_Lamp_wg(nat,fun(real,real),Na),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(real,Xa,Sa)) ).

% DERIV_pow
tff(fact_7456_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xa: A,Sa: set(A)] :
          ( ( Xa != zero_zero(A) )
         => has_field_derivative(A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xa)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% DERIV_inverse
tff(fact_7457_DERIV__local__max,axiom,
    ! [F2: fun(real,real),L: real,Xa: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),Y3))),D2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,Y3)),aa(real,real,F2,Xa)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_7458_DERIV__local__min,axiom,
    ! [F2: fun(real,real),L: real,Xa: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),Y3))),D2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,Xa)),aa(real,real,F2,Y3)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_7459_DERIV__ln__divide,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => has_field_derivative(real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Xa),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_ln_divide
tff(fact_7460_at__neq__bot,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [A2: A] : topolo174197925503356063within(A,A2,top_top(set(A))) != bot_bot(filter(A)) ) ).

% at_neq_bot
tff(fact_7461_DERIV__const__ratio__const2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X: real] : has_field_derivative(real,F2,K,topolo174197925503356063within(real,X,top_top(set(real))))
       => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2))),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_7462_DERIV__const__ratio__const,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X: real] : has_field_derivative(real,F2,K,topolo174197925503356063within(real,X,top_top(set(real))))
       => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_7463_DERIV__local__const,axiom,
    ! [F2: fun(real,real),L: real,Xa: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),Y3))),D2)
             => ( aa(real,real,F2,Xa) = aa(real,real,F2,Y3) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_7464_DERIV__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,Xa: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),H3))),aa(real,real,F2,Xa)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_7465_DERIV__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,Xa: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xa)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),H3))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_7466_DERIV__ln,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => has_field_derivative(real,ln_ln(real),aa(real,real,inverse_inverse(real),Xa),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_ln
tff(fact_7467_DERIV__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : has_field_derivative(A,exp(A),aa(A,A,exp(A),Xa),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ).

% DERIV_exp
tff(fact_7468_DERIV__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : has_field_derivative(A,sin(A),aa(A,A,cos(A),Xa),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ).

% DERIV_sin
tff(fact_7469_DERIV__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Xa: A,Z: A] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z),top_top(set(A))))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_wh(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ).

% DERIV_shift
tff(fact_7470_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K: A,Xa: A] : has_field_derivative(A,aTP_Lamp_wi(A,fun(A,A),K),aa(A,A,uminus_uminus(A),aa(A,A,sin(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),K))),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ).

% DERIV_cos_add
tff(fact_7471_DERIV__mirror,axiom,
    ! [F2: fun(real,real),Y: real,Xa: real] :
      ( has_field_derivative(real,F2,Y,topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),Xa),top_top(set(real))))
    <=> has_field_derivative(real,aTP_Lamp_wj(fun(real,real),fun(real,real),F2),aa(real,real,uminus_uminus(real),Y),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_mirror
tff(fact_7472_DERIV__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : has_field_derivative(A,cos(A),aa(A,A,uminus_uminus(A),aa(A,A,sin(A),Xa)),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ).

% DERIV_cos
tff(fact_7473_DERIV__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,Xa: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xa)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),H3))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_7474_DERIV__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,Xa: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D3)
                 => 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),Xa),H3))),aa(real,real,F2,Xa)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_7475_DERIV__fun__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,Xa: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,Xa,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_wk(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,Xa))),M),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ).

% DERIV_fun_exp
tff(fact_7476_DERIV__fun__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,Xa: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,Xa,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_wl(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(A,A,sin(A),aa(A,A,G,Xa)))),M),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ).

% DERIV_fun_cos
tff(fact_7477_DERIV__fun__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,Xa: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,Xa,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_wm(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),aa(A,A,G,Xa))),M),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ).

% DERIV_fun_sin
tff(fact_7478_DERIV__chain__s,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Sa: set(A),G: fun(A,A),G4: fun(A,A),F2: fun(A,A),F4: A,Xa: A] :
          ( ! [X: A] :
              ( member(A,X,Sa)
             => has_field_derivative(A,G,aa(A,A,G4,X),topolo174197925503356063within(A,X,top_top(set(A)))) )
         => ( has_field_derivative(A,F2,F4,topolo174197925503356063within(A,Xa,top_top(set(A))))
           => ( member(A,aa(A,A,F2,Xa),Sa)
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_wn(fun(A,A),fun(fun(A,A),fun(A,A)),G),F2),aa(A,A,aa(A,fun(A,A),times_times(A),F4),aa(A,A,G4,aa(A,A,F2,Xa))),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ) ) ).

% DERIV_chain_s
tff(fact_7479_DERIV__chain3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [G: fun(A,A),G4: fun(A,A),F2: fun(A,A),F4: A,Xa: A] :
          ( ! [X: A] : has_field_derivative(A,G,aa(A,A,G4,X),topolo174197925503356063within(A,X,top_top(set(A))))
         => ( has_field_derivative(A,F2,F4,topolo174197925503356063within(A,Xa,top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_wn(fun(A,A),fun(fun(A,A),fun(A,A)),G),F2),aa(A,A,aa(A,fun(A,A),times_times(A),F4),aa(A,A,G4,aa(A,A,F2,Xa))),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ) ).

% DERIV_chain3
tff(fact_7480_DERIV__chain2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),Xa: A,Db: A,Sa: set(A)] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,Xa),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xa,Sa))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_wn(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,Xa,Sa)) ) ) ) ).

% DERIV_chain2
tff(fact_7481_DERIV__chain_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A,Sa: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,Xa,Sa))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,aa(A,A,F2,Xa),top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_wo(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),E4),D4),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% DERIV_chain'
tff(fact_7482_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2)
             => ? [Y2: real] :
                  ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),zero_zero(real)) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_7483_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2)
             => ? [Y2: real] :
                  ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y2) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,A2)),aa(real,real,F2,B2)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_7484_MVT2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),F4: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2)
             => has_field_derivative(real,F2,aa(real,real,F4,X),topolo174197925503356063within(real,X,top_top(set(real)))) ) )
       => ? [Z3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z3)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),B2)
            & ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),aa(real,real,F4,Z3)) ) ) ) ) ).

% MVT2
tff(fact_7485_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A,Sa: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,Xa,Sa))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,Xa,Sa))
           => ( ( aa(A,A,G,Xa) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_wp(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,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D4),aa(A,A,G,Xa))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,Xa)),E4))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G,Xa)),aa(A,A,G,Xa))),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ) ).

% DERIV_divide
tff(fact_7486_DERIV__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A,Sa: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,Xa,Sa))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,Xa,Sa))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_wq(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),minus_minus(A),D4),E4),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% DERIV_diff
tff(fact_7487_field__differentiable__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F4: A,F3: filter(A),G: fun(A,A),G4: A] :
          ( has_field_derivative(A,F2,F4,F3)
         => ( has_field_derivative(A,G,G4,F3)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_wq(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),minus_minus(A),F4),G4),F3) ) ) ) ).

% field_differentiable_diff
tff(fact_7488_has__field__derivative__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,F3: filter(A),C2: real] :
          ( has_field_derivative(A,F2,D4,F3)
         => has_field_derivative(A,aa(real,fun(A,A),aTP_Lamp_wr(fun(A,A),fun(real,fun(A,A)),F2),C2),aa(A,A,real_V8093663219630862766scaleR(A,C2),D4),F3) ) ) ).

% has_field_derivative_scaleR_right
tff(fact_7489_DERIV__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A,Sa: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,Xa,Sa))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,Xa,Sa))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_ws(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),D4),E4),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% DERIV_add
tff(fact_7490_field__differentiable__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F4: A,F3: filter(A),G: fun(A,A),G4: A] :
          ( has_field_derivative(A,F2,F4,F3)
         => ( has_field_derivative(A,G,G4,F3)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_ws(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),F4),G4),F3) ) ) ) ).

% field_differentiable_add
tff(fact_7491_field__differentiable__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F4: A,F3: filter(A)] :
          ( has_field_derivative(A,F2,F4,F3)
         => has_field_derivative(A,aTP_Lamp_wt(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),F4),F3) ) ) ).

% field_differentiable_minus
tff(fact_7492_DERIV__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A,Sa: set(A)] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,Xa,Sa))
         => has_field_derivative(A,aTP_Lamp_wt(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),D4),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% DERIV_minus
tff(fact_7493_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K: A,F3: filter(A)] : has_field_derivative(A,aTP_Lamp_wu(A,fun(A,A),K),zero_zero(A),F3) ) ).

% DERIV_const
tff(fact_7494_DERIV__ident,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: filter(A)] : has_field_derivative(A,aTP_Lamp_wv(A,A),one_one(A),F3) ) ).

% DERIV_ident
tff(fact_7495_DERIV__cmult__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A,Sa: set(A),C2: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,Xa,Sa))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_ww(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),D4),C2),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% DERIV_cmult_right
tff(fact_7496_DERIV__cmult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A,Sa: set(A),C2: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,Xa,Sa))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_wx(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D4),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% DERIV_cmult
tff(fact_7497_has__field__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xa: A,Sa: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xa,Sa))
         => has_field_derivative(A,aTP_Lamp_wy(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cosh(A),aa(A,A,G,Xa))),Db),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% has_field_derivative_sinh
tff(fact_7498_has__field__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xa: A,Sa: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xa,Sa))
         => has_field_derivative(A,aTP_Lamp_wz(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sinh(A),aa(A,A,G,Xa))),Db),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% has_field_derivative_cosh
tff(fact_7499_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A,Sa: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,Xa,Sa))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,Xa,Sa))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_xa(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,Xa)),E4)),aa(A,A,aa(A,fun(A,A),times_times(A),D4),aa(A,A,G,Xa))),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% DERIV_mult'
tff(fact_7500_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,Xa: A,Sa: set(A),G: fun(A,A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,Xa,Sa))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xa,Sa))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_xa(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,Xa))),aa(A,A,aa(A,fun(A,A),times_times(A),Db),aa(A,A,F2,Xa))),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% DERIV_mult
tff(fact_7501_DERIV__cdivide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A,Sa: set(A),C2: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,Xa,Sa))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_xb(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),D4),C2),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% DERIV_cdivide
tff(fact_7502_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A,Sa: set(A)] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,Xa,Sa))
         => ( ( aa(A,A,F2,Xa) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_xc(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,Xa))),D4)),aa(A,A,inverse_inverse(A),aa(A,A,F2,Xa)))),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% DERIV_inverse'
tff(fact_7503_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A,Sa: set(A),Na: nat] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,Xa,Sa))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_xd(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),D4),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,Xa)),Na))),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% DERIV_power_Suc
tff(fact_7504_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A,Sa: set(A),Na: nat] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,Xa,Sa))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_xe(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),D4),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% DERIV_power
tff(fact_7505_DERIV__const__average,axiom,
    ! [A2: real,B2: real,V: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X: real] : has_field_derivative(real,V,K,topolo174197925503356063within(real,X,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),A2),B2)),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,A2)),aa(real,real,V,B2))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ) ).

% DERIV_const_average
tff(fact_7506_DERIV__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [S: set(A),F2: fun(B,fun(A,B)),F4: fun(C,fun(A,B)),Xa: C,F3: filter(B)] :
          ( ! [N: A] :
              ( member(A,N,S)
             => has_field_derivative(B,aa(A,fun(B,B),aTP_Lamp_xf(fun(B,fun(A,B)),fun(A,fun(B,B)),F2),N),aa(A,B,aa(C,fun(A,B),F4,Xa),N),F3) )
         => has_field_derivative(B,aa(fun(B,fun(A,B)),fun(B,B),aTP_Lamp_xg(set(A),fun(fun(B,fun(A,B)),fun(B,B)),S),F2),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(C,fun(A,B),F4,Xa)),S),F3) ) ) ).

% DERIV_sum
tff(fact_7507_DERIV__at__within__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,Xa: A,S: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Xa),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Z)),S)))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_xh(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,Xa,S)) ) ) ).

% DERIV_at_within_shift
tff(fact_7508_termdiffs__strong__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xa: A] :
          ( ! [Y3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),C2),Y3))
         => has_field_derivative(A,aTP_Lamp_xi(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_il(fun(nat,A),fun(A,fun(nat,A)),C2),Xa)),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ).

% termdiffs_strong_converges_everywhere
tff(fact_7509_at__within__Icc__at,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,Xa: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),B2)
           => ( topolo174197925503356063within(A,Xa,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,Xa,top_top(set(A))) ) ) ) ) ).

% at_within_Icc_at
tff(fact_7510_DERIV__fun__pow,axiom,
    ! [G: fun(real,real),M: real,Xa: real,Na: nat] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_xj(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,aa(real,fun(nat,real),power_power(real),aa(real,real,G,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))))),M),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_fun_pow
tff(fact_7511_at__discrete,axiom,
    ! [A: $tType] :
      ( topolo8865339358273720382pology(A)
     => ! [Xa: A,S: set(A)] : topolo174197925503356063within(A,Xa,S) = bot_bot(filter(A)) ) ).

% at_discrete
tff(fact_7512_trivial__limit__at__left__bot,axiom,
    ! [A: $tType] :
      ( ( order_bot(A)
        & topolo1944317154257567458pology(A) )
     => ( topolo174197925503356063within(A,bot_bot(A),set_ord_lessThan(A,bot_bot(A))) = bot_bot(filter(A)) ) ) ).

% trivial_limit_at_left_bot
tff(fact_7513_trivial__limit__at__left__real,axiom,
    ! [A: $tType] :
      ( ( dense_order(A)
        & no_bot(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xa: A] : topolo174197925503356063within(A,Xa,set_ord_lessThan(A,Xa)) != bot_bot(filter(A)) ) ).

% trivial_limit_at_left_real
tff(fact_7514_at__within__union,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xa: A,S: set(A),T3: set(A)] : topolo174197925503356063within(A,Xa,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),topolo174197925503356063within(A,Xa,S)),topolo174197925503356063within(A,Xa,T3)) ) ).

% at_within_union
tff(fact_7515_DERIV__cmult__Id,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,Xa: A,Sa: set(A)] : has_field_derivative(A,aa(A,fun(A,A),times_times(A),C2),C2,topolo174197925503356063within(A,Xa,Sa)) ) ).

% DERIV_cmult_Id
tff(fact_7516_has__real__derivative__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,Xa: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,S))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),H3),S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xa)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),H3))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_7517_has__real__derivative__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,Xa: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,S))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),H3),S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D3)
                   => 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),Xa),H3))),aa(real,real,F2,Xa)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_7518_has__real__derivative__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,Xa: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,S))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( member(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),H3),S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xa)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),H3))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_7519_has__real__derivative__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,Xa: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,S))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( member(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),H3),S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),H3))),aa(real,real,F2,Xa)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_7520_at__within__Icc__at__left,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( topolo174197925503356063within(A,B2,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2)) ) ) ) ).

% at_within_Icc_at_left
tff(fact_7521_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,Xa: A,Sa: set(A),G: fun(A,A),E: A] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,Xa,Sa))
         => ( has_field_derivative(A,G,E,topolo174197925503356063within(A,Xa,Sa))
           => ( ( aa(A,A,G,Xa) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_wp(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,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D2),aa(A,A,G,Xa))),aa(A,A,aa(A,fun(A,A),times_times(A),E),aa(A,A,F2,Xa)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,G,Xa)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ) ).

% DERIV_quotient
tff(fact_7522_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,Xa: A,Sa: set(A)] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,Xa,Sa))
         => ( ( aa(A,A,F2,Xa) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_xc(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,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,Xa)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_7523_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K6: real,C2: fun(nat,A),F2: fun(A,A),F4: A,Z: A] :
          ( ! [Z3: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z3)),K6)
             => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),aa(A,A,F2,Z3)) )
         => ( has_field_derivative(A,F2,F4,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K6)
             => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_il(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),F4) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_7524_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_xk(real,fun(real,real),R),aa(real,real,aa(real,fun(real,real),times_times(real),R),powr(real,Z,aa(real,real,aa(real,fun(real,real),minus_minus(real),R),one_one(real)))),topolo174197925503356063within(real,Z,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_7525_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K6: real,C2: fun(nat,A),Z: A] :
          ( ! [Z3: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z3)),K6)
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K6)
           => has_field_derivative(A,aTP_Lamp_xi(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_il(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),topolo174197925503356063within(A,Z,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_7526_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,Xa: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,K6))
           => has_field_derivative(A,aTP_Lamp_xi(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_il(fun(nat,A),fun(A,fun(nat,A)),C2),Xa)),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_7527_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,Xa: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_il(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_xl(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,K6))
               => has_field_derivative(A,aTP_Lamp_xi(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_il(fun(nat,A),fun(A,fun(nat,A)),C2),Xa)),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_7528_DERIV__fun__powr,axiom,
    ! [G: fun(real,real),M: real,Xa: real,R: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G,Xa))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_xm(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,Xa),aa(real,real,aa(real,fun(real,real),minus_minus(real),R),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),M),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_7529_DERIV__log,axiom,
    ! [Xa: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => has_field_derivative(real,log(B2),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,ln_ln(real),B2)),Xa)),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_7530_DERIV__powr,axiom,
    ! [G: fun(real,real),M: real,Xa: real,F2: fun(real,real),R: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G,Xa))
       => ( has_field_derivative(real,F2,R,topolo174197925503356063within(real,Xa,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(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,Xa),aa(real,real,F2,Xa))),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,Xa)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),M),aa(real,real,F2,Xa))),aa(real,real,G,Xa)))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_7531_DERIV__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( aa(A,A,cos(A),Xa) != zero_zero(A) )
         => has_field_derivative(A,tan(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cos(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ).

% DERIV_tan
tff(fact_7532_DERIV__real__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => has_field_derivative(real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xa))),aa(num,real,numeral_numeral(real),bit0(one2))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_real_sqrt
tff(fact_7533_DERIV__arctan,axiom,
    ! [Xa: 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,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ).

% DERIV_arctan
tff(fact_7534_arsinh__real__has__field__derivative,axiom,
    ! [Xa: real,A3: 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,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,Xa,A3)) ).

% arsinh_real_has_field_derivative
tff(fact_7535_DERIV__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( aa(A,A,sin(A),Xa) != zero_zero(A) )
         => has_field_derivative(A,cot(A),aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sin(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ).

% DERIV_cot
tff(fact_7536_has__field__derivative__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Xa: A,Db: A,Sa: set(A)] :
          ( ( aa(A,A,cosh(A),aa(A,A,G,Xa)) != zero_zero(A) )
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xa,Sa))
           => has_field_derivative(A,aTP_Lamp_xo(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tanh(A),aa(A,A,G,Xa))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Db),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_7537_DERIV__real__sqrt__generic,axiom,
    ! [Xa: real,D4: real] :
      ( ( Xa != zero_zero(real) )
     => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( D4 = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xa))),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
       => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
           => ( D4 = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xa)))),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
         => has_field_derivative(real,sqrt,D4,topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_7538_arcosh__real__has__field__derivative,axiom,
    ! [Xa: real,A3: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => has_field_derivative(real,arcosh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,Xa,A3)) ) ).

% arcosh_real_has_field_derivative
tff(fact_7539_artanh__real__has__field__derivative,axiom,
    ! [Xa: real,A3: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => has_field_derivative(real,artanh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,Xa,A3)) ) ).

% artanh_real_has_field_derivative
tff(fact_7540_slice__head,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs))
       => ( hd(A,slice(A,From,To,Xs)) = aa(nat,A,nth(A,Xs),From) ) ) ) ).

% slice_head
tff(fact_7541_DERIV__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => has_field_derivative(real,root(Na),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ).

% DERIV_real_root
tff(fact_7542_DERIV__arccos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => has_field_derivative(real,arccos,aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ).

% DERIV_arccos
tff(fact_7543_DERIV__arcsin,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => has_field_derivative(real,arcsin,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ).

% DERIV_arcsin
tff(fact_7544_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Xa: real,Na: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M4: nat,X: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),X),topolo174197925503356063within(real,X,top_top(set(real))))
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xa))
            & ( aa(real,real,F2,Xa) = 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_xp(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_7545_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Xa: real,Na: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
        & ! [M4: nat,X: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),X),topolo174197925503356063within(real,X,top_top(set(real)))) )
     => ? [T4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xa))
          & ( aa(real,real,F2,Xa) = 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_xp(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_7546_DERIV__odd__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( ( Xa != zero_zero(real) )
       => has_field_derivative(real,root(Na),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ).

% DERIV_odd_real_root
tff(fact_7547_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 )
         => ( ! [M4: nat,T4: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),H),T4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),zero_zero(real)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
           => ? [T4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),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_xq(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),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Na))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_7548_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 )
       => ( ! [M4: nat,T4: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),H) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ? [T4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T4)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),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_xq(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),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Na))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_7549_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 )
         => ( ! [M4: nat,T4: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),H) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
           => ? [T4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),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_xq(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),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Na))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_7550_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Na: nat,Xa: 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)
       => ( ( Xa != zero_zero(real) )
         => ( ! [M4: nat,X: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),X),topolo174197925503356063within(real,X,top_top(set(real))))
           => ? [T4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T4))
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xa))
                & ( aa(real,real,F2,Xa) = 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_xp(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_7551_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Na: nat,Xa: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M4: nat,T4: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xa)) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xa))
            & ( aa(real,real,F2,Xa) = 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_xp(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_7552_Taylor__down,axiom,
    ! [Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: 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 )
       => ( ! [M4: nat,T4: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),C2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C2),B2)
             => ? [T4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),T4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),C2)
                  & ( aa(real,real,F2,A2) = 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_xr(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A2),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),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),C2)),Na))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_7553_Taylor__up,axiom,
    ! [Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: 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 )
       => ( ! [M4: nat,T4: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),C2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),B2)
             => ? [T4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),T4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),B2)
                  & ( aa(real,real,F2,B2) = 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_xr(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B2),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),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),C2)),Na))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_7554_Taylor,axiom,
    ! [Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M4: nat,T4: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),C2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C2),B2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Xa)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),B2)
                 => ( ( Xa != C2 )
                   => ? [T4: real] :
                        ( $ite(
                            aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),C2),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),T4)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),C2) ),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),T4)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),Xa) ) )
                        & ( aa(real,real,F2,Xa) = 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_xs(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C2),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T4)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),C2)),Na))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_7555_Maclaurin__lemma2,axiom,
    ! [Na: nat,H: real,Diff: fun(nat,fun(real,real)),K: nat,B3: real] :
      ( ! [M4: nat,T4: real] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),H) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
     => ( ( Na = aa(nat,nat,suc,K) )
       => ! [M2: nat,T8: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),Na)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T8)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T8),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_xu(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Na),Diff),B3),M2),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T8)),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_xv(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M2),T8)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,M2))))),aa(real,real,aa(real,fun(real,real),times_times(real),B3),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),T8),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,M2)))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,M2))))))),topolo174197925503356063within(real,T8,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_7556_DERIV__arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => has_field_derivative(real,aTP_Lamp_xw(real,real),suminf(real,aTP_Lamp_xx(real,fun(nat,real),Xa)),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_7557_DERIV__even__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
         => has_field_derivative(real,root(Na),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ) ).

% DERIV_even_real_root
tff(fact_7558_DERIV__power__series_H,axiom,
    ! [R2: real,F2: fun(nat,real),X0: real] :
      ( ! [X: real] :
          ( member(real,X,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R2),R2))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_xy(fun(nat,real),fun(real,fun(nat,real)),F2),X)) )
     => ( member(real,X0,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R2),R2))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
         => has_field_derivative(real,aTP_Lamp_ya(fun(nat,real),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),aTP_Lamp_xy(fun(nat,real),fun(real,fun(nat,real)),F2),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_7559_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xa: A,G4: fun(A,real),Sa: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G,Xa)),one_one(real))
           => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xa,Sa))
             => has_derivative(A,real,aTP_Lamp_yb(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_yc(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xa),G4),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_7560_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or5935395276787703475ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),U) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_7561_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( hd(nat,upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_7562_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_7563_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or5935395276787703475ssThan(A,A2,B2) = bot_bot(set(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_7564_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A2,B2) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_7565_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ finite_finite2(A,set_or5935395276787703475ssThan(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ioo_iff
tff(fact_7566_has__derivative__compose,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),Xa: A,Sa: set(A),G: fun(B,C),G4: fun(B,C)] :
          ( has_derivative(A,B,F2,F4,topolo174197925503356063within(A,Xa,Sa))
         => ( has_derivative(B,C,G,G4,topolo174197925503356063within(B,aa(A,B,F2,Xa),top_top(set(B))))
           => has_derivative(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_yd(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),aa(fun(B,C),fun(A,C),aTP_Lamp_yd(fun(A,B),fun(fun(B,C),fun(A,C)),F4),G4),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_derivative_compose
tff(fact_7567_has__derivative__zero__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(A,B),Xa: A] :
          ( has_derivative(A,B,aTP_Lamp_ye(A,B),F3,topolo174197925503356063within(A,Xa,top_top(set(A))))
         => ! [X5: A] : aa(A,B,F3,X5) = zero_zero(B) ) ) ).

% has_derivative_zero_unique
tff(fact_7568_has__derivative__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,real),F4: fun(A,real),Xa: A,Sa: set(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,real,F2,F4,topolo174197925503356063within(A,Xa,Sa))
         => ( has_derivative(A,B,G,G4,topolo174197925503356063within(A,Xa,Sa))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yf(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_yg(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F4),Xa),G),G4),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_derivative_scaleR
tff(fact_7569_has__derivative__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),Xa: A,Sa: set(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F4,topolo174197925503356063within(A,Xa,Sa))
         => ( has_derivative(A,B,G,G4,topolo174197925503356063within(A,Xa,Sa))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yh(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_yi(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F4),Xa),G),G4),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_derivative_mult
tff(fact_7570_has__derivative__in__compose2,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [T: set(A),G: fun(A,B),G4: fun(A,fun(A,B)),F2: fun(C,A),Sa: set(C),Xa: C,F4: fun(C,A)] :
          ( ! [X: A] :
              ( member(A,X,T)
             => has_derivative(A,B,G,aa(A,fun(A,B),G4,X),topolo174197925503356063within(A,X,T)) )
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,F2),Sa)),T)
           => ( member(C,Xa,Sa)
             => ( has_derivative(C,A,F2,F4,topolo174197925503356063within(C,Xa,Sa))
               => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_yj(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_yk(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),G4),F2),Xa),F4),topolo174197925503356063within(C,Xa,Sa)) ) ) ) ) ) ).

% has_derivative_in_compose2
tff(fact_7571_has__derivative__in__compose,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),Xa: A,Sa: set(A),G: fun(B,C),G4: fun(B,C)] :
          ( has_derivative(A,B,F2,F4,topolo174197925503356063within(A,Xa,Sa))
         => ( has_derivative(B,C,G,G4,topolo174197925503356063within(B,aa(A,B,F2,Xa),aa(set(A),set(B),image(A,B,F2),Sa)))
           => has_derivative(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_yd(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),aa(fun(B,C),fun(A,C),aTP_Lamp_yd(fun(A,B),fun(fun(B,C),fun(A,C)),F4),G4),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_derivative_in_compose
tff(fact_7572_has__field__derivative__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,F3: filter(A)] :
          ( has_field_derivative(A,F2,D4,F3)
        <=> has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D4),F3) ) ) ).

% has_field_derivative_def
tff(fact_7573_has__derivative__imp__has__field__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: fun(A,A),F3: filter(A),D7: A] :
          ( has_derivative(A,A,F2,D4,F3)
         => ( ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X),D7) = aa(A,A,D4,X)
           => has_field_derivative(A,F2,D7,F3) ) ) ) ).

% has_derivative_imp_has_field_derivative
tff(fact_7574_has__field__derivative__imp__has__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,F3: filter(A)] :
          ( has_field_derivative(A,F2,D4,F3)
         => has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D4),F3) ) ) ).

% has_field_derivative_imp_has_derivative
tff(fact_7575_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ finite_finite2(A,set_or5935395276787703475ssThan(A,A2,B2)) ) ) ).

% infinite_Ioo
tff(fact_7576_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)),F4: fun(A,fun(B,C)),F3: filter(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => has_derivative(B,C,aa(A,fun(B,C),F2,I3),aa(A,fun(B,C),F4,I3),F3) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ym(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ym(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F4),F3) ) ) ).

% has_derivative_sum
tff(fact_7577_has__derivative__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G: fun(A,B),G4: fun(A,B),F3: filter(A),Y: B] :
          ( has_derivative(A,B,G,G4,F3)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_yn(fun(A,B),fun(B,fun(A,B)),G),Y),aa(B,fun(A,B),aTP_Lamp_yn(fun(A,B),fun(B,fun(A,B)),G4),Y),F3) ) ) ).

% has_derivative_mult_left
tff(fact_7578_has__derivative__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G: fun(A,B),G4: fun(A,B),F3: filter(A),Xa: B] :
          ( has_derivative(A,B,G,G4,F3)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_yo(fun(A,B),fun(B,fun(A,B)),G),Xa),aa(B,fun(A,B),aTP_Lamp_yo(fun(A,B),fun(B,fun(A,B)),G4),Xa),F3) ) ) ).

% has_derivative_mult_right
tff(fact_7579_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C2: B,F3: filter(A)] : has_derivative(A,B,aTP_Lamp_yp(B,fun(A,B),C2),aTP_Lamp_ye(A,B),F3) ) ).

% has_derivative_const
tff(fact_7580_has__derivative__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),F3: filter(A)] :
          ( has_derivative(A,B,F2,F4,F3)
         => has_derivative(A,B,aTP_Lamp_yq(fun(A,B),fun(A,B),F2),aTP_Lamp_yq(fun(A,B),fun(A,B),F4),F3) ) ) ).

% has_derivative_minus
tff(fact_7581_has__derivative__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),F3: filter(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F4,F3)
         => ( has_derivative(A,B,G,G4,F3)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_yr(fun(A,B),fun(fun(A,B),fun(A,B)),F4),G4),F3) ) ) ) ).

% has_derivative_add
tff(fact_7582_has__derivative__scaleR__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,B),G4: fun(A,B),F3: filter(A),R: real] :
          ( has_derivative(A,B,G,G4,F3)
         => has_derivative(A,B,aa(real,fun(A,B),aTP_Lamp_ys(fun(A,B),fun(real,fun(A,B)),G),R),aa(real,fun(A,B),aTP_Lamp_ys(fun(A,B),fun(real,fun(A,B)),G4),R),F3) ) ) ).

% has_derivative_scaleR_right
tff(fact_7583_has__derivative__scaleR__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),G4: fun(A,real),F3: filter(A),Xa: B] :
          ( has_derivative(A,real,G,G4,F3)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_yt(fun(A,real),fun(B,fun(A,B)),G),Xa),aa(B,fun(A,B),aTP_Lamp_yt(fun(A,real),fun(B,fun(A,B)),G4),Xa),F3) ) ) ).

% has_derivative_scaleR_left
tff(fact_7584_has__derivative__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: filter(A)] : has_derivative(A,A,aTP_Lamp_yu(A,A),aTP_Lamp_yu(A,A),F3) ) ).

% has_derivative_ident
tff(fact_7585_has__derivative__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),G4: fun(A,real),F3: filter(A)] :
          ( has_derivative(A,real,G,G4,F3)
         => has_derivative(A,B,aTP_Lamp_yv(fun(A,real),fun(A,B),G),aTP_Lamp_yv(fun(A,real),fun(A,B),G4),F3) ) ) ).

% has_derivative_of_real
tff(fact_7586_has__derivative__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),F3: filter(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F4,F3)
         => ( has_derivative(A,B,G,G4,F3)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yw(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_yw(fun(A,B),fun(fun(A,B),fun(A,B)),F4),G4),F3) ) ) ) ).

% has_derivative_diff
tff(fact_7587_has__derivative__exp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),Xa: A,Sa: set(A)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xa,Sa))
         => has_derivative(A,real,aTP_Lamp_yx(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_yy(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G4),Xa),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% has_derivative_exp
tff(fact_7588_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or5935395276787703475ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_7589_ivl__disj__int__two_I5_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(5)
tff(fact_7590_ivl__disj__int__two_I4_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(4)
tff(fact_7591_ivl__disj__int__two_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(1)
tff(fact_7592_ivl__disj__int__one_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_atMost(A,L)),set_or5935395276787703475ssThan(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(1)
tff(fact_7593_tanh__real__bounds,axiom,
    ! [Xa: real] : member(real,aa(real,real,tanh(real),Xa),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real))) ).

% tanh_real_bounds
tff(fact_7594_has__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xa: A,Sa: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,Xa,Sa))
         => has_derivative(A,A,aTP_Lamp_wz(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),aa(A,A,sinh(A),aa(A,A,G,Xa))),Db)),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% has_derivative_cosh
tff(fact_7595_has__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xa: A,Sa: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,Xa,Sa))
         => has_derivative(A,A,aTP_Lamp_wy(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),aa(A,A,cosh(A),aa(A,A,G,Xa))),Db)),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% has_derivative_sinh
tff(fact_7596_has__derivative__sin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),Xa: A,Sa: set(A)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xa,Sa))
         => has_derivative(A,real,aTP_Lamp_yz(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_za(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G4),Xa),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% has_derivative_sin
tff(fact_7597_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_7598_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_7599_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_7600_atLeastAtMost__diff__ends,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A2,B2) ) ).

% atLeastAtMost_diff_ends
tff(fact_7601_ivl__disj__un__one_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_atMost(A,L)),set_or5935395276787703475ssThan(A,L,U)) = set_ord_lessThan(A,U) ) ) ) ).

% ivl_disj_un_one(1)
tff(fact_7602_has__derivative__divide_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),Xa: A,S: set(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F4,topolo174197925503356063within(A,Xa,S))
         => ( has_derivative(A,B,G,G4,topolo174197925503356063within(A,Xa,S))
           => ( ( aa(A,B,G,Xa) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zb(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_zc(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F4),Xa),G),G4),topolo174197925503356063within(A,Xa,S)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_7603_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xa: A,S: set(A)] :
          ( ( Xa != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_zd(A,fun(A,A),Xa),topolo174197925503356063within(A,Xa,S)) ) ) ).

% has_derivative_inverse'
tff(fact_7604_has__derivative__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(B,A),Xa: B,F4: fun(B,A),S: set(B)] :
          ( ( aa(B,A,F2,Xa) != zero_zero(A) )
         => ( has_derivative(B,A,F2,F4,topolo174197925503356063within(B,Xa,S))
           => has_derivative(B,A,aTP_Lamp_ze(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_zf(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),F2),Xa),F4),topolo174197925503356063within(B,Xa,S)) ) ) ) ).

% has_derivative_inverse
tff(fact_7605_DERIV__compose__FDERIV,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(real,real),F4: real,G: fun(A,real),Xa: A,G4: fun(A,real),Sa: set(A)] :
          ( has_field_derivative(real,F2,F4,topolo174197925503356063within(real,aa(A,real,G,Xa),top_top(set(real))))
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xa,Sa))
           => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zg(fun(real,real),fun(fun(A,real),fun(A,real)),F2),G),aa(fun(A,real),fun(A,real),aTP_Lamp_zh(real,fun(fun(A,real),fun(A,real)),F4),G4),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% DERIV_compose_FDERIV
tff(fact_7606_has__derivative__cos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),Xa: A,Sa: set(A)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xa,Sa))
         => has_derivative(A,real,aTP_Lamp_zi(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_zj(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G4),Xa),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% has_derivative_cos
tff(fact_7607_DERIV__isconst3,axiom,
    ! [A2: real,B2: real,Xa: real,Y: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( member(real,Xa,set_or5935395276787703475ssThan(real,A2,B2))
       => ( member(real,Y,set_or5935395276787703475ssThan(real,A2,B2))
         => ( ! [X: real] :
                ( member(real,X,set_or5935395276787703475ssThan(real,A2,B2))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X,top_top(set(real)))) )
           => ( aa(real,real,F2,Xa) = aa(real,real,F2,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_7608_ivl__disj__un__singleton_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(3)
tff(fact_7609_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_7610_has__derivative__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),Xa: A,S: set(A),Na: nat] :
          ( has_derivative(A,B,F2,F4,topolo174197925503356063within(A,Xa,S))
         => has_derivative(A,B,aa(nat,fun(A,B),aTP_Lamp_zk(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_zl(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),F2),F4),Xa),Na),topolo174197925503356063within(A,Xa,S)) ) ) ).

% has_derivative_power
tff(fact_7611_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xa: A,G4: fun(A,real),Sa: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xa))
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xa,Sa))
           => has_derivative(A,real,aTP_Lamp_zm(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_zn(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xa),G4),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_derivative_ln
tff(fact_7612_has__derivative__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),Xa: A,S: set(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F4,topolo174197925503356063within(A,Xa,S))
         => ( has_derivative(A,B,G,G4,topolo174197925503356063within(A,Xa,S))
           => ( ( aa(A,B,G,Xa) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zo(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_zp(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F4),Xa),G),G4),topolo174197925503356063within(A,Xa,S)) ) ) ) ) ).

% has_derivative_divide
tff(fact_7613_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)),F4: fun(A,fun(B,C)),Xa: B,S: set(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => has_derivative(B,C,aa(A,fun(B,C),F2,I3),aa(A,fun(B,C),F4,I3),topolo174197925503356063within(B,Xa,S)) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_zr(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_zt(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),I5),F2),F4),Xa),topolo174197925503356063within(B,Xa,S)) ) ) ).

% has_derivative_prod
tff(fact_7614_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),Xa: A,X7: set(A),F2: fun(A,real),F4: fun(A,real)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xa,X7))
         => ( has_derivative(A,real,F2,F4,topolo174197925503356063within(A,Xa,X7))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xa))
             => ( member(A,Xa,X7)
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zu(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_zv(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G),G4),Xa),F2),F4),topolo174197925503356063within(A,Xa,X7)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_7615_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xa: A,G4: fun(A,real),Sa: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xa))
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xa,Sa))
           => has_derivative(A,real,aTP_Lamp_zw(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_zx(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xa),G4),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_7616_has__derivative__arctan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),Xa: A,Sa: set(A)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xa,Sa))
         => has_derivative(A,real,aTP_Lamp_zy(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_zz(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G4),Xa),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% has_derivative_arctan
tff(fact_7617_has__derivative__tan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xa: A,G4: fun(A,real),Sa: set(A)] :
          ( ( aa(real,real,cos(real),aa(A,real,G,Xa)) != zero_zero(real) )
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xa,Sa))
           => has_derivative(A,real,aTP_Lamp_aaa(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_aab(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xa),G4),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_derivative_tan
tff(fact_7618_DERIV__series_H,axiom,
    ! [F2: fun(real,fun(nat,real)),F4: fun(real,fun(nat,real)),X0: real,A2: real,B2: real,L5: fun(nat,real)] :
      ( ! [N: nat] : has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_aac(fun(real,fun(nat,real)),fun(nat,fun(real,real)),F2),N),aa(nat,real,aa(real,fun(nat,real),F4,X0),N),topolo174197925503356063within(real,X0,top_top(set(real))))
     => ( ! [X: real] :
            ( member(real,X,set_or5935395276787703475ssThan(real,A2,B2))
           => summable(real,aa(real,fun(nat,real),F2,X)) )
       => ( member(real,X0,set_or5935395276787703475ssThan(real,A2,B2))
         => ( summable(real,aa(real,fun(nat,real),F4,X0))
           => ( summable(real,L5)
             => ( ! [N: nat,X: real,Y3: real] :
                    ( member(real,X,set_or5935395276787703475ssThan(real,A2,B2))
                   => ( member(real,Y3,set_or5935395276787703475ssThan(real,A2,B2))
                     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),F2,X),N)),aa(nat,real,aa(real,fun(nat,real),F2,Y3),N)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,L5,N)),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y3)))) ) )
               => has_field_derivative(real,aTP_Lamp_aad(fun(real,fun(nat,real)),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),F4,X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_7619_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xa: A,G4: fun(A,real),Sa: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G,Xa)),one_one(real))
           => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xa,Sa))
             => has_derivative(A,real,aTP_Lamp_aae(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_aaf(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xa),G4),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_7620_has__derivative__floor,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [G: fun(B,real),Xa: B,F2: fun(real,A),G4: fun(B,real),Sa: set(B)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,aa(B,real,G,Xa),top_top(set(real))),F2)
         => ( ~ member(A,aa(real,A,F2,aa(B,real,G,Xa)),ring_1_Ints(A))
           => ( has_derivative(B,real,G,G4,topolo174197925503356063within(B,Xa,Sa))
             => has_derivative(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_aag(fun(B,real),fun(fun(real,A),fun(B,real)),G),F2),aTP_Lamp_aah(fun(B,real),fun(B,real),G4),topolo174197925503356063within(B,Xa,Sa)) ) ) ) ) ).

% has_derivative_floor
tff(fact_7621_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,Xa: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_xl(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,K6))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aaj(fun(nat,A),fun(A,fun(A,A)),C2),Xa),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_7622_tendsto__const,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [K: B,F3: filter(A)] : filterlim(A,B,aTP_Lamp_aak(B,fun(A,B),K),topolo7230453075368039082e_nhds(B,K),F3) ) ).

% tendsto_const
tff(fact_7623_tendsto__ident__at,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A,Sa: set(A)] : filterlim(A,A,aTP_Lamp_aal(A,A),topolo7230453075368039082e_nhds(A,A2),topolo174197925503356063within(A,A2,Sa)) ) ).

% tendsto_ident_at
tff(fact_7624_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F3: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_aam(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C2)),F3)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_7625_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F3: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_aan(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),L)),F3)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_7626_power__tendsto__0__iff,axiom,
    ! [A: $tType,Na: nat,F2: fun(A,real),F3: filter(A)] :
      ( aa(nat,$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_aao(nat,fun(fun(A,real),fun(A,real)),Na),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% power_tendsto_0_iff
tff(fact_7627_isCont__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_aap(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% isCont_Pair
tff(fact_7628_filterlim__at__within__If,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,B),G2: filter(B),Xa: A,A3: set(A),P: fun(A,$o),G: fun(A,B)] :
          ( filterlim(A,B,F2,G2,topolo174197925503356063within(A,Xa,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,P))))
         => ( filterlim(A,B,G,G2,topolo174197925503356063within(A,Xa,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,aTP_Lamp_aaq(fun(A,$o),fun(A,$o),P)))))
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_aar(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),F2),P),G),G2,topolo174197925503356063within(A,Xa,A3)) ) ) ) ).

% filterlim_at_within_If
tff(fact_7629_isCont__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [A2: A,F2: fun(A,B),Na: nat] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(nat,fun(A,B),aTP_Lamp_aas(fun(A,B),fun(nat,fun(A,B)),F2),Na)) ) ) ).

% isCont_power
tff(fact_7630_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,A),A2: A,D4: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aat(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aau(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_7631_LIM__fun__less__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [R4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
            & ! [X5: real] :
                ( ( ( X5 != C2 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C2),X5))),R4) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,X5)),zero_zero(real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_7632_LIM__fun__not__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( ( L != zero_zero(real) )
       => ? [R4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
            & ! [X5: real] :
                ( ( ( X5 != C2 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C2),X5))),R4) )
               => ( aa(real,real,F2,X5) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_7633_LIM__fun__gt__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [R4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
            & ! [X5: real] :
                ( ( ( X5 != C2 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C2),X5))),R4) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,F2,X5)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_7634_isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A2),top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A2))),D6) )
                     => ( aa(A,B,F2,X) != aa(A,B,F2,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aav(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% isCont_LIM_compose2
tff(fact_7635_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A2))),D6) )
                     => ( aa(A,B,F2,X) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aav(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_compose2
tff(fact_7636_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R2: real,A2: A,F2: fun(A,B),G: fun(A,B),L: B] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
         => ( ! [X: A] :
                ( ( X != A2 )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A2))),R2)
                 => ( aa(A,B,F2,X) = aa(A,B,G,X) ) ) )
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_equal2
tff(fact_7637_LIM__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S6)
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),A2))),S6) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X4)),L5))),R5) ) ) ) ) ) ).

% LIM_eq
tff(fact_7638_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B),L5: B] :
          ( ! [R4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A2))),S7) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X)),L5))),R4) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_7639_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,R: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => ? [S2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S2)
                & ! [X5: A] :
                    ( ( ( X5 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),A2))),S2) )
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X5)),L5))),R) ) ) ) ) ) ).

% LIM_D
tff(fact_7640_LIM__offset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,K: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_aaw(fun(A,B),fun(A,fun(A,B)),F2),K),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),K),top_top(set(A)))) ) ) ).

% LIM_offset
tff(fact_7641_isCont__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_aax(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_diff
tff(fact_7642_isCont__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,top_top(set(A))),exp(A)) ) ).

% isCont_exp
tff(fact_7643_isCont__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,top_top(set(A))),cosh(A)) ) ).

% isCont_cosh
tff(fact_7644_isCont__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_aay(fun(A,B),fun(A,real),F2)) ) ) ).

% isCont_norm
tff(fact_7645_isCont__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,top_top(set(A))),cos(A)) ) ).

% isCont_cos
tff(fact_7646_isCont__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_aaz(fun(A,real),fun(A,B),G)) ) ) ).

% isCont_of_real
tff(fact_7647_isCont__arctan,axiom,
    ! [Xa: real] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),arctan) ).

% isCont_arctan
tff(fact_7648_isCont__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,top_top(set(A))),sin(A)) ) ).

% isCont_sin
tff(fact_7649_isCont__real__sqrt,axiom,
    ! [Xa: real] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),sqrt) ).

% isCont_real_sqrt
tff(fact_7650_isCont__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,top_top(set(A))),sinh(A)) ) ).

% isCont_sinh
tff(fact_7651_isCont__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,real),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_aba(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_scaleR
tff(fact_7652_isCont__real__root,axiom,
    ! [Xa: real,Na: nat] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),root(Na)) ).

% isCont_real_root
tff(fact_7653_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A,L5: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_abb(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_7654_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_abb(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_7655_LIM__isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_abb(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_isCont_iff
tff(fact_7656_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Xa: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xa,top_top(set(A))),F2)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_abc(A,fun(fun(A,B),fun(A,B)),Xa),F2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,Xa)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_7657_isCont__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_abd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_add
tff(fact_7658_isCont__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_abe(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_minus
tff(fact_7659_LIM__not__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo8386298272705272623_space(B)
        & zero(A)
        & topological_t2_space(A) )
     => ! [K: A,A2: B] :
          ( ( K != zero_zero(A) )
         => ~ filterlim(B,A,aTP_Lamp_abf(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(B,A2,top_top(set(B)))) ) ) ).

% LIM_not_zero
tff(fact_7660_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),Xa: C,G: fun(A,B),Sa: set(C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(C,A,F2,Xa),top_top(set(A))),G)
         => ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,Xa,Sa),F2)
           => topolo3448309680560233919inuous(C,B,topolo174197925503356063within(C,Xa,Sa),aa(fun(A,B),fun(C,B),aTP_Lamp_abg(fun(C,A),fun(fun(A,B),fun(C,B)),F2),G)) ) ) ) ).

% continuous_within_compose3
tff(fact_7661_isCont__tendsto__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [L: A,G: fun(A,B),F2: fun(C,A),F3: filter(C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,L,top_top(set(A))),G)
         => ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_abh(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),F3) ) ) ) ).

% isCont_tendsto_compose
tff(fact_7662_LIM__const__not__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo8386298272705272623_space(B)
        & topological_t2_space(A) )
     => ! [K: A,L5: A,A2: B] :
          ( ( K != L5 )
         => ~ filterlim(B,A,aTP_Lamp_abi(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,L5),topolo174197925503356063within(B,A2,top_top(set(B)))) ) ) ).

% LIM_const_not_eq
tff(fact_7663_tendsto__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [G: fun(A,B),L: A,F2: fun(C,A),F3: filter(C)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),topolo174197925503356063within(A,L,top_top(set(A))))
         => ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_abj(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),F3) ) ) ) ).

% tendsto_compose
tff(fact_7664_LIM__const__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo8386298272705272623_space(A) )
     => ! [K: B,L5: B,A2: A] :
          ( filterlim(A,B,aTP_Lamp_abk(B,fun(A,B),K),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( K = L5 ) ) ) ).

% LIM_const_eq
tff(fact_7665_isCont__o2,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4958980785337419405_space(C)
        & topological_t2_space(A) )
     => ! [A2: A,F2: fun(A,B),G: fun(B,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,aa(A,B,F2,A2),top_top(set(B))),G)
           => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(B,C),fun(A,C),aTP_Lamp_abl(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G)) ) ) ) ).

% isCont_o2
tff(fact_7666_filterlim__at__If,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,B),G2: filter(B),Xa: A,P: fun(A,$o),G: fun(A,B)] :
          ( filterlim(A,B,F2,G2,topolo174197925503356063within(A,Xa,collect(A,P)))
         => ( filterlim(A,B,G,G2,topolo174197925503356063within(A,Xa,collect(A,aTP_Lamp_aaq(fun(A,$o),fun(A,$o),P))))
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_aar(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),F2),P),G),G2,topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ) ).

% filterlim_at_If
tff(fact_7667_isCont__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_abm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_mult
tff(fact_7668_has__field__derivativeD,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A,S: set(A)] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,Xa,S))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_abn(fun(A,A),fun(A,fun(A,A)),F2),Xa),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,Xa,S)) ) ) ).

% has_field_derivativeD
tff(fact_7669_has__field__derivative__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A,S: set(A)] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,Xa,S))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_abn(fun(A,A),fun(A,fun(A,A)),F2),Xa),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,Xa,S)) ) ) ).

% has_field_derivative_iff
tff(fact_7670_continuous__at__within__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [A2: A,Sa: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,Sa),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,Sa),aTP_Lamp_abo(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_inverse
tff(fact_7671_continuous__ident,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xa: A,S: set(A)] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,S),aTP_Lamp_abp(A,A)) ) ).

% continuous_ident
tff(fact_7672_continuous__at__within__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,Sa: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,Sa),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,Sa),G)
           => ( ( aa(A,B,G,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,Sa),aa(fun(A,B),fun(A,B),aTP_Lamp_abq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_7673_continuous__within__compose2,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4958980785337419405_space(C)
        & topological_t2_space(A) )
     => ! [Xa: A,Sa: set(A),F2: fun(A,B),G: fun(B,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xa,Sa),F2)
         => ( topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,aa(A,B,F2,Xa),aa(set(A),set(B),image(A,B,F2),Sa)),G)
           => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,Xa,Sa),aa(fun(B,C),fun(A,C),aTP_Lamp_abl(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G)) ) ) ) ).

% continuous_within_compose2
tff(fact_7674_isCont__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [A3: set(A),A2: B,F2: fun(A,fun(B,C))] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,A2,top_top(set(B))),aa(A,fun(B,C),F2,X)) )
         => topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,A2,top_top(set(B))),aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_abs(set(A),fun(fun(A,fun(B,C)),fun(B,C)),A3),F2)) ) ) ).

% isCont_sum
tff(fact_7675_continuous__within__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A,Sa: set(A)] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Z,Sa),sin(A)) ) ).

% continuous_within_sin
tff(fact_7676_continuous__within__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A,Sa: set(A)] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Z,Sa),cos(A)) ) ).

% continuous_within_cos
tff(fact_7677_tendsto__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [F3: filter(A),F2: fun(A,B),A2: B,B2: B] :
          ( ( F3 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
           => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),F3)
             => ( A2 = B2 ) ) ) ) ) ).

% tendsto_unique
tff(fact_7678_tendsto__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),A2: B] : filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),bot_bot(filter(A))) ) ).

% tendsto_bot
tff(fact_7679_nhds__neq__bot,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A] : topolo7230453075368039082e_nhds(A,A2) != bot_bot(filter(A)) ) ).

% nhds_neq_bot
tff(fact_7680_continuous__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B)] : topolo3448309680560233919inuous(A,B,bot_bot(filter(A)),F2) ) ).

% continuous_bot
tff(fact_7681_continuous__trivial__limit,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Net: filter(A),F2: fun(A,B)] :
          ( ( Net = bot_bot(filter(A)) )
         => topolo3448309680560233919inuous(A,B,Net,F2) ) ) ).

% continuous_trivial_limit
tff(fact_7682_tendsto__const__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [F3: filter(A),A2: B,B2: B] :
          ( ( F3 != bot_bot(filter(A)) )
         => ( filterlim(A,B,aTP_Lamp_abt(B,fun(A,B),A2),topolo7230453075368039082e_nhds(B,B2),F3)
          <=> ( A2 = B2 ) ) ) ) ).

% tendsto_const_iff
tff(fact_7683_continuous__at__within__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,Sa: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,Sa),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,Sa),aTP_Lamp_abu(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_sgn
tff(fact_7684_isCont__cos_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_abv(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_cos'
tff(fact_7685_isCont__sin_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_abw(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_sin'
tff(fact_7686_isCont__exp_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_abx(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_exp'
tff(fact_7687_tendsto__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,C),B2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,C,G,topolo7230453075368039082e_nhds(C,B2),F3)
           => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_aby(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),A2),B2)),F3) ) ) ) ).

% tendsto_Pair
tff(fact_7688_continuous__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,C,F3,G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),F3,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_aap(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% continuous_Pair
tff(fact_7689_filterlim__inf,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F24: filter(B),F33: filter(B),F13: filter(A)] :
      ( filterlim(A,B,F2,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F24),F33),F13)
    <=> ( filterlim(A,B,F2,F24,F13)
        & filterlim(A,B,F2,F33,F13) ) ) ).

% filterlim_inf
tff(fact_7690_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,B),F3: filter(A),Na: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_abz(fun(A,B),fun(nat,fun(A,B)),F2),Na),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_null_power
tff(fact_7691_tendsto__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A),Na: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_aca(fun(A,B),fun(nat,fun(A,B)),F2),Na),topolo7230453075368039082e_nhds(B,aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),Na)),F3) ) ) ).

% tendsto_power
tff(fact_7692_tendsto__power__strong,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,nat),B2: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,nat,G,topolo7230453075368039082e_nhds(nat,B2),F3)
           => filterlim(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_acb(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),B2)),F3) ) ) ) ).

% tendsto_power_strong
tff(fact_7693_continuous__power_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,nat)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,nat,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,nat),fun(A,B),aTP_Lamp_acc(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G)) ) ) ) ).

% continuous_power'
tff(fact_7694_continuous__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),F2: fun(A,B),Na: nat] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aa(nat,fun(A,B),aTP_Lamp_aas(fun(A,B),fun(nat,fun(A,B)),F2),Na)) ) ) ).

% continuous_power
tff(fact_7695_Lim__transform__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),G: fun(A,B),F3: filter(A),A2: B] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
          <=> filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F3) ) ) ) ).

% Lim_transform_eq
tff(fact_7696_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_ace(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ).

% LIM_zero_cancel
tff(fact_7697_Lim__transform2,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F3) ) ) ) ).

% Lim_transform2
tff(fact_7698_Lim__transform,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [G: fun(A,B),A2: B,F3: filter(A),F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acf(fun(A,B),fun(fun(A,B),fun(A,B)),G),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3) ) ) ) ).

% Lim_transform
tff(fact_7699_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_ace(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ).

% LIM_zero_iff
tff(fact_7700_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_ace(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% LIM_zero
tff(fact_7701_continuous__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_acg(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_diff
tff(fact_7702_tendsto__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ach(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),F3) ) ) ) ).

% tendsto_diff
tff(fact_7703_tendsto__of__int__floor,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( ring_1(C)
        & topolo4958980785337419405_space(C)
        & archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ member(B,L,ring_1_Ints(B))
           => filterlim(A,C,aTP_Lamp_aci(fun(A,B),fun(A,C),F2),topolo7230453075368039082e_nhds(C,aa(int,C,ring_1_of_int(C),archim6421214686448440834_floor(B,L))),F3) ) ) ) ).

% tendsto_of_int_floor
tff(fact_7704_tendsto__of__int__ceiling,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( ring_1(C)
        & topolo4958980785337419405_space(C)
        & archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ member(B,L,ring_1_Ints(B))
           => filterlim(A,C,aTP_Lamp_acj(fun(A,B),fun(A,C),F2),topolo7230453075368039082e_nhds(C,aa(int,C,ring_1_of_int(C),archimedean_ceiling(B,L))),F3) ) ) ) ).

% tendsto_of_int_ceiling
tff(fact_7705_tendsto__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( ( A2 != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_ack(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,inverse_inverse(B),A2)),F3) ) ) ) ).

% tendsto_inverse
tff(fact_7706_tendsto__exp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aTP_Lamp_acl(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,exp(B),A2)),F3) ) ) ).

% tendsto_exp
tff(fact_7707_continuous__exp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_abx(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_exp
tff(fact_7708_continuous__cosh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_acm(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_cosh
tff(fact_7709_tendsto__cosh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aTP_Lamp_acn(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,cosh(B),A2)),F3) ) ) ).

% tendsto_cosh
tff(fact_7710_tendsto__Complex,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => filterlim(A,complex,aa(fun(A,real),fun(A,complex),aTP_Lamp_aco(fun(A,real),fun(fun(A,real),fun(A,complex)),F2),G),topolo7230453075368039082e_nhds(complex,complex2(A2,B2)),F3) ) ) ).

% tendsto_Complex
tff(fact_7711_continuous__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_aay(fun(A,B),fun(A,real),F2)) ) ) ).

% continuous_norm
tff(fact_7712_tendsto__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,real,aTP_Lamp_acp(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,real_V7770717601297561774m_norm(B,A2)),F3) ) ) ).

% tendsto_norm
tff(fact_7713_tendsto__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aTP_Lamp_abv(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,cos(B),A2)),F3) ) ) ).

% tendsto_cos
tff(fact_7714_tendsto__of__real__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,real),C2: real,F3: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_acq(fun(A,real),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(real,B,real_Vector_of_real(B),C2)),F3)
        <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3) ) ) ).

% tendsto_of_real_iff
tff(fact_7715_continuous__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [F3: filter(A),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,G)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_aaz(fun(A,real),fun(A,B),G)) ) ) ).

% continuous_of_real
tff(fact_7716_tendsto__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),A2: real,F3: filter(A)] :
          ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,A2),F3)
         => filterlim(A,B,aTP_Lamp_acr(fun(A,real),fun(A,B),G),topolo7230453075368039082e_nhds(B,aa(real,B,real_Vector_of_real(B),A2)),F3) ) ) ).

% tendsto_of_real
tff(fact_7717_tendsto__arsinh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => filterlim(A,real,aTP_Lamp_acs(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arsinh(real),A2)),F3) ) ).

% tendsto_arsinh
tff(fact_7718_filterlim__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(A,B),F33: filter(B),F24: filter(A),F2: fun(C,A),F13: filter(C)] :
      ( filterlim(A,B,G,F33,F24)
     => ( filterlim(C,A,F2,F24,F13)
       => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_act(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),F33,F13) ) ) ).

% filterlim_compose
tff(fact_7719_filterlim__ident,axiom,
    ! [A: $tType,F3: filter(A)] : filterlim(A,A,aTP_Lamp_ma(A,A),F3,F3) ).

% filterlim_ident
tff(fact_7720_tendsto__arctan,axiom,
    ! [A: $tType,F2: fun(A,real),Xa: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Xa),F3)
     => filterlim(A,real,aTP_Lamp_acu(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arctan,Xa)),F3) ) ).

% tendsto_arctan
tff(fact_7721_tendsto__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aTP_Lamp_abw(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,sin(B),A2)),F3) ) ) ).

% tendsto_sin
tff(fact_7722_tendsto__real__sqrt,axiom,
    ! [A: $tType,F2: fun(A,real),Xa: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Xa),F3)
     => filterlim(A,real,aTP_Lamp_acv(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,sqrt,Xa)),F3) ) ).

% tendsto_real_sqrt
tff(fact_7723_continuous__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_abw(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_sin
tff(fact_7724_continuous__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_abv(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_cos
tff(fact_7725_continuous__sinh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_acw(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_sinh
tff(fact_7726_tendsto__sinh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aTP_Lamp_acx(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,sinh(B),A2)),F3) ) ) ).

% tendsto_sinh
tff(fact_7727_tendsto__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acy(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,real_V8093663219630862766scaleR(B,A2),B2)),F3) ) ) ) ).

% tendsto_scaleR
tff(fact_7728_continuous__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: filter(A),F2: fun(A,real),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_aba(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_scaleR
tff(fact_7729_tendsto__real__root,axiom,
    ! [A: $tType,F2: fun(A,real),Xa: real,F3: filter(A),Na: nat] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Xa),F3)
     => filterlim(A,real,aa(nat,fun(A,real),aTP_Lamp_acz(fun(A,real),fun(nat,fun(A,real)),F2),Na),topolo7230453075368039082e_nhds(real,aa(real,real,root(Na),Xa)),F3) ) ).

% tendsto_real_root
tff(fact_7730_tendsto__add__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ada(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_add_zero
tff(fact_7731_continuous__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_abd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_add
tff(fact_7732_tendsto__add__const__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [C2: B,F2: fun(A,B),D2: B,F3: filter(A)] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adb(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)),F3)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,D2),F3) ) ) ).

% tendsto_add_const_iff
tff(fact_7733_tendsto__add,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ada(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),A2),B2)),F3) ) ) ) ).

% tendsto_add
tff(fact_7734_tendsto__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aTP_Lamp_adc(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),A2)),F3) ) ) ).

% tendsto_minus
tff(fact_7735_tendsto__minus__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_adc(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),A2)),F3)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3) ) ) ).

% tendsto_minus_cancel
tff(fact_7736_tendsto__minus__cancel__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),Y: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),Y)),F3)
        <=> filterlim(A,B,aTP_Lamp_adc(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,Y),F3) ) ) ).

% tendsto_minus_cancel_left
tff(fact_7737_continuous__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_add(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_minus
tff(fact_7738_tendsto__rabs,axiom,
    ! [A: $tType,F2: fun(A,real),L: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,L),F3)
     => filterlim(A,real,aTP_Lamp_ade(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,abs_abs(real),L)),F3) ) ).

% tendsto_rabs
tff(fact_7739_tendsto__rabs__zero,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => filterlim(A,real,aTP_Lamp_ade(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero
tff(fact_7740_tendsto__rabs__zero__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_ade(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
    <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero_iff
tff(fact_7741_tendsto__rabs__zero__cancel,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_ade(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero_cancel
tff(fact_7742_tendsto__ln,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( ( A2 != zero_zero(real) )
       => filterlim(A,real,aTP_Lamp_im(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,ln_ln(real),A2)),F3) ) ) ).

% tendsto_ln
tff(fact_7743_tendsto__powr,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( ( A2 != zero_zero(real) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adf(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F3) ) ) ) ).

% tendsto_powr
tff(fact_7744_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F3: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( aa(A,A,cos(A),A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_adg(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),A2)),F3) ) ) ) ).

% tendsto_tan
tff(fact_7745_tendsto__norm__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,real,aTP_Lamp_acp(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_norm_zero
tff(fact_7746_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_acp(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_norm_zero_iff
tff(fact_7747_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_acp(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_norm_zero_cancel
tff(fact_7748_tendsto__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( ( aa(B,B,cosh(B),A2) != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_adh(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,tanh(B),A2)),F3) ) ) ) ).

% tendsto_tanh
tff(fact_7749_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F3: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( aa(A,A,sin(A),A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_adi(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),A2)),F3) ) ) ) ).

% tendsto_cot
tff(fact_7750_tendsto__arcosh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
       => filterlim(A,real,aTP_Lamp_adj(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F3) ) ) ).

% tendsto_arcosh
tff(fact_7751_tendsto__log,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
         => ( ( A2 != one_one(real) )
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
             => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adk(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,aa(real,real,log(A2),B2)),F3) ) ) ) ) ) ).

% tendsto_log
tff(fact_7752_tendsto__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ( L != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_adl(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,sgn_sgn(B,L)),F3) ) ) ) ).

% tendsto_sgn
tff(fact_7753_tendsto__artanh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),A2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),one_one(real))
         => filterlim(A,real,aTP_Lamp_adm(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A2)),F3) ) ) ) ).

% tendsto_artanh
tff(fact_7754_continuous__max,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_adn(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_max
tff(fact_7755_tendsto__max,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [X7: fun(A,B),Xa: B,Net: filter(A),Y6: fun(A,B),Y: B] :
          ( filterlim(A,B,X7,topolo7230453075368039082e_nhds(B,Xa),Net)
         => ( filterlim(A,B,Y6,topolo7230453075368039082e_nhds(B,Y),Net)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ado(fun(A,B),fun(fun(A,B),fun(A,B)),X7),Y6),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),ord_max(B),Xa),Y)),Net) ) ) ) ).

% tendsto_max
tff(fact_7756_continuous__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),C2: B] : topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_adp(B,fun(A,B),C2)) ) ).

% continuous_const
tff(fact_7757_tendsto__divide__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_adq(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_divide_zero
tff(fact_7758_tendsto__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F3)
           => ( ( B2 != zero_zero(B) )
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adr(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),A2),B2)),F3) ) ) ) ) ).

% tendsto_divide
tff(fact_7759_tendsto__mult__one,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,one_one(B)),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,one_one(B)),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ads(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,one_one(B)),F3) ) ) ) ).

% tendsto_mult_one
tff(fact_7760_continuous__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_abm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_mult
tff(fact_7761_continuous__mult_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_adt(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_mult'
tff(fact_7762_continuous__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),F2: fun(A,B),C2: B] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aa(B,fun(A,B),aTP_Lamp_adu(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_mult_left
tff(fact_7763_continuous__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),F2: fun(A,B),C2: B] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aa(B,fun(A,B),aTP_Lamp_adv(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_mult_right
tff(fact_7764_tendsto__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_adw(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),L),C2)),F3) ) ) ).

% tendsto_mult_right
tff(fact_7765_tendsto__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_adx(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),C2),L)),F3) ) ) ).

% tendsto_mult_left
tff(fact_7766_tendsto__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ady(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),A2),B2)),F3) ) ) ) ).

% tendsto_mult
tff(fact_7767_tendsto__mult__right__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_adz(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_mult_right_zero
tff(fact_7768_tendsto__mult__left__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_aea(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_mult_left_zero
tff(fact_7769_tendsto__mult__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aeb(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_mult_zero
tff(fact_7770_tendsto__min,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [X7: fun(A,B),Xa: B,Net: filter(A),Y6: fun(A,B),Y: B] :
          ( filterlim(A,B,X7,topolo7230453075368039082e_nhds(B,Xa),Net)
         => ( filterlim(A,B,Y6,topolo7230453075368039082e_nhds(B,Y),Net)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aec(fun(A,B),fun(fun(A,B),fun(A,B)),X7),Y6),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),ord_min(B),Xa),Y)),Net) ) ) ) ).

% tendsto_min
tff(fact_7771_continuous__min,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_aed(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_min
tff(fact_7772_tendsto__one__prod_H,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(A),F2: fun(B,fun(A,C)),F3: filter(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_aee(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I3),topolo7230453075368039082e_nhds(C,one_one(C)),F3) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_aef(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,one_one(C)),F3) ) ) ).

% tendsto_one_prod'
tff(fact_7773_tendsto__null__sum,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),F2: fun(B,fun(A,C)),F3: filter(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_aeg(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I3),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_aeh(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) ) ) ).

% tendsto_null_sum
tff(fact_7774_continuous__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [I5: set(A),F3: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => topolo3448309680560233919inuous(B,C,F3,aa(A,fun(B,C),F2,I3)) )
         => topolo3448309680560233919inuous(B,C,F3,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_abs(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_sum
tff(fact_7775_tendsto__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),F2: fun(A,fun(B,C)),A2: fun(A,C),F3: filter(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => filterlim(B,C,aa(A,fun(B,C),F2,I3),topolo7230453075368039082e_nhds(C,aa(A,C,A2,I3)),F3) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aej(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,aa(set(A),C,groups7311177749621191930dd_sum(A,C,A2),I5)),F3) ) ) ).

% tendsto_sum
tff(fact_7776_tendsto__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [S: set(A),F2: fun(A,fun(B,C)),L5: fun(A,C),F3: filter(B)] :
          ( ! [I3: A] :
              ( member(A,I3,S)
             => filterlim(B,C,aa(A,fun(B,C),F2,I3),topolo7230453075368039082e_nhds(C,aa(A,C,L5,I3)),F3) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ael(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S),F2),topolo7230453075368039082e_nhds(C,groups7121269368397514597t_prod(A,C,L5,S)),F3) ) ) ).

% tendsto_prod
tff(fact_7777_tendsto__prod_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(A),F2: fun(A,fun(B,C)),A2: fun(A,C),F3: filter(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => filterlim(B,C,aa(A,fun(B,C),F2,I3),topolo7230453075368039082e_nhds(C,aa(A,C,A2,I3)),F3) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aen(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,groups7121269368397514597t_prod(A,C,A2,I5)),F3) ) ) ).

% tendsto_prod'
tff(fact_7778_continuous__prod,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [S: set(A),F3: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( member(A,I3,S)
             => topolo3448309680560233919inuous(B,C,F3,aa(A,fun(B,C),F2,I3)) )
         => topolo3448309680560233919inuous(B,C,F3,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aep(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S),F2)) ) ) ).

% continuous_prod
tff(fact_7779_continuous__prod_H,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [I5: set(A),F3: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => topolo3448309680560233919inuous(B,C,F3,aa(A,fun(B,C),F2,I3)) )
         => topolo3448309680560233919inuous(B,C,F3,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aer(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_prod'
tff(fact_7780_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_aes(nat,fun(A,A),Na)) ) ).

% isCont_pochhammer
tff(fact_7781_isCont__arsinh,axiom,
    ! [Xa: real] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),arsinh(real)) ).

% isCont_arsinh
tff(fact_7782_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,Xa,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aet(fun(A,A),fun(A,fun(A,A)),F2),Xa),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_7783_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,Xa,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aet(fun(A,A),fun(A,fun(A,A)),F2),Xa),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_7784_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_aeu(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_7785_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_7786_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K: real,F2: fun(A,B),K6: real] :
          ( aa(real,$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),K6),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_7787_isCont__inverse__function2,axiom,
    ! [A2: real,Xa: real,B2: real,G: fun(real,real),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),B2)
       => ( ! [Z3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z3),B2)
               => ( aa(real,real,G,aa(real,real,F2,Z3)) = Z3 ) ) )
         => ( ! [Z3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z3),B2)
                 => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,top_top(set(real))),F2) ) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,Xa),top_top(set(real))),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_7788_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,Xa: A] :
          ( has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D4),topolo174197925503356063within(A,Xa,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aet(fun(A,A),fun(A,fun(A,A)),F2),Xa),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_7789_isCont__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => ( ( aa(A,B,G,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_abq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% isCont_divide
tff(fact_7790_isCont__ln,axiom,
    ! [Xa: real] :
      ( ( Xa != zero_zero(real) )
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),ln_ln(real)) ) ).

% isCont_ln
tff(fact_7791_isCont__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_abu(fun(A,B),fun(A,B),F2)) ) ) ) ).

% isCont_sgn
tff(fact_7792_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F3: filter(B),A2: A] :
          ( filterlim(A,B,F2,F3,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_aev(fun(A,B),fun(A,fun(A,B)),F2),A2),F3,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_7793_continuous__within__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Sa: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,Sa),F2)
         => ( ( aa(A,A,cos(A),aa(A,A,F2,Xa)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,Sa),aTP_Lamp_adg(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_tan
tff(fact_7794_continuous__within__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Sa: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,Sa),F2)
         => ( ( aa(A,A,sin(A),aa(A,A,F2,Xa)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,Sa),aTP_Lamp_adi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_cot
tff(fact_7795_continuous__at__within__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Xa: A,A3: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xa,A3),F2)
         => ( ( aa(B,B,cosh(B),aa(A,B,F2,Xa)) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xa,A3),aTP_Lamp_aew(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_tanh
tff(fact_7796_CARAT__DERIV,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A,Xa: A] :
          ( has_field_derivative(A,F2,L,topolo174197925503356063within(A,Xa,top_top(set(A))))
        <=> ? [G3: fun(A,A)] :
              ( ! [Z6: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,F2,Z6)),aa(A,A,F2,Xa)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G3,Z6)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z6),Xa))
              & topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,top_top(set(A))),G3)
              & ( aa(A,A,G3,Xa) = L ) ) ) ) ).

% CARAT_DERIV
tff(fact_7797_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X,top_top(set(real))),F2) )
           => ? [M8: A] :
                ( ! [X5: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X5)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X5),B2) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F2,X5)),M8) )
                & ! [N7: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N7),M8)
                   => ? [X: real] :
                        ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
                        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),N7),aa(real,A,F2,X)) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_7798_filterlim__shift__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),D2: A,F3: filter(B),A2: A] :
          ( filterlim(A,B,comp(A,B,A,F2,aa(A,fun(A,A),plus_plus(A),D2)),F3,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D2),top_top(set(A))))
        <=> filterlim(A,B,F2,F3,topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% filterlim_shift_iff
tff(fact_7799_filterlim__shift,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F3: filter(B),A2: A,D2: A] :
          ( filterlim(A,B,F2,F3,topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,comp(A,B,A,F2,aa(A,fun(A,A),plus_plus(A),D2)),F3,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D2),top_top(set(A)))) ) ) ).

% filterlim_shift
tff(fact_7800_isCont__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( aa(A,A,cos(A),Xa) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,top_top(set(A))),tan(A)) ) ) ).

% isCont_tan
tff(fact_7801_isCont__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( aa(A,A,sin(A),Xa) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,top_top(set(A))),cot(A)) ) ) ).

% isCont_cot
tff(fact_7802_isCont__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( aa(A,A,cosh(A),Xa) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,top_top(set(A))),tanh(A)) ) ) ).

% isCont_tanh
tff(fact_7803_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Sa: real,A2: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Sa)
         => ( ! [X: A] :
                ( ( X != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),Sa)
                 => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),A2),X)),aa(A,A,F2,X)) ) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0_strong
tff(fact_7804_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Sa: real,A2: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Sa)
         => ( ! [X: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),Sa)
               => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),A2),X)),aa(A,A,F2,X)) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_7805_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_aex(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_7806_isCont__tan_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( aa(A,A,cos(A),aa(A,A,F2,A2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_adg(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_tan'
tff(fact_7807_isCont__arcosh,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),arcosh(real)) ) ).

% isCont_arcosh
tff(fact_7808_LIM__cos__div__sin,axiom,
    filterlim(real,real,aTP_Lamp_aey(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_7809_isCont__cot_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( aa(A,A,sin(A),aa(A,A,F2,A2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_adi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_cot'
tff(fact_7810_DERIV__inverse__function,axiom,
    ! [F2: fun(real,real),D4: real,G: fun(real,real),Xa: real,A2: real,B2: real] :
      ( has_field_derivative(real,F2,D4,topolo174197925503356063within(real,aa(real,real,G,Xa),top_top(set(real))))
     => ( ( D4 != zero_zero(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),B2)
           => ( ! [Y3: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Y3)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y3),B2)
                   => ( aa(real,real,F2,aa(real,real,G,Y3)) = Y3 ) ) )
             => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),G)
               => has_field_derivative(real,G,aa(real,real,inverse_inverse(real),D4),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_7811_isCont__polynom,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: A,C2: fun(nat,A),Na: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aa(nat,fun(A,A),aTP_Lamp_aez(fun(nat,A),fun(nat,fun(A,A)),C2),Na)) ) ).

% isCont_polynom
tff(fact_7812_isCont__powser__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xa: A] :
          ( ! [Y3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),C2),Y3))
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,top_top(set(A))),aTP_Lamp_xi(fun(nat,A),fun(A,A),C2)) ) ) ).

% isCont_powser_converges_everywhere
tff(fact_7813_isCont__arccos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),arccos) ) ) ).

% isCont_arccos
tff(fact_7814_isCont__arcsin,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),arcsin) ) ) ).

% isCont_arcsin
tff(fact_7815_LIM__less__bound,axiom,
    ! [B2: real,Xa: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),B2),Xa)
     => ( ! [X: real] :
            ( member(real,X,set_or5935395276787703475ssThan(real,B2,Xa))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,X)) )
       => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),F2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,Xa)) ) ) ) ).

% LIM_less_bound
tff(fact_7816_isCont__artanh,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),artanh(real)) ) ) ).

% isCont_artanh
tff(fact_7817_isCont__inverse__function,axiom,
    ! [D2: real,Xa: 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,aa(real,fun(real,real),minus_minus(real),Z3),Xa))),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,aa(real,fun(real,real),minus_minus(real),Z3),Xa))),D2)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,top_top(set(real))),F2) )
         => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,Xa),top_top(set(real))),G) ) ) ) ).

% isCont_inverse_function
tff(fact_7818_GMVT_H,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),G: fun(real,real),G4: fun(real,real),F4: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [Z3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z3),B2)
             => 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),A2),Z3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z3),B2)
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,top_top(set(real))),G) ) )
         => ( ! [Z3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),B2)
                 => has_field_derivative(real,G,aa(real,real,G4,Z3),topolo174197925503356063within(real,Z3,top_top(set(real)))) ) )
           => ( ! [Z3: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z3)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),B2)
                   => has_field_derivative(real,F2,aa(real,real,F4,Z3),topolo174197925503356063within(real,Z3,top_top(set(real)))) ) )
             => ? [C5: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),C5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),C5),B2)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2))),aa(real,real,G4,C5)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,G,B2)),aa(real,real,G,A2))),aa(real,real,F4,C5)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_7819_floor__has__real__derivative,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Xa: real,F2: fun(real,A)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,Xa,top_top(set(real))),F2)
         => ( ~ member(A,aa(real,A,F2,Xa),ring_1_Ints(A))
           => has_field_derivative(real,aTP_Lamp_afa(fun(real,A),fun(real,real),F2),zero_zero(real),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ) ).

% floor_has_real_derivative
tff(fact_7820_isCont__powser_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [A2: A,F2: fun(A,B),C2: fun(nat,B),K6: B] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( summable(B,aa(B,fun(nat,B),aTP_Lamp_afb(fun(nat,B),fun(B,fun(nat,B)),C2),K6))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,A2))),real_V7770717601297561774m_norm(B,K6))
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(nat,B),fun(A,B),aTP_Lamp_afd(fun(A,B),fun(fun(nat,B),fun(A,B)),F2),C2)) ) ) ) ) ).

% isCont_powser'
tff(fact_7821_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,Xa: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,K6))
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,top_top(set(A))),aTP_Lamp_xi(fun(nat,A),fun(A,A),C2)) ) ) ) ).

% isCont_powser
tff(fact_7822_summable__Leibniz_I3_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,A2,zero_zero(nat))),zero_zero(real))
         => ! [N9: nat] : member(real,suminf(real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2)),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N9)),one_one(nat)))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N9))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_7823_summable__Leibniz_I2_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(nat,real,A2,zero_zero(nat)))
         => ! [N9: nat] : member(real,suminf(real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2)),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N9))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N9)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_7824_trivial__limit__sequentially,axiom,
    at_top(nat) != bot_bot(filter(nat)) ).

% trivial_limit_sequentially
tff(fact_7825_tendsto__zero__mult__left__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aff(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_left_iff
tff(fact_7826_tendsto__zero__mult__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_afg(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_right_iff
tff(fact_7827_tendsto__zero__divide__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_afh(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_divide_iff
tff(fact_7828_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_afi(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% seq_offset_neg
tff(fact_7829_continuous__real__sqrt,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_afj(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_real_sqrt
tff(fact_7830_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),F3: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_so(fun(nat,A),fun(nat,A),F2),F3,at_top(nat))
    <=> filterlim(nat,A,F2,F3,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_7831_continuous__arctan,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_afk(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_arctan
tff(fact_7832_continuous__arsinh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_afl(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_arsinh
tff(fact_7833_continuous__real__root,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),Na: nat] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aa(nat,fun(A,real),aTP_Lamp_afm(fun(A,real),fun(nat,fun(A,real)),F2),Na)) ) ) ).

% continuous_real_root
tff(fact_7834_continuous__rabs,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_afn(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_rabs
tff(fact_7835_LIMSEQ__offset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),K: nat,A2: A] :
          ( filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afo(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_offset
tff(fact_7836_LIMSEQ__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),A2: A,K: nat] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
         => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afo(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_ignore_initial_segment
tff(fact_7837_LIMSEQ__const__iff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [K: A,L: A] :
          ( filterlim(nat,A,aTP_Lamp_afp(A,fun(nat,A),K),topolo7230453075368039082e_nhds(A,L),at_top(nat))
        <=> ( K = L ) ) ) ).

% LIMSEQ_const_iff
tff(fact_7838_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_afq(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_Suc
tff(fact_7839_LIMSEQ__imp__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A] :
          ( filterlim(nat,A,aTP_Lamp_afq(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_7840_approx__from__above__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ? [U3: fun(nat,A)] :
              ( ! [N9: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(nat,A,U3,N9))
              & filterlim(nat,A,U3,topolo7230453075368039082e_nhds(A,Xa),at_top(nat)) ) ) ) ).

% approx_from_above_dense_linorder
tff(fact_7841_approx__from__below__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ? [U3: fun(nat,A)] :
              ( ! [N9: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,U3,N9)),Xa)
              & filterlim(nat,A,U3,topolo7230453075368039082e_nhds(A,Xa),at_top(nat)) ) ) ) ).

% approx_from_below_dense_linorder
tff(fact_7842_isCont__rabs,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_afn(fun(A,real),fun(A,real),F2)) ) ) ).

% isCont_rabs
tff(fact_7843_trivial__limit__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_top(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_top_linorder
tff(fact_7844_continuous__at__within__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,Sa: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,Sa),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,Sa),G)
           => ( ( aa(A,real,F2,A2) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,Sa),aa(fun(A,real),fun(A,real),aTP_Lamp_afr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_at_within_powr
tff(fact_7845_continuous__within__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xa: A,Sa: set(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Xa,Sa),F2)
         => ( ( aa(A,real,F2,Xa) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Xa,Sa),aTP_Lamp_afs(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_within_ln
tff(fact_7846_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_aft(nat,fun(nat,nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_7847_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_7848_LIMSEQ__lessThan__iff__atMost,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(set(nat),A),Xa: A] :
          ( filterlim(nat,A,aTP_Lamp_afu(fun(set(nat),A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
        <=> filterlim(nat,A,aTP_Lamp_afv(fun(set(nat),A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,Xa),at_top(nat)) ) ) ).

% LIMSEQ_lessThan_iff_atMost
tff(fact_7849_LIMSEQ__root,axiom,
    filterlim(nat,real,aTP_Lamp_afw(nat,real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ).

% LIMSEQ_root
tff(fact_7850_isCont__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => ( ( aa(A,real,F2,A2) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_afr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% isCont_powr
tff(fact_7851_isCont__ln_H,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xa: A,F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Xa,top_top(set(A))),F2)
         => ( ( aa(A,real,F2,Xa) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Xa,top_top(set(A))),aTP_Lamp_afs(fun(A,real),fun(A,real),F2)) ) ) ) ).

% isCont_ln'
tff(fact_7852_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A] : filterlim(nat,A,aTP_Lamp_afx(A,fun(nat,A),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_7853_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X7: fun(nat,A),Xa: A,L: nat] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),L)
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afy(fun(nat,A),fun(nat,fun(nat,A)),X7),L),topolo7230453075368039082e_nhds(A,Xa),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_7854_lim__inverse__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_afz(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_inverse_n
tff(fact_7855_LIMSEQ__SEQ__conv,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,X7: fun(A,B),L5: B] :
          ( ! [S8: fun(nat,A)] :
              ( ( ! [N2: nat] : aa(nat,A,S8,N2) != A2
                & filterlim(nat,A,S8,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aga(fun(A,B),fun(fun(nat,A),fun(nat,B)),X7),S8),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) )
        <=> filterlim(A,B,X7,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIMSEQ_SEQ_conv
tff(fact_7856_LIMSEQ__SEQ__conv1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ! [S9: fun(nat,A)] :
              ( ( ! [N: nat] : aa(nat,A,S9,N) != A2
                & filterlim(nat,A,S9,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_agb(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),S9),topolo7230453075368039082e_nhds(B,L),at_top(nat)) ) ) ) ).

% LIMSEQ_SEQ_conv1
tff(fact_7857_LIMSEQ__SEQ__conv2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,F2: fun(A,B),L: B] :
          ( ! [S4: fun(nat,A)] :
              ( ( ! [N9: nat] : aa(nat,A,S4,N9) != A2
                & filterlim(nat,A,S4,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aga(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,A2,top_top(set(A)))) ) ) ).

% LIMSEQ_SEQ_conv2
tff(fact_7858_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_agc(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable'
tff(fact_7859_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_agd(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable
tff(fact_7860_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_age(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => ? [L2: real] :
                ( ! [N9: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N9)),L2)
                & filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L2),at_top(nat))
                & ! [N9: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L2),aa(nat,real,G,N9))
                & filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_7861_LIMSEQ__inverse__zero,axiom,
    ! [X7: fun(nat,real)] :
      ( ! [R4: real] :
        ? [N7: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),R4),aa(nat,real,X7,N)) )
     => filterlim(nat,real,aTP_Lamp_agf(fun(nat,real),fun(nat,real),X7),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_7862_lim__inverse__n_H,axiom,
    filterlim(nat,real,aTP_Lamp_agg(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% lim_inverse_n'
tff(fact_7863_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim(nat,real,aTP_Lamp_agh(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_7864_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_agi(real,fun(nat,real),C2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_7865_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R: real] : filterlim(nat,real,aTP_Lamp_agj(real,fun(nat,real),R),topolo7230453075368039082e_nhds(real,R),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_7866_sums__def,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),Sa: A] :
          ( aa(A,$o,sums(A,F2),Sa)
        <=> filterlim(nat,A,aTP_Lamp_agk(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,Sa),at_top(nat)) ) ) ).

% sums_def
tff(fact_7867_sums__def__le,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),Sa: A] :
          ( aa(A,$o,sums(A,F2),Sa)
        <=> filterlim(nat,A,aTP_Lamp_agl(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,Sa),at_top(nat)) ) ) ).

% sums_def_le
tff(fact_7868_continuous__at__within__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,Sa: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,Sa),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,Sa),G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,A2))
             => ( ( aa(A,real,F2,A2) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,A2))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,Sa),aa(fun(A,real),fun(A,real),aTP_Lamp_agm(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_7869_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)
       => ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [N9: 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,N9)),E2)) )
         => filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_7870_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_agn(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_7871_LIMSEQ__n__over__Suc__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_ago(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_n_over_Suc_n
tff(fact_7872_LIMSEQ__Suc__n__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_agp(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_Suc_n_over_n
tff(fact_7873_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_agc(fun(nat,A),fun(nat,A),F2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),C2)) ) ) ).

% telescope_sums'
tff(fact_7874_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_agd(fun(nat,A),fun(nat,A),F2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% telescope_sums
tff(fact_7875_LIMSEQ__realpow__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),Xa),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_7876_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xa: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_agq(real,fun(real,fun(nat,real)),Xa),A2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_7877_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,aa(real,fun(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_7878_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,aa(real,fun(nat,real),power_power(real),C2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_7879_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => filterlim(nat,real,aTP_Lamp_agr(real,fun(nat,real),Xa),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_7880_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),Sa: A] :
          ( aa(A,$o,sums(A,F2),Sa)
        <=> filterlim(nat,A,aTP_Lamp_ags(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,Sa),at_top(nat)) ) ) ).

% sums_def'
tff(fact_7881_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R: real] : filterlim(nat,real,aTP_Lamp_agt(real,fun(nat,real),R),topolo7230453075368039082e_nhds(real,R),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_7882_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),Xa: real] :
          ( filterlim(nat,real,aTP_Lamp_agu(fun(nat,A),fun(nat,real),F2),topolo7230453075368039082e_nhds(real,Xa),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
           => summable(A,F2) ) ) ) ).

% root_test_convergence
tff(fact_7883_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_agv(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,suminf(A,F2)),at_top(nat)) ) ) ).

% summable_LIMSEQ
tff(fact_7884_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_agw(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,suminf(A,F2)),at_top(nat)) ) ) ).

% summable_LIMSEQ'
tff(fact_7885_isCont__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,A2))
             => ( ( aa(A,real,F2,A2) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,A2))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_agm(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_7886_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A),L5: A,R: real] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => ? [No: nat] :
              ! [N9: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N9)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X7,N9)),L5))),R) ) ) ) ) ).

% LIMSEQ_D
tff(fact_7887_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A),L5: A] :
          ( ! [R4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
             => ? [No2: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X7,N)),L5))),R4) ) )
         => filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_7888_LIMSEQ__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X7: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No3: nat] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X7,N2)),L5))),R5) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_7889_LIMSEQ__power__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),one_one(real))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),Xa),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_power_zero
tff(fact_7890_tendsto__exp__limit__sequentially,axiom,
    ! [Xa: real] : filterlim(nat,real,aTP_Lamp_agx(real,fun(nat,real),Xa),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),Xa)),at_top(nat)) ).

% tendsto_exp_limit_sequentially
tff(fact_7891_tendsto__at__iff__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: B,Xa: A,Sa: set(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),topolo174197925503356063within(A,Xa,Sa))
        <=> ! [X9: fun(nat,A)] :
              ( ! [I2: nat] : member(A,aa(nat,A,X9,I2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Sa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))
             => ( filterlim(nat,A,X9,topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
               => filterlim(nat,B,comp(A,B,nat,F2,X9),topolo7230453075368039082e_nhds(B,A2),at_top(nat)) ) ) ) ) ).

% tendsto_at_iff_sequentially
tff(fact_7892_tendsto__power__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,nat),F3: filter(A),Xa: B] :
          ( filterlim(A,nat,F2,at_top(nat),F3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,Xa)),one_one(real))
           => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_agy(fun(A,nat),fun(B,fun(A,B)),F2),Xa),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_power_zero
tff(fact_7893_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] : filterlim(nat,real,aTP_Lamp_agz(real,fun(nat,real),R),topolo7230453075368039082e_nhds(real,R),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_7894_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_7895_summable__Leibniz_I1_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => summable(real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2)) ) ) ).

% summable_Leibniz(1)
tff(fact_7896_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Df: A,Z: A,Sa: fun(nat,A),A2: A] :
          ( has_field_derivative(A,F2,Df,topolo174197925503356063within(A,Z,top_top(set(A))))
         => ( filterlim(nat,A,Sa,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
           => ( ! [N: nat] : aa(nat,A,Sa,N) != zero_zero(A)
             => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_aha(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),F2),Z),Sa),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
               => ( Df = A2 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_7897_powser__times__n__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),one_one(real))
         => filterlim(nat,A,aTP_Lamp_ahb(A,fun(nat,A),Xa),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_7898_lim__n__over__pown,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,Xa))
         => filterlim(nat,A,aTP_Lamp_ahc(A,fun(nat,A),Xa),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_7899_summable,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => summable(real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2)) ) ) ) ).

% summable
tff(fact_7900_cos__diff__limit__1,axiom,
    ! [Theta: fun(nat,real),Theta2: real] :
      ( filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_ahd(fun(nat,real),fun(real,fun(nat,real)),Theta),Theta2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ~ ! [K2: fun(nat,int)] : ~ filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_ahe(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K2),topolo7230453075368039082e_nhds(real,Theta2),at_top(nat)) ) ).

% cos_diff_limit_1
tff(fact_7901_cos__limit__1,axiom,
    ! [Theta: fun(nat,real)] :
      ( filterlim(nat,real,aTP_Lamp_ahf(fun(nat,real),fun(nat,real),Theta),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ? [K2: fun(nat,int)] : filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_ahe(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% cos_limit_1
tff(fact_7902_summable__Leibniz_I4_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => filterlim(nat,real,aTP_Lamp_ahg(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(4)
tff(fact_7903_zeroseq__arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => filterlim(nat,real,aTP_Lamp_aw(real,fun(nat,real),Xa),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_7904_summable__Leibniz_H_I2_J,axiom,
    ! [A2: fun(nat,real),Na: nat] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))),suminf(real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_7905_summable__Leibniz_H_I3_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => filterlim(nat,real,aTP_Lamp_ahg(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_7906_sums__alternating__upper__lower,axiom,
    ! [A2: fun(nat,real)] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
         => ? [L2: real] :
              ( ! [N9: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N9)))),L2)
              & filterlim(nat,real,aTP_Lamp_ahg(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L2),at_top(nat))
              & ! [N9: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L2),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N9)),one_one(nat)))))
              & filterlim(nat,real,aTP_Lamp_ahh(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_7907_summable__Leibniz_I5_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => filterlim(nat,real,aTP_Lamp_ahh(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_7908_summable__Leibniz_H_I4_J,axiom,
    ! [A2: fun(nat,real),Na: nat] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),suminf(real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),one_one(nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_7909_summable__Leibniz_H_I5_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => filterlim(nat,real,aTP_Lamp_ahh(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_7910_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),Xa: A] :
          ( has_derivative(A,B,F2,F4,topolo174197925503356063within(A,Xa,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F4)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_ahi(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F4),Xa),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_7911_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),D4: fun(A,B),Xa: A] :
          ( has_derivative(A,B,F2,D4,topolo174197925503356063within(A,Xa,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D4)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_ahj(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),D4),Xa),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_7912_exp__at__top,axiom,
    filterlim(real,real,exp(real),at_top(real),at_top(real)) ).

% exp_at_top
tff(fact_7913_bounded__linear_Ocontinuous,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(C),G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo3448309680560233919inuous(C,A,F3,G)
           => topolo3448309680560233919inuous(C,B,F3,aa(fun(C,A),fun(C,B),aTP_Lamp_ahk(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear.continuous
tff(fact_7914_bounded__linear_Otendsto,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,B),G: fun(C,A),A2: A,F3: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,A2),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ahl(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),F3) ) ) ) ).

% bounded_linear.tendsto
tff(fact_7915_ln__at__top,axiom,
    filterlim(real,real,ln_ln(real),at_top(real),at_top(real)) ).

% ln_at_top
tff(fact_7916_sqrt__at__top,axiom,
    filterlim(real,real,sqrt,at_top(real),at_top(real)) ).

% sqrt_at_top
tff(fact_7917_filterlim__at__top__add__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,at_top(real),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahm(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ).

% filterlim_at_top_add_at_top
tff(fact_7918_filterlim__at__top__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,at_top(real),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahn(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ).

% filterlim_at_top_mult_at_top
tff(fact_7919_sinh__real__at__top,axiom,
    filterlim(real,real,sinh(real),at_top(real),at_top(real)) ).

% sinh_real_at_top
tff(fact_7920_arsinh__real__at__top,axiom,
    filterlim(real,real,arsinh(real),at_top(real),at_top(real)) ).

% arsinh_real_at_top
tff(fact_7921_filterlim__nat__sequentially,axiom,
    filterlim(int,nat,nat2,at_top(nat),at_top(int)) ).

% filterlim_nat_sequentially
tff(fact_7922_arcosh__real__at__top,axiom,
    filterlim(real,real,arcosh(real),at_top(real),at_top(real)) ).

% arcosh_real_at_top
tff(fact_7923_filterlim__int__sequentially,axiom,
    filterlim(nat,int,semiring_1_of_nat(int),at_top(int),at_top(nat)) ).

% filterlim_int_sequentially
tff(fact_7924_cosh__real__at__top,axiom,
    filterlim(real,real,cosh(real),at_top(real),at_top(real)) ).

% cosh_real_at_top
tff(fact_7925_bounded__linear_Otendsto__zero,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(C,A),F3: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ahl(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% bounded_linear.tendsto_zero
tff(fact_7926_bounded__linear_OisCont,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),A2: C,G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,A2,top_top(set(C))),G)
           => topolo3448309680560233919inuous(C,B,topolo174197925503356063within(C,A2,top_top(set(C))),aa(fun(C,A),fun(C,B),aTP_Lamp_ahk(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear.isCont
tff(fact_7927_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_yw(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_sub
tff(fact_7928_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_yj(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear_compose
tff(fact_7929_bounded__linear__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => real_V3181309239436604168linear(A,A,aTP_Lamp_yu(A,A)) ) ).

% bounded_linear_ident
tff(fact_7930_bounded__linear_Osums,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X7: fun(nat,A),A2: A] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( aa(A,$o,sums(A,X7),A2)
           => aa(B,$o,sums(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aho(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X7)),aa(A,B,F2,A2)) ) ) ) ).

% bounded_linear.sums
tff(fact_7931_bounded__linear_Osummable,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X7: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( summable(A,X7)
           => summable(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aho(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X7)) ) ) ) ).

% bounded_linear.summable
tff(fact_7932_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_7933_bounded__linear__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] : real_V3181309239436604168linear(real,A,aTP_Lamp_ahp(A,fun(real,A),Xa)) ) ).

% bounded_linear_scaleR_left
tff(fact_7934_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_ys(fun(A,B),fun(real,fun(A,B)),G),R)) ) ) ).

% bounded_linear_const_scaleR
tff(fact_7935_bounded__linear__scaleR__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,real),Xa: B] :
          ( real_V3181309239436604168linear(A,real,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_yt(fun(A,real),fun(B,fun(A,B)),G),Xa)) ) ) ).

% bounded_linear_scaleR_const
tff(fact_7936_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_7937_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_yr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_add
tff(fact_7938_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_yq(fun(A,B),fun(A,B),F2)) ) ) ).

% bounded_linear_minus
tff(fact_7939_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_ye(A,B)) ) ).

% bounded_linear_zero
tff(fact_7940_bounded__linear__mult__left,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_ahq(A,fun(A,A),Y)) ) ).

% bounded_linear_mult_left
tff(fact_7941_bounded__linear__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),Xa: B] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_yo(fun(A,B),fun(B,fun(A,B)),G),Xa)) ) ) ).

% bounded_linear_const_mult
tff(fact_7942_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_yn(fun(A,B),fun(B,fun(A,B)),G),Y)) ) ) ).

% bounded_linear_mult_const
tff(fact_7943_bounded__linear__mult__right,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xa: A] : real_V3181309239436604168linear(A,A,aa(A,fun(A,A),times_times(A),Xa)) ) ).

% bounded_linear_mult_right
tff(fact_7944_bounded__linear__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_ahr(A,fun(A,A),Y)) ) ).

% bounded_linear_divide
tff(fact_7945_bounded__linear_Osuminf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X7: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( summable(A,X7)
           => ( aa(A,B,F2,suminf(A,X7)) = suminf(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aho(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X7)) ) ) ) ) ).

% bounded_linear.suminf
tff(fact_7946_bounded__linear_OCauchy,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X7: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo3814608138187158403Cauchy(A,X7)
           => topolo3814608138187158403Cauchy(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aho(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X7)) ) ) ) ).

% bounded_linear.Cauchy
tff(fact_7947_bounded__linear__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [I5: set(A),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => real_V3181309239436604168linear(B,C,aa(A,fun(B,C),F2,I3)) )
         => real_V3181309239436604168linear(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ym(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% bounded_linear_sum
tff(fact_7948_bounded__linear_Obounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K8: real] :
            ! [X5: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X5))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X5)),K8)) ) ) ).

% bounded_linear.bounded
tff(fact_7949_real__bounded__linear,axiom,
    ! [F2: fun(real,real)] :
      ( real_V3181309239436604168linear(real,real,F2)
    <=> ? [C6: real] :
        ! [X4: real] : aa(real,real,F2,X4) = aa(real,real,aa(real,fun(real,real),times_times(real),X4),C6) ) ).

% real_bounded_linear
tff(fact_7950_bounded__linear_Ohas__derivative,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,B),G: fun(C,A),G4: fun(C,A),F3: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( has_derivative(C,A,G,G4,F3)
           => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_yj(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),aa(fun(C,A),fun(C,B),aTP_Lamp_yj(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G4),F3) ) ) ) ).

% bounded_linear.has_derivative
tff(fact_7951_filterlim__tendsto__add__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahm(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ).

% filterlim_tendsto_add_at_top
tff(fact_7952_filterlim__real__sequentially,axiom,
    filterlim(nat,real,semiring_1_of_nat(real),at_top(real),at_top(nat)) ).

% filterlim_real_sequentially
tff(fact_7953_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,$o),ord_less_eq(real),zero_zero(real)),K8)
              & ! [X5: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X5))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X5)),K8)) ) ) ) ).

% bounded_linear.nonneg_bounded
tff(fact_7954_has__derivative__within__singleton__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B),Xa: A] :
          ( has_derivative(A,B,F2,G,topolo174197925503356063within(A,Xa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))
        <=> real_V3181309239436604168linear(A,B,G) ) ) ).

% has_derivative_within_singleton_iff
tff(fact_7955_filterlim__pow__at__top,axiom,
    ! [A: $tType,Na: nat,F2: fun(A,real),F3: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(A,real,F2,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aao(nat,fun(fun(A,real),fun(A,real)),Na),F2),at_top(real),F3) ) ) ).

% filterlim_pow_at_top
tff(fact_7956_tanh__real__at__top,axiom,
    filterlim(real,real,tanh(real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(real)) ).

% tanh_real_at_top
tff(fact_7957_real__tendsto__divide__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahs(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% real_tendsto_divide_at_top
tff(fact_7958_artanh__real__at__left__1,axiom,
    filterlim(real,real,artanh(real),at_top(real),topolo174197925503356063within(real,one_one(real),set_ord_lessThan(real,one_one(real)))) ).

% artanh_real_at_left_1
tff(fact_7959_tendsto__inverse__0__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F3)
     => filterlim(A,real,aTP_Lamp_aht(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_inverse_0_at_top
tff(fact_7960_filterlim__int__of__nat__at__topD,axiom,
    ! [A: $tType,F2: fun(int,A),F3: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_ahu(fun(int,A),fun(nat,A),F2),F3,at_top(nat))
     => filterlim(int,A,F2,F3,at_top(int)) ) ).

% filterlim_int_of_nat_at_topD
tff(fact_7961_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,$o),ord_less(real),zero_zero(real)),K8)
              & ! [X5: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X5))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X5)),K8)) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_7962_bounded__linear__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),K6: real] :
          ( ! [X: A,Y3: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X)),aa(A,B,F2,Y3))
         => ( ! [R4: real,X: A] : aa(A,B,F2,aa(A,A,real_V8093663219630862766scaleR(A,R4),X)) = aa(B,B,real_V8093663219630862766scaleR(B,R4),aa(A,B,F2,X))
           => ( ! [X: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),K6))
             => real_V3181309239436604168linear(A,B,F2) ) ) ) ) ).

% bounded_linear_intro
tff(fact_7963_filterlim__sequentially__iff__filterlim__real,axiom,
    ! [A: $tType,F2: fun(A,nat),F3: filter(A)] :
      ( filterlim(A,nat,F2,at_top(nat),F3)
    <=> filterlim(A,real,aTP_Lamp_nl(fun(A,nat),fun(A,real),F2),at_top(real),F3) ) ).

% filterlim_sequentially_iff_filterlim_real
tff(fact_7964_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahv(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_7965_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahn(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_7966_tendsto__neg__powr,axiom,
    ! [A: $tType,Sa: real,F2: fun(A,real),F3: filter(A)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Sa),zero_zero(real))
     => ( filterlim(A,real,F2,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahw(real,fun(fun(A,real),fun(A,real)),Sa),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_neg_powr
tff(fact_7967_ln__x__over__x__tendsto__0,axiom,
    filterlim(real,real,aTP_Lamp_ahx(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% ln_x_over_x_tendsto_0
tff(fact_7968_tendsto__at__topI__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [F2: fun(real,A),Y: A] :
          ( ! [X10: fun(nat,real)] :
              ( filterlim(nat,real,X10,at_top(real),at_top(nat))
             => filterlim(nat,A,aa(fun(nat,real),fun(nat,A),aTP_Lamp_ahy(fun(real,A),fun(fun(nat,real),fun(nat,A)),F2),X10),topolo7230453075368039082e_nhds(A,Y),at_top(nat)) )
         => filterlim(real,A,F2,topolo7230453075368039082e_nhds(A,Y),at_top(real)) ) ) ).

% tendsto_at_topI_sequentially
tff(fact_7969_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim(real,real,aTP_Lamp_ahz(nat,fun(real,real),K),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% tendsto_power_div_exp_0
tff(fact_7970_tendsto__exp__limit__at__top,axiom,
    ! [Xa: real] : filterlim(real,real,aTP_Lamp_aia(real,fun(real,real),Xa),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),Xa)),at_top(real)) ).

% tendsto_exp_limit_at_top
tff(fact_7971_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_7972_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_7973_has__derivative__iff__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),Xa: A,Sa: set(A)] :
          ( has_derivative(A,B,F2,F4,topolo174197925503356063within(A,Xa,Sa))
        <=> ( real_V3181309239436604168linear(A,B,F4)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_aib(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),F4),Xa),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_derivative_iff_norm
tff(fact_7974_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),X)
         => ? [Y2: real] :
              ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X,top_top(set(real))))
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),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,B2)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_7975_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F4: fun(A,B),Xa: A,F2: fun(A,B),Sa: set(A)] :
          ( real_V3181309239436604168linear(A,B,F4)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_aic(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F4),Xa),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xa,Sa))
           => has_derivative(A,B,F2,F4,topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_derivativeI
tff(fact_7976_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),Xa: A,Sa: set(A)] :
          ( has_derivative(A,B,F2,F4,topolo174197925503356063within(A,Xa,Sa))
        <=> ( real_V3181309239436604168linear(A,B,F4)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_aid(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F4),Xa),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_derivative_at_within
tff(fact_7977_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),Xa: A] :
          ( has_derivative(A,B,F2,F4,topolo174197925503356063within(A,Xa,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F4)
            & ? [E3: fun(A,B)] :
                ( ! [H4: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),H4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,Xa)),aa(A,B,F4,H4))),aa(A,B,E3,H4))
                & filterlim(A,real,aTP_Lamp_aie(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_7978_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),Xa: A,Sa: set(A)] :
          ( has_derivative(A,B,F2,F4,topolo174197925503356063within(A,Xa,Sa))
        <=> ( real_V3181309239436604168linear(A,B,F4)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_ahi(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F4),Xa),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_derivative_within
tff(fact_7979_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),F3: filter(A)] :
          ( has_derivative(A,B,F2,F4,F3)
        <=> ( real_V3181309239436604168linear(A,B,F4)
            & filterlim(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_aif(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F2),F4),F3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% has_derivative_def
tff(fact_7980_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Xa: A,S: set(A),F2: fun(A,B),F4: fun(A,B)] :
          ( member(A,Xa,S)
         => ( topolo1002775350975398744n_open(A,S)
           => ( has_derivative(A,B,F2,F4,topolo174197925503356063within(A,Xa,S))
            <=> ( real_V3181309239436604168linear(A,B,F4)
                & ? [E3: fun(A,B)] :
                    ( ! [H4: A] :
                        ( member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),H4),S)
                       => ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),H4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,Xa)),aa(A,B,F4,H4))),aa(A,B,E3,H4)) ) )
                    & filterlim(A,real,aTP_Lamp_aie(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_7981_lim__const,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A] : topolo3827282254853284352ce_Lim(nat,A,at_top(nat),aTP_Lamp_afp(A,fun(nat,A),A2)) = A2 ) ).

% lim_const
tff(fact_7982_open__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo1002775350975398744n_open(A,bot_bot(set(A))) ) ).

% open_empty
tff(fact_7983_open__Un,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T3: set(A)] :
          ( topolo1002775350975398744n_open(A,S)
         => ( topolo1002775350975398744n_open(A,T3)
           => topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)) ) ) ) ).

% open_Un
tff(fact_7984_hausdorff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
         => ? [U4: set(A),V5: set(A)] :
              ( topolo1002775350975398744n_open(A,U4)
              & topolo1002775350975398744n_open(A,V5)
              & member(A,Xa,U4)
              & member(A,Y,V5)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U4),V5) = bot_bot(set(A)) ) ) ) ) ).

% hausdorff
tff(fact_7985_separation__t2,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
        <=> ? [U5: set(A),V6: set(A)] :
              ( topolo1002775350975398744n_open(A,U5)
              & topolo1002775350975398744n_open(A,V6)
              & member(A,Xa,U5)
              & member(A,Y,V6)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U5),V6) = bot_bot(set(A)) ) ) ) ) ).

% separation_t2
tff(fact_7986_not__open__singleton,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [Xa: A] : ~ topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) ) ).

% not_open_singleton
tff(fact_7987_open__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A),Xa: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S)
         => ( member(A,Xa,S)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
             => ? [B6: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),B6)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,Xa,B6)),S) ) ) ) ) ) ).

% open_right
tff(fact_7988_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,$o),aTP_Lamp_aig(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ) ) ).

% open_Collect_conj
tff(fact_7989_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,$o),aTP_Lamp_aih(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ) ) ).

% open_Collect_disj
tff(fact_7990_open__Collect__const,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: $o] : topolo1002775350975398744n_open(A,collect(A,aTP_Lamp_aii($o,fun(A,$o),(P)))) ) ).

% open_Collect_const
tff(fact_7991_Lim__ident__at,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xa: A,Sa: set(A)] :
          ( ( topolo174197925503356063within(A,Xa,Sa) != bot_bot(filter(A)) )
         => ( topolo3827282254853284352ce_Lim(A,A,topolo174197925503356063within(A,Xa,Sa),aTP_Lamp_abp(A,A)) = Xa ) ) ) ).

% Lim_ident_at
tff(fact_7992_tendsto__Lim,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Net: filter(A),F2: fun(A,B),L: B] :
          ( ( Net != bot_bot(filter(A)) )
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),Net)
           => ( topolo3827282254853284352ce_Lim(A,B,Net,F2) = L ) ) ) ) ).

% tendsto_Lim
tff(fact_7993_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => ( ( aa(A,B,G,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_abp(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_abq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_divide
tff(fact_7994_continuous__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_abp(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_abo(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_inverse
tff(fact_7995_continuous__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_abp(A,A)))),F3) ) ) ).

% continuous_def
tff(fact_7996_continuous__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_abp(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_abu(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_sgn
tff(fact_7997_t2__space__class_OLim__def,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(A)
     => ! [A3: filter(B),F2: fun(B,A)] : topolo3827282254853284352ce_Lim(B,A,A3,F2) = the(A,aa(fun(B,A),fun(A,$o),aTP_Lamp_aij(filter(B),fun(fun(B,A),fun(A,$o)),A3),F2)) ) ).

% t2_space_class.Lim_def
tff(fact_7998_at__within__nhd,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xa: A,S: set(A),T3: set(A),U6: set(A)] :
          ( member(A,Xa,S)
         => ( topolo1002775350975398744n_open(A,S)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T3),S)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U6),S)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) )
             => ( topolo174197925503356063within(A,Xa,T3) = topolo174197925503356063within(A,Xa,U6) ) ) ) ) ) ).

% at_within_nhd
tff(fact_7999_continuous__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( topolo3448309680560233919inuous(A,real,F3,G)
           => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_abp(A,A))) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,F3,aa(fun(A,real),fun(A,real),aTP_Lamp_afr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_powr
tff(fact_8000_continuous__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_abp(A,A))) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_afs(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_ln
tff(fact_8001_at__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A] :
          ( ( topolo174197925503356063within(A,A2,top_top(set(A))) = bot_bot(filter(A)) )
        <=> topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) ) ) ).

% at_eq_bot_iff
tff(fact_8002_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_agv(fun(nat,A),fun(nat,A),F2)) ) ).

% suminf_eq_lim
tff(fact_8003_Topological__Spaces_Olim__def,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X7: fun(nat,A)] : topolo3827282254853284352ce_Lim(nat,A,at_top(nat),X7) = the(A,aTP_Lamp_aik(fun(nat,A),fun(A,$o),X7)) ) ).

% Topological_Spaces.lim_def
tff(fact_8004_continuous__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F3,F2)
         => ( ( aa(A,A,cos(A),aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_ail(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F3,aTP_Lamp_adg(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_tan
tff(fact_8005_continuous__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F3,F2)
         => ( ( aa(A,A,sin(A),aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_ail(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F3,aTP_Lamp_adi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_cot
tff(fact_8006_continuous__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( aa(B,B,cosh(B),aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_abp(A,A)))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_aew(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_tanh
tff(fact_8007_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_abp(A,A))))
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_aim(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh
tff(fact_8008_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( topolo3448309680560233919inuous(A,real,F3,G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_abp(A,A))))
             => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_abp(A,A))) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_abp(A,A))))
                 => topolo3448309680560233919inuous(A,real,F3,aa(fun(A,real),fun(A,real),aTP_Lamp_agm(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_8009_continuous__artanh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( member(real,aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_abp(A,A))),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_ain(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_artanh
tff(fact_8010_tendsto__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A2: B,S: set(B),F2: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A2)
         => ( member(B,A2,S)
           => ( topolo1002775350975398744n_open(B,S)
             => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,S))
              <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_aio(B,fun(fun(B,C),fun(B,C)),A2),F2),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_8011_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E: real,F4: fun(A,B),Sa: set(A),Xa: A,F2: fun(A,B),H5: fun(A,real)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
         => ( real_V3181309239436604168linear(A,B,F4)
           => ( ! [Y3: A] :
                  ( member(A,Y3,Sa)
                 => ( ( Y3 != Xa )
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y3,Xa)),E)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,Y3)),aa(A,B,F2,Xa))),aa(A,B,F4,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y3),Xa))))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y3),Xa)))),aa(A,real,H5,Y3)) ) ) )
             => ( filterlim(A,real,H5,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xa,Sa))
               => has_derivative(A,B,F2,F4,topolo174197925503356063within(A,Xa,Sa)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_8012_dist__add__cancel2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [B2: A,A2: A,C2: A] : real_V557655796197034286t_dist(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)) = real_V557655796197034286t_dist(A,B2,C2) ) ).

% dist_add_cancel2
tff(fact_8013_dist__add__cancel,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: A] : real_V557655796197034286t_dist(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) = real_V557655796197034286t_dist(A,B2,C2) ) ).

% dist_add_cancel
tff(fact_8014_zero__less__dist__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xa,Y))
        <=> ( Xa != Y ) ) ) ).

% zero_less_dist_iff
tff(fact_8015_div__add__self2__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [Xa: A,B2: B,A2: B] :
          ( nO_MATCH(A,B,Xa,B2)
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),one_one(B)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_8016_div__add__self1__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [Xa: A,B2: B,A2: B] :
          ( nO_MATCH(A,B,Xa,B2)
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),one_one(B)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_8017_dist__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: real,A2: A,Y: real] : real_V557655796197034286t_dist(A,aa(A,A,real_V8093663219630862766scaleR(A,Xa),A2),aa(A,A,real_V8093663219630862766scaleR(A,Y),A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),Y))),real_V7770717601297561774m_norm(A,A2)) ) ).

% dist_scaleR
tff(fact_8018_open__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S: set(A)] :
          ( topolo1002775350975398744n_open(A,S)
        <=> ! [X4: A] :
              ( member(A,X4,S)
             => ? [E3: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
                  & ! [Y5: A] :
                      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y5,X4)),E3)
                     => member(A,Y5,S) ) ) ) ) ) ).

% open_dist
tff(fact_8019_open__ball,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,D2: real] : topolo1002775350975398744n_open(A,collect(A,aa(real,fun(A,$o),aTP_Lamp_aip(A,fun(real,fun(A,$o)),Xa),D2))) ) ).

% open_ball
tff(fact_8020_continuous__dist,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,real,F3,aa(fun(A,B),fun(A,real),aTP_Lamp_aiq(fun(A,B),fun(fun(A,B),fun(A,real)),F2),G)) ) ) ) ).

% continuous_dist
tff(fact_8021_dist__pos__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xa,Y)) ) ) ).

% dist_pos_lt
tff(fact_8022_dist__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Y: A] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xa,Y)),zero_zero(real)) ) ).

% dist_not_less_zero
tff(fact_8023_dist__triangle__less__add,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X12: A,Y: A,E1: real,X22: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X12,Y)),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X22,Y)),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X12,X22)),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% dist_triangle_less_add
tff(fact_8024_dist__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Z: A,Y: A,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xa,Z)),real_V557655796197034286t_dist(A,Y,Z))),E)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xa,Y)),E) ) ) ).

% dist_triangle_lt
tff(fact_8025_dist__commute__lessI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,Xa: A,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,Xa)),E)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xa,Y)),E) ) ) ).

% dist_commute_lessI
tff(fact_8026_filterlim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [G: fun(A,B),G2: filter(B),Xa: A,S: set(A),F3: filter(B),D2: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,G2,topolo174197925503356063within(A,Xa,S))
         => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),G2),F3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
             => ( ! [X8: A] :
                    ( member(A,X8,S)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X8,Xa))
                     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X8,Xa)),D2)
                       => ( aa(A,B,F2,X8) = aa(A,B,G,X8) ) ) ) )
               => filterlim(A,B,F2,F3,topolo174197925503356063within(A,Xa,S)) ) ) ) ) ) ).

% filterlim_transform_within
tff(fact_8027_has__field__derivative__transform__within,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F4: A,A2: A,S: set(A),D2: real,G: fun(A,A)] :
          ( has_field_derivative(A,F2,F4,topolo174197925503356063within(A,A2,S))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( member(A,A2,S)
             => ( ! [X: A] :
                    ( member(A,X,S)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A2)),D2)
                     => ( aa(A,A,F2,X) = aa(A,A,G,X) ) ) )
               => has_field_derivative(A,G,F4,topolo174197925503356063within(A,A2,S)) ) ) ) ) ) ).

% has_field_derivative_transform_within
tff(fact_8028_has__derivative__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),Xa: A,Sa: set(A),D2: real,G: fun(A,B)] :
          ( has_derivative(A,B,F2,F4,topolo174197925503356063within(A,Xa,Sa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( member(A,Xa,Sa)
             => ( ! [X8: A] :
                    ( member(A,X8,Sa)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X8,Xa)),D2)
                     => ( aa(A,B,F2,X8) = aa(A,B,G,X8) ) ) )
               => has_derivative(A,B,G,F4,topolo174197925503356063within(A,Xa,Sa)) ) ) ) ) ) ).

% has_derivative_transform_within
tff(fact_8029_Cauchy__def,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X7)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M9: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M3)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,M3),aa(nat,A,X7,N2))),E3) ) ) ) ) ) ).

% Cauchy_def
tff(fact_8030_Cauchy__altdef2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Sa: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,Sa)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [N10: nat] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N10),N2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,Sa,N2),aa(nat,A,Sa,N10))),E3) ) ) ) ) ).

% Cauchy_altdef2
tff(fact_8031_metric__CauchyD,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),E: real] :
          ( topolo3814608138187158403Cauchy(A,X7)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
           => ? [M8: nat] :
              ! [M2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M2)
               => ! [N9: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N9)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,M2),aa(nat,A,X7,N9))),E) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_8032_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [M10: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),M4)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,M4),aa(nat,A,X7,N))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X7) ) ) ).

% metric_CauchyI
tff(fact_8033_tendsto__dist,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),G: fun(A,B),M: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,M),F3)
           => filterlim(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_air(fun(A,B),fun(fun(A,B),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,real_V557655796197034286t_dist(B,L,M)),F3) ) ) ) ).

% tendsto_dist
tff(fact_8034_scale__right__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A,Y: A,A2: real] :
          ( nO_MATCH(A,real,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y),A2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ) ) ).

% scale_right_distrib_NO_MATCH
tff(fact_8035_scale__right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A,Y: A,A2: real] :
          ( nO_MATCH(A,real,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y),A2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ) ) ).

% scale_right_diff_distrib_NO_MATCH
tff(fact_8036_distrib__left__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xa: A,Y: A,A2: B,B2: B,C2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y),A2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_8037_distrib__right__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xa: A,Y: A,C2: B,A2: B,B2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y),C2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_8038_right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xa: A,Y: A,A2: B,B2: B,C2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y),A2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) ) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_8039_left__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xa: A,Y: A,C2: B,A2: B,B2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y),C2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_8040_dist__triangle__half__l,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X12: A,Y: A,E: real,X22: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X12,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),bit0(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X22,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X12,X22)),E) ) ) ) ).

% dist_triangle_half_l
tff(fact_8041_dist__triangle__half__r,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,X12: A,E: real,X22: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,X12)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),bit0(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,X22)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X12,X22)),E) ) ) ) ).

% dist_triangle_half_r
tff(fact_8042_dist__triangle__third,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X12: A,X22: A,E: real,X32: A,X42: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X12,X22)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X22,X32)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X32,X42)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X12,X42)),E) ) ) ) ) ).

% dist_triangle_third
tff(fact_8043_Lim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L: B,Xa: A,S: set(A),D2: real,G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Xa,S))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( ! [X8: A] :
                  ( member(A,X8,S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X8,Xa))
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X8,Xa)),D2)
                     => ( aa(A,B,F2,X8) = aa(A,B,G,X8) ) ) ) )
             => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Xa,S)) ) ) ) ) ).

% Lim_transform_within
tff(fact_8044_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,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M9: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M3)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,F2,M3),aa(nat,A,F2,N2))),E3) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_8045_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [M10: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),M4)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,M4),aa(nat,A,X7,N))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X7) ) ) ).

% CauchyI'
tff(fact_8046_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,aa(int,fun(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_8047_tendsto__dist__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
        <=> filterlim(A,real,aa(B,fun(A,real),aTP_Lamp_ais(fun(A,B),fun(B,fun(A,real)),F2),L),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_dist_iff
tff(fact_8048_LIM__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S6)
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),S6) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X4),L5)),R5) ) ) ) ) ) ).

% LIM_def
tff(fact_8049_metric__LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,R: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => ? [S2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S2)
                & ! [X5: A] :
                    ( ( ( X5 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X5,A2)),S2) )
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X5),L5)),R) ) ) ) ) ) ).

% metric_LIM_D
tff(fact_8050_metric__LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [A2: A,F2: fun(A,B),L5: B] :
          ( ! [R4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A2)),S7) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X),L5)),R4) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% metric_LIM_I
tff(fact_8051_metric__LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),L: B,A2: A,R2: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ( ! [X: A] :
                  ( ( X != A2 )
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A2)),R2)
                   => ( aa(A,B,F2,X) = aa(A,B,G,X) ) ) )
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_equal2
tff(fact_8052_lim__sequentially,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No3: nat] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,N2),L5)),R5) ) ) ) ) ).

% lim_sequentially
tff(fact_8053_metric__LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),L5: A] :
          ( ! [R4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
             => ? [No2: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,N),L5)),R4) ) )
         => filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_8054_metric__LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),L5: A,R: real] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => ? [No: nat] :
              ! [N9: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N9)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,N9),L5)),R) ) ) ) ) ).

% metric_LIMSEQ_D
tff(fact_8055_power__minus_H,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: A,Na: nat] :
          ( nO_MATCH(A,A,one_one(A),Xa)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xa)),Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)) ) ) ) ).

% power_minus'
tff(fact_8056_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X7)
        <=> ! [J2: nat] :
            ? [M9: nat] :
            ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M3)
             => ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),N2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,M3),aa(nat,A,X7,N2))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J2)))) ) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_8057_scale__left__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A,Y: A,C2: B,A2: real,B2: real] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y),C2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),Xa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xa)) ) ) ) ).

% scale_left_distrib_NO_MATCH
tff(fact_8058_metric__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A2)),D6) )
                     => ( aa(A,B,F2,X) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ait(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_compose2
tff(fact_8059_scale__left__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A,Y: A,C2: B,A2: real,B2: real] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y),C2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2)),Xa) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xa)) ) ) ) ).

% scale_left_diff_distrib_NO_MATCH
tff(fact_8060_metric__isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A2),top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A2)),D6) )
                     => ( aa(A,B,F2,X) != aa(A,B,F2,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ait(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_isCont_LIM_compose2
tff(fact_8061_LIMSEQ__iff__nz,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X7: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),No3)
                  & ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X7,N2),L5)),R5) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_8062_LIM__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A2: B,F2: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A2)
         => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,top_top(set(B))))
          <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_aio(B,fun(fun(B,C),fun(B,C)),A2),F2),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_8063_filterlim__pow__at__bot__even,axiom,
    ! [Na: nat,F2: fun(real,real),F3: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(real,real,F2,at_bot(real),F3)
       => ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aiu(nat,fun(fun(real,real),fun(real,real)),Na),F2),at_top(real),F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_8064_tendsto__exp__limit__at__right,axiom,
    ! [Xa: real] : filterlim(real,real,aTP_Lamp_aiv(real,fun(real,real),Xa),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),Xa)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% tendsto_exp_limit_at_right
tff(fact_8065_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_greaterThan(A,K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),I) ) ) ).

% greaterThan_iff
tff(fact_8066_ivl__disj__int__one_I7_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,U)),set_ord_greaterThan(A,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(7)
tff(fact_8067_trivial__limit__at__right__top,axiom,
    ! [A: $tType] :
      ( ( order_top(A)
        & topolo1944317154257567458pology(A) )
     => ( topolo174197925503356063within(A,top_top(A),set_ord_greaterThan(A,top_top(A))) = bot_bot(filter(A)) ) ) ).

% trivial_limit_at_right_top
tff(fact_8068_trivial__limit__at__right__real,axiom,
    ! [A: $tType] :
      ( ( dense_order(A)
        & no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xa: A] : topolo174197925503356063within(A,Xa,set_ord_greaterThan(A,Xa)) != bot_bot(filter(A)) ) ).

% trivial_limit_at_right_real
tff(fact_8069_trivial__limit__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_bot(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_bot_linorder
tff(fact_8070_greaterThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xa: A] : set_ord_greaterThan(A,Xa) != bot_bot(set(A)) ) ).

% greaterThan_non_empty
tff(fact_8071_less__filter__def,axiom,
    ! [A: $tType,F3: filter(A),F7: filter(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less(filter(A)),F3),F7)
    <=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),F7)
        & ~ aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F7),F3) ) ) ).

% less_filter_def
tff(fact_8072_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_8073_at__within__Icc__at__right,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( topolo174197925503356063within(A,A2,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2)) ) ) ) ).

% at_within_Icc_at_right
tff(fact_8074_arsinh__real__at__bot,axiom,
    filterlim(real,real,arsinh(real),at_bot(real),at_bot(real)) ).

% arsinh_real_at_bot
tff(fact_8075_sinh__real__at__bot,axiom,
    filterlim(real,real,sinh(real),at_bot(real),at_bot(real)) ).

% sinh_real_at_bot
tff(fact_8076_artanh__real__at__right__1,axiom,
    filterlim(real,real,artanh(real),at_bot(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),one_one(real)),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),one_one(real))))) ).

% artanh_real_at_right_1
tff(fact_8077_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_8078_cosh__real__at__bot,axiom,
    filterlim(real,real,cosh(real),at_top(real),at_bot(real)) ).

% cosh_real_at_bot
tff(fact_8079_filterlim__uminus__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_bot(real),F3)
    <=> filterlim(A,real,aTP_Lamp_aiw(fun(A,real),fun(A,real),F2),at_top(real),F3) ) ).

% filterlim_uminus_at_bot
tff(fact_8080_filterlim__uminus__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F3)
    <=> filterlim(A,real,aTP_Lamp_aiw(fun(A,real),fun(A,real),F2),at_bot(real),F3) ) ).

% filterlim_uminus_at_top
tff(fact_8081_filterlim__at__bot__mirror,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,at_bot(real))
    <=> filterlim(real,A,aTP_Lamp_aix(fun(real,A),fun(real,A),F2),F3,at_top(real)) ) ).

% filterlim_at_bot_mirror
tff(fact_8082_filterlim__at__top__mirror,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_aix(fun(real,A),fun(real,A),F2),F3,at_bot(real)) ) ).

% filterlim_at_top_mirror
tff(fact_8083_less__separate,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ? [A5: A,B6: A] :
              ( member(A,Xa,set_ord_lessThan(A,A5))
              & member(A,Y,set_ord_greaterThan(A,B6))
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,A5)),set_ord_greaterThan(A,B6)) = bot_bot(set(A)) ) ) ) ) ).

% less_separate
tff(fact_8084_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_8085_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_8086_filterlim__at__right__to__0,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A),A2: real] :
      ( filterlim(real,A,F2,F3,topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
    <=> filterlim(real,A,aa(real,fun(real,A),aTP_Lamp_aiy(fun(real,A),fun(real,fun(real,A)),F2),A2),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% filterlim_at_right_to_0
tff(fact_8087_filterlim__at__left__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A),A2: real] :
      ( filterlim(real,A,F2,F3,topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
    <=> filterlim(real,A,aTP_Lamp_aix(fun(real,A),fun(real,A),F2),F3,topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A2),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),A2)))) ) ).

% filterlim_at_left_to_right
tff(fact_8088_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_8089_exp__at__bot,axiom,
    filterlim(real,real,exp(real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_bot(real)) ).

% exp_at_bot
tff(fact_8090_filterlim__times__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),P2: B,F13: filter(A),C2: B,L: B] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,P2,set_ord_greaterThan(B,P2)),F13)
         => ( aa(B,$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_aiz(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo174197925503356063within(B,L,set_ord_greaterThan(B,L)),F13) ) ) ) ) ).

% filterlim_times_pos
tff(fact_8091_filterlim__at__top__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_aja(fun(real,A),fun(real,A),F2),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% filterlim_at_top_to_right
tff(fact_8092_filterlim__at__right__to__top,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
    <=> filterlim(real,A,aTP_Lamp_aja(fun(real,A),fun(real,A),F2),F3,at_top(real)) ) ).

% filterlim_at_right_to_top
tff(fact_8093_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_8094_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_8095_log__inj,axiom,
    ! [B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => inj_on(real,real,log(B2),set_ord_greaterThan(real,zero_zero(real))) ) ).

% log_inj
tff(fact_8096_tanh__real__at__bot,axiom,
    filterlim(real,real,tanh(real),topolo7230453075368039082e_nhds(real,aa(real,real,uminus_uminus(real),one_one(real))),at_bot(real)) ).

% tanh_real_at_bot
tff(fact_8097_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_bot(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahn(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F3) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_8098_tendsto__arcosh__at__left__1,axiom,
    filterlim(real,real,arcosh(real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,one_one(real),set_ord_greaterThan(real,one_one(real)))) ).

% tendsto_arcosh_at_left_1
tff(fact_8099_tendsto__at__botI__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [F2: fun(real,A),Y: A] :
          ( ! [X10: fun(nat,real)] :
              ( filterlim(nat,real,X10,at_bot(real),at_top(nat))
             => filterlim(nat,A,aa(fun(nat,real),fun(nat,A),aTP_Lamp_ahy(fun(real,A),fun(fun(nat,real),fun(nat,A)),F2),X10),topolo7230453075368039082e_nhds(A,Y),at_top(nat)) )
         => filterlim(real,A,F2,topolo7230453075368039082e_nhds(A,Y),at_bot(real)) ) ) ).

% tendsto_at_botI_sequentially
tff(fact_8100_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_8101_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahn(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F3) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_8102_isCont__If__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,G: fun(A,B),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2)),G)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,G,A2)),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2)))
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ajb(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),G),F2)) ) ) ) ).

% isCont_If_ge
tff(fact_8103_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2)
         => ? [Y2: real] :
              ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X,top_top(set(real))))
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y2) ) )
     => ( 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,B2)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_8104_filterlim__pow__at__bot__odd,axiom,
    ! [Na: nat,F2: fun(real,real),F3: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(real,real,F2,at_bot(real),F3)
       => ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aiu(nat,fun(fun(real,real),fun(real,real)),Na),F2),at_bot(real),F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_8105_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_8106_lhopital__left__at__top,axiom,
    ! [G: fun(real,real),Xa: real,G4: fun(real,real),F2: fun(real,real),F4: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,Xa,set_ord_lessThan(real,Xa)))
     => ( eventually(real,aTP_Lamp_ajc(fun(real,real),fun(real,$o),G4),topolo174197925503356063within(real,Xa,set_ord_lessThan(real,Xa)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ajd(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F4),topolo174197925503356063within(real,Xa,set_ord_lessThan(real,Xa)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ajd(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G4),topolo174197925503356063within(real,Xa,set_ord_lessThan(real,Xa)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aje(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F4),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xa,set_ord_lessThan(real,Xa)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aje(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xa,set_ord_lessThan(real,Xa))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_8107_lhopital__right__0__at__top,axiom,
    ! [G: fun(real,real),G4: fun(real,real),F2: fun(real,real),F4: fun(real,real),Xa: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
     => ( eventually(real,aTP_Lamp_ajc(fun(real,real),fun(real,$o),G4),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ajd(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F4),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_ajd(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G4),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aje(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F4),topolo7230453075368039082e_nhds(real,Xa),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aje(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Xa),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_8108_eventually__const,axiom,
    ! [A: $tType,F3: filter(A),P: $o] :
      ( ( F3 != bot_bot(filter(A)) )
     => ( eventually(A,aTP_Lamp_sr($o,fun(A,$o),(P)),F3)
      <=> (P) ) ) ).

% eventually_const
tff(fact_8109_eventually__at__bot__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N10: A] :
            ! [N2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N2),N10)
             => aa(A,$o,P,N2) ) ) ) ).

% eventually_at_bot_dense
tff(fact_8110_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_ajf(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_8111_eventually__at__bot__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_ajg(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_at_bot_not_equal
tff(fact_8112_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_ajh(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_8113_eventually__at__right__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),Xa: A] :
          ( eventually(A,P,topolo174197925503356063within(A,Xa,set_ord_greaterThan(A,Xa)))
        <=> ? [B10: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),B10)
              & ! [Y5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y5)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),B10)
                   => aa(A,$o,P,Y5) ) ) ) ) ) ).

% eventually_at_right_field
tff(fact_8114_eventually__at__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Xa: A,Y: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( eventually(A,P,topolo174197925503356063within(A,Xa,set_ord_greaterThan(A,Xa)))
          <=> ? [B10: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),B10)
                & ! [Y5: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),B10)
                     => aa(A,$o,P,Y5) ) ) ) ) ) ) ).

% eventually_at_right
tff(fact_8115_eventually__at__right__less,axiom,
    ! [A: $tType] :
      ( ( no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xa: A] : eventually(A,aa(A,fun(A,$o),ord_less(A),Xa),topolo174197925503356063within(A,Xa,set_ord_greaterThan(A,Xa))) ) ).

% eventually_at_right_less
tff(fact_8116_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_8117_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z8: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_aji(fun(A,B),fun(B,fun(A,$o)),F2),Z8),F3) ) ) ).

% filterlim_at_bot
tff(fact_8118_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z8: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_ajj(fun(A,B),fun(B,fun(A,$o)),F2),Z8),F3) ) ) ).

% filterlim_at_top
tff(fact_8119_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z8: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z8),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aji(fun(A,B),fun(B,fun(A,$o)),F2),Z8),F3) ) ) ) ).

% filterlim_at_bot_le
tff(fact_8120_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z8: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),Z8)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ajj(fun(A,B),fun(B,fun(A,$o)),F2),Z8),F3) ) ) ) ).

% filterlim_at_top_ge
tff(fact_8121_filterlim__at__top__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,at_top(B),F3)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajk(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F3)
           => filterlim(A,B,G,at_top(B),F3) ) ) ) ).

% filterlim_at_top_mono
tff(fact_8122_filterlim__mono__eventually,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F3: filter(B),G2: filter(A),F7: filter(B),G5: filter(A),F4: fun(A,B)] :
      ( filterlim(A,B,F2,F3,G2)
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),F3),F7)
       => ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),G5),G2)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajl(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),F4),G5)
           => filterlim(A,B,F4,F7,G5) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_8123_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_ajm(fun(A,real),fun(fun(A,real),fun(A,$o)),F2),G),Net)
     => ( eventually(A,aa(fun(A,real),fun(A,$o),aTP_Lamp_ajm(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_8124_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_ajn(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),Net)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajn(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_8125_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z8: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_ajo(fun(A,B),fun(B,fun(A,$o)),F2),Z8),F3) ) ) ).

% filterlim_at_bot_dense
tff(fact_8126_sequentially__imp__eventually__at,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [A2: A,P: fun(A,$o)] :
          ( ! [F8: fun(nat,A)] :
              ( ( ! [N9: nat] : aa(nat,A,F8,N9) != A2
                & filterlim(nat,A,F8,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ajp(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F8),at_top(nat)) )
         => eventually(A,P,topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% sequentially_imp_eventually_at
tff(fact_8127_filterlim__at__within__not__equal,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t2_space(B)
     => ! [F2: fun(A,B),A2: B,Sa: set(B),F3: filter(A),B2: B] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,A2,Sa),F3)
         => eventually(A,aa(B,fun(A,$o),aa(set(B),fun(B,fun(A,$o)),aTP_Lamp_ajq(fun(A,B),fun(set(B),fun(B,fun(A,$o))),F2),Sa),B2),F3) ) ) ).

% filterlim_at_within_not_equal
tff(fact_8128_sequentially__imp__eventually__within,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Sa: set(A),A2: A,P: fun(A,$o)] :
          ( ! [F8: fun(nat,A)] :
              ( ( ! [N9: nat] :
                    ( member(A,aa(nat,A,F8,N9),Sa)
                    & ( aa(nat,A,F8,N9) != A2 ) )
                & filterlim(nat,A,F8,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ajp(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F8),at_top(nat)) )
         => eventually(A,P,topolo174197925503356063within(A,A2,Sa)) ) ) ).

% sequentially_imp_eventually_within
tff(fact_8129_filterlim__at,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),B2: B,Sa: set(B),F3: filter(A)] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,B2,Sa),F3)
        <=> ( eventually(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_ajr(fun(A,B),fun(B,fun(set(B),fun(A,$o))),F2),B2),Sa),F3)
            & filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),F3) ) ) ) ).

% filterlim_at
tff(fact_8130_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z8: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_ajs(fun(A,B),fun(B,fun(A,$o)),F2),Z8),F3) ) ) ).

% filterlim_at_top_dense
tff(fact_8131_Lim__transform__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajt(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F3)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),F3) ) ) ) ).

% Lim_transform_eventually
tff(fact_8132_eventually__compose__filterlim,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F3: filter(A),F2: fun(B,A),G2: filter(B)] :
      ( eventually(A,P,F3)
     => ( filterlim(B,A,F2,F3,G2)
       => eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_aju(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F2),G2) ) ) ).

% eventually_compose_filterlim
tff(fact_8133_filterlim__cong,axiom,
    ! [A: $tType,B: $tType,F13: filter(A),F14: filter(A),F24: filter(B),F25: filter(B),F2: fun(B,A),G: fun(B,A)] :
      ( ( F13 = F14 )
     => ( ( F24 = F25 )
       => ( eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_ajv(fun(B,A),fun(fun(B,A),fun(B,$o)),F2),G),F24)
         => ( filterlim(B,A,F2,F13,F24)
          <=> filterlim(B,A,G,F14,F25) ) ) ) ) ).

% filterlim_cong
tff(fact_8134_filterlim__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F24: filter(B),F13: filter(A)] :
      ( filterlim(A,B,F2,F24,F13)
    <=> ! [P6: fun(B,$o)] :
          ( eventually(B,P6,F24)
         => eventually(A,aa(fun(B,$o),fun(A,$o),aTP_Lamp_ajw(fun(A,B),fun(fun(B,$o),fun(A,$o)),F2),P6),F13) ) ) ).

% filterlim_iff
tff(fact_8135_eventually__nhds__iff__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [P: fun(A,$o),A2: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,A2))
        <=> ! [F6: fun(nat,A)] :
              ( filterlim(nat,A,F6,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ajp(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F6),at_top(nat)) ) ) ) ).

% eventually_nhds_iff_sequentially
tff(fact_8136_topological__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [L: A,F2: fun(B,A),F3: filter(B)] :
          ( ! [S4: set(A)] :
              ( topolo1002775350975398744n_open(A,S4)
             => ( member(A,L,S4)
               => eventually(B,aa(set(A),fun(B,$o),aTP_Lamp_ajx(fun(B,A),fun(set(A),fun(B,$o)),F2),S4),F3) ) )
         => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ).

% topological_tendstoI
tff(fact_8137_topological__tendstoD,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),S: set(B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( topolo1002775350975398744n_open(B,S)
           => ( member(B,L,S)
             => eventually(A,aa(set(B),fun(A,$o),aTP_Lamp_ajy(fun(A,B),fun(set(B),fun(A,$o)),F2),S),F3) ) ) ) ) ).

% topological_tendstoD
tff(fact_8138_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_ajz(fun(A,B),fun(B,fun(A,$o)),F2),L),Net)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),Net) ) ) ).

% tendsto_eventually
tff(fact_8139_tendsto__imp__eventually__ne,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t1_space(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A),C8: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( ( C2 != C8 )
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aka(fun(A,B),fun(B,fun(A,$o)),F2),C8),F3) ) ) ) ).

% tendsto_imp_eventually_ne
tff(fact_8140_tendsto__discrete,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo8865339358273720382pology(B)
     => ! [F2: fun(A,B),Y: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F3)
        <=> eventually(A,aa(B,fun(A,$o),aTP_Lamp_akb(fun(A,B),fun(B,fun(A,$o)),F2),Y),F3) ) ) ).

% tendsto_discrete
tff(fact_8141_tendsto__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),G: fun(A,B),F3: filter(A),C2: B] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajt(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F3)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
          <=> filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F3) ) ) ) ).

% tendsto_cong
tff(fact_8142_tendsto__def,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
        <=> ! [S8: set(B)] :
              ( topolo1002775350975398744n_open(B,S8)
             => ( member(B,L,S8)
               => eventually(A,aa(set(B),fun(A,$o),aTP_Lamp_ajy(fun(A,B),fun(set(B),fun(A,$o)),F2),S8),F3) ) ) ) ) ).

% tendsto_def
tff(fact_8143_order__tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Xa: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xa),F3)
        <=> ( ! [L4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L4),Xa)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_akc(fun(A,B),fun(B,fun(A,$o)),F2),L4),F3) )
            & ! [U2: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Xa),U2)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_akd(fun(A,B),fun(B,fun(A,$o)),F2),U2),F3) ) ) ) ) ).

% order_tendsto_iff
tff(fact_8144_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Y: A,F2: fun(B,A),F3: filter(B)] :
          ( ! [A5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A5),Y)
             => eventually(B,aa(A,fun(B,$o),aTP_Lamp_ake(fun(B,A),fun(A,fun(B,$o)),F2),A5),F3) )
         => ( ! [A5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A5)
               => eventually(B,aa(A,fun(B,$o),aTP_Lamp_akf(fun(B,A),fun(A,fun(B,$o)),F2),A5),F3) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y),F3) ) ) ) ).

% order_tendstoI
tff(fact_8145_order__tendstoD_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F3: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),Y)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_akc(fun(A,B),fun(B,fun(A,$o)),F2),A2),F3) ) ) ) ).

% order_tendstoD(1)
tff(fact_8146_order__tendstoD_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F3: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),A2)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_akd(fun(A,B),fun(B,fun(A,$o)),F2),A2),F3) ) ) ) ).

% order_tendstoD(2)
tff(fact_8147_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N10: A] :
            ! [N2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N10),N2)
             => aa(A,$o,P,N2) ) ) ) ).

% eventually_at_top_dense
tff(fact_8148_eventually__at__top__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_akg(A,fun(A,$o),C2),at_top(A)) ) ).

% eventually_at_top_not_equal
tff(fact_8149_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_8150_eventually__at__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,Xa: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( eventually(A,P,topolo174197925503356063within(A,Xa,set_ord_lessThan(A,Xa)))
          <=> ? [B10: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B10),Xa)
                & ! [Y5: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B10),Y5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),Xa)
                     => aa(A,$o,P,Y5) ) ) ) ) ) ) ).

% eventually_at_left
tff(fact_8151_eventually__at__left__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),Xa: A] :
          ( eventually(A,P,topolo174197925503356063within(A,Xa,set_ord_lessThan(A,Xa)))
        <=> ? [B10: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B10),Xa)
              & ! [Y5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B10),Y5)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),Xa)
                   => aa(A,$o,P,Y5) ) ) ) ) ) ).

% eventually_at_left_field
tff(fact_8152_eventually__nhds__in__open,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A),Xa: A] :
          ( topolo1002775350975398744n_open(A,Sa)
         => ( member(A,Xa,Sa)
           => eventually(A,aTP_Lamp_akh(set(A),fun(A,$o),Sa),topolo7230453075368039082e_nhds(A,Xa)) ) ) ) ).

% eventually_nhds_in_open
tff(fact_8153_eventually__at__filter,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),A2: A,Sa: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,Sa))
        <=> eventually(A,aa(set(A),fun(A,$o),aa(A,fun(set(A),fun(A,$o)),aTP_Lamp_aki(fun(A,$o),fun(A,fun(set(A),fun(A,$o))),P),A2),Sa),topolo7230453075368039082e_nhds(A,A2)) ) ) ).

% eventually_at_filter
tff(fact_8154_eventually__nhds__top,axiom,
    ! [A: $tType] :
      ( ( order_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),top_top(A))
         => ( eventually(A,P,topolo7230453075368039082e_nhds(A,top_top(A)))
          <=> ? [B10: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B10),top_top(A))
                & ! [Z6: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B10),Z6)
                   => aa(A,$o,P,Z6) ) ) ) ) ) ).

% eventually_nhds_top
tff(fact_8155_eventually__eventually,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),Xa: A] :
          ( eventually(A,aTP_Lamp_akj(fun(A,$o),fun(A,$o),P),topolo7230453075368039082e_nhds(A,Xa))
        <=> eventually(A,P,topolo7230453075368039082e_nhds(A,Xa)) ) ) ).

% eventually_eventually
tff(fact_8156_t1__space__nhds,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
         => eventually(A,aTP_Lamp_akk(A,fun(A,$o),Y),topolo7230453075368039082e_nhds(A,Xa)) ) ) ).

% t1_space_nhds
tff(fact_8157_has__field__derivative__cong__ev,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xa: A,Y: A,S: set(A),F2: fun(A,A),G: fun(A,A),U: A,V: A,T: set(A)] :
          ( ( Xa = Y )
         => ( eventually(A,aa(fun(A,A),fun(A,$o),aa(fun(A,A),fun(fun(A,A),fun(A,$o)),aTP_Lamp_akl(set(A),fun(fun(A,A),fun(fun(A,A),fun(A,$o))),S),F2),G),topolo7230453075368039082e_nhds(A,Xa))
           => ( ( U = V )
             => ( ( S = T )
               => ( member(A,Xa,S)
                 => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,Xa,S))
                  <=> has_field_derivative(A,G,V,topolo174197925503356063within(A,Y,T)) ) ) ) ) ) ) ) ).

% has_field_derivative_cong_ev
tff(fact_8158_has__field__derivative__cong__eventually,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),G: fun(A,A),Xa: A,S: set(A),U: A] :
          ( eventually(A,aa(fun(A,A),fun(A,$o),aTP_Lamp_akm(fun(A,A),fun(fun(A,A),fun(A,$o)),F2),G),topolo174197925503356063within(A,Xa,S))
         => ( ( aa(A,A,F2,Xa) = aa(A,A,G,Xa) )
           => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,Xa,S))
            <=> has_field_derivative(A,G,U,topolo174197925503356063within(A,Xa,S)) ) ) ) ) ).

% has_field_derivative_cong_eventually
tff(fact_8159_not__eventually__impI,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F3)
     => ( ~ eventually(A,Q,F3)
       => ~ eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_av(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3) ) ) ).

% not_eventually_impI
tff(fact_8160_eventually__conj__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F3: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3)
    <=> ( eventually(A,P,F3)
        & eventually(A,Q,F3) ) ) ).

% eventually_conj_iff
tff(fact_8161_eventually__rev__mp,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F3)
     => ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_av(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3)
       => eventually(A,Q,F3) ) ) ).

% eventually_rev_mp
tff(fact_8162_eventually__subst,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F3: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akn(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3)
     => ( eventually(A,P,F3)
      <=> eventually(A,Q,F3) ) ) ).

% eventually_subst
tff(fact_8163_eventually__elim2,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o),R2: fun(A,$o)] :
      ( eventually(A,P,F3)
     => ( eventually(A,Q,F3)
       => ( ! [I3: A] :
              ( aa(A,$o,P,I3)
             => ( aa(A,$o,Q,I3)
               => aa(A,$o,R2,I3) ) )
         => eventually(A,R2,F3) ) ) ) ).

% eventually_elim2
tff(fact_8164_eventually__conj,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F3)
     => ( eventually(A,Q,F3)
       => eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3) ) ) ).

% eventually_conj
tff(fact_8165_eventually__True,axiom,
    ! [A: $tType,F3: filter(A)] : eventually(A,aTP_Lamp_rl(A,$o),F3) ).

% eventually_True
tff(fact_8166_eventually__mp,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F3: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_av(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3)
     => ( eventually(A,P,F3)
       => eventually(A,Q,F3) ) ) ).

% eventually_mp
tff(fact_8167_eventually__frequently__const__simps_I3_J,axiom,
    ! [A: $tType,P: fun(A,$o),C3: $o,F3: filter(A)] :
      ( eventually(A,aa($o,fun(A,$o),aTP_Lamp_ako(fun(A,$o),fun($o,fun(A,$o)),P),(C3)),F3)
    <=> ( eventually(A,P,F3)
        | (C3) ) ) ).

% eventually_frequently_const_simps(3)
tff(fact_8168_eventually__frequently__const__simps_I4_J,axiom,
    ! [A: $tType,C3: $o,P: fun(A,$o),F3: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akp($o,fun(fun(A,$o),fun(A,$o)),(C3)),P),F3)
    <=> ( (C3)
        | eventually(A,P,F3) ) ) ).

% eventually_frequently_const_simps(4)
tff(fact_8169_eventually__frequently__const__simps_I6_J,axiom,
    ! [A: $tType,C3: $o,P: fun(A,$o),F3: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akq($o,fun(fun(A,$o),fun(A,$o)),(C3)),P),F3)
    <=> ( (C3)
       => eventually(A,P,F3) ) ) ).

% eventually_frequently_const_simps(6)
tff(fact_8170_has__derivative__transform__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: fun(A,B),Xa: A,Sa: set(A),G: fun(A,B)] :
          ( has_derivative(A,B,F2,F4,topolo174197925503356063within(A,Xa,Sa))
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_akr(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),topolo174197925503356063within(A,Xa,Sa))
           => ( ( aa(A,B,F2,Xa) = aa(A,B,G,Xa) )
             => ( member(A,Xa,Sa)
               => has_derivative(A,B,G,F4,topolo174197925503356063within(A,Xa,Sa)) ) ) ) ) ) ).

% has_derivative_transform_eventually
tff(fact_8171_False__imp__not__eventually,axiom,
    ! [A: $tType,P: fun(A,$o),Net: filter(A)] :
      ( ! [X: A] : ~ aa(A,$o,P,X)
     => ( ( Net != bot_bot(filter(A)) )
       => ~ eventually(A,P,Net) ) ) ).

% False_imp_not_eventually
tff(fact_8172_eventually__const__iff,axiom,
    ! [A: $tType,P: $o,F3: filter(A)] :
      ( eventually(A,aTP_Lamp_sr($o,fun(A,$o),(P)),F3)
    <=> ( (P)
        | ( F3 = bot_bot(filter(A)) ) ) ) ).

% eventually_const_iff
tff(fact_8173_trivial__limit__def,axiom,
    ! [A: $tType,F3: filter(A)] :
      ( ( F3 = bot_bot(filter(A)) )
    <=> eventually(A,aTP_Lamp_ab(A,$o),F3) ) ).

% trivial_limit_def
tff(fact_8174_eventually__happens_H,axiom,
    ! [A: $tType,F3: filter(A),P: fun(A,$o)] :
      ( ( F3 != bot_bot(filter(A)) )
     => ( eventually(A,P,F3)
       => ? [X_1: A] : aa(A,$o,P,X_1) ) ) ).

% eventually_happens'
tff(fact_8175_eventually__happens,axiom,
    ! [A: $tType,P: fun(A,$o),Net: filter(A)] :
      ( eventually(A,P,Net)
     => ( ( Net = bot_bot(filter(A)) )
        | ? [X_1: A] : aa(A,$o,P,X_1) ) ) ).

% eventually_happens
tff(fact_8176_eventually__bot,axiom,
    ! [A: $tType,P: fun(A,$o)] : eventually(A,P,bot_bot(filter(A))) ).

% eventually_bot
tff(fact_8177_eventually__Lim__ident__at,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [P: fun(A,fun(A,$o)),Xa: A,X7: set(A)] :
          ( eventually(A,aa(A,fun(A,$o),P,topolo3827282254853284352ce_Lim(A,A,topolo174197925503356063within(A,Xa,X7),aTP_Lamp_abp(A,A))),topolo174197925503356063within(A,Xa,X7))
        <=> eventually(A,aa(A,fun(A,$o),P,Xa),topolo174197925503356063within(A,Xa,X7)) ) ) ).

% eventually_Lim_ident_at
tff(fact_8178_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_cl(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uu)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))) ).

% ATP.lambda_1
tff(fact_8179_ATP_Olambda__2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_aeu(A,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,exp(A),Uu)),one_one(A))),Uu) ) ).

% ATP.lambda_2
tff(fact_8180_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_gm(nat,real),Uu) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_3
tff(fact_8181_ATP_Olambda__4,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_oz(real,$o),Uu)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uu)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uu),aa(num,real,numeral_numeral(real),bit0(one2)))
        & ( aa(real,real,cos(real),Uu) = zero_zero(real) ) ) ) ).

% ATP.lambda_4
tff(fact_8182_ATP_Olambda__5,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_agp(nat,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu))),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_5
tff(fact_8183_ATP_Olambda__6,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_gv(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,cos_coeff,Uu)),aa(nat,real,aa(real,fun(nat,real),power_power(real),zero_zero(real)),Uu)) ).

% ATP.lambda_6
tff(fact_8184_ATP_Olambda__7,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: nat] : aa(nat,fun(int,int),aTP_Lamp_vh(nat,fun(int,int)),Uu) = comp(int,int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Uu),aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% ATP.lambda_7
tff(fact_8185_ATP_Olambda__8,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_ago(nat,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu))) ) ).

% ATP.lambda_8
tff(fact_8186_ATP_Olambda__9,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: nat] : aa(nat,fun(int,int),aTP_Lamp_uy(nat,fun(int,int)),Uu) = comp(int,int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Uu),aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A)))) ) ).

% ATP.lambda_9
tff(fact_8187_ATP_Olambda__10,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_aey(real,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,cos(real),Uu)),aa(real,real,sin(real),Uu)) ).

% ATP.lambda_10
tff(fact_8188_ATP_Olambda__11,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_ahx(real,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Uu)),Uu) ).

% ATP.lambda_11
tff(fact_8189_ATP_Olambda__12,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Uu: fun(int,int)] :
          ( aa(fun(int,int),$o,aTP_Lamp_vc(fun(int,int),$o),Uu)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A))) ) ) ).

% ATP.lambda_12
tff(fact_8190_ATP_Olambda__13,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Uu: fun(int,int)] :
          ( aa(fun(int,int),$o,aTP_Lamp_vb(fun(int,int),$o),Uu)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))) ) ) ).

% ATP.lambda_13
tff(fact_8191_ATP_Olambda__14,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Uu: word(B)] :
          ( aa(word(B),$o,aTP_Lamp_tb(word(B),$o),Uu)
        <=> aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),Uu),aa(nat,word(B),aa(word(B),fun(nat,word(B)),power_power(word(B)),aa(num,word(B),numeral_numeral(word(B)),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% ATP.lambda_14
tff(fact_8192_ATP_Olambda__15,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_uj(int,int),Uu) = aa(int,int,aa(int,fun(int,int),divide_divide(int),Uu),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% ATP.lambda_15
tff(fact_8193_ATP_Olambda__16,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [Uu: fun(int,A)] : aa(fun(int,A),fun(int,A),aTP_Lamp_vd(fun(int,A),fun(int,A)),Uu) = comp(int,A,int,Uu,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(B,type2(B)))) ) ).

% ATP.lambda_16
tff(fact_8194_ATP_Olambda__17,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_afw(nat,real),Uu) = aa(real,real,root(Uu),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_17
tff(fact_8195_ATP_Olambda__18,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_sm(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_18
tff(fact_8196_ATP_Olambda__19,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_tt(int,int),Uu) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),Uu) ).

% ATP.lambda_19
tff(fact_8197_ATP_Olambda__20,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_un(int,int),Uu) = aa(int,int,aa(int,fun(int,int),minus_minus(int),Uu),one_one(int)) ).

% ATP.lambda_20
tff(fact_8198_ATP_Olambda__21,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_um(int,int),Uu) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),one_one(int)) ).

% ATP.lambda_21
tff(fact_8199_ATP_Olambda__22,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_fq(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_22
tff(fact_8200_ATP_Olambda__23,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_uw(A,set(A)),Uu) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu),bot_bot(set(A))) ).

% ATP.lambda_23
tff(fact_8201_ATP_Olambda__24,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_agg(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_24
tff(fact_8202_ATP_Olambda__25,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_agn(nat,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_25
tff(fact_8203_ATP_Olambda__26,axiom,
    ! [Uu: int] :
      ( aa(int,$o,aTP_Lamp_rr(int,$o),Uu)
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Uu) ) ).

% ATP.lambda_26
tff(fact_8204_ATP_Olambda__27,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: nat] : aa(nat,fun(int,int),aTP_Lamp_uz(nat,fun(int,int)),Uu) = aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),Uu)) ) ).

% ATP.lambda_27
tff(fact_8205_ATP_Olambda__28,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_xw(real,real),Uu) = suminf(real,aTP_Lamp_bf(real,fun(nat,real),Uu)) ).

% ATP.lambda_28
tff(fact_8206_ATP_Olambda__29,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_ur(fun(A,B),set(product_prod(A,B))),Uu) = collect(product_prod(A,B),product_case_prod(A,B,$o,aTP_Lamp_uq(fun(A,B),fun(A,fun(B,$o)),Uu))) ).

% ATP.lambda_29
tff(fact_8207_ATP_Olambda__30,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_agh(nat,real),Uu) = aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uu))) ).

% ATP.lambda_30
tff(fact_8208_ATP_Olambda__31,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_lo(nat,A),Uu) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uu)) ) ).

% ATP.lambda_31
tff(fact_8209_ATP_Olambda__32,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_afz(nat,A),Uu) = aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_32
tff(fact_8210_ATP_Olambda__33,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_rk(num,option(num)),Uu) = some(num,aa(num,num,bit1,Uu)) ).

% ATP.lambda_33
tff(fact_8211_ATP_Olambda__34,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_ri(num,option(num)),Uu) = some(num,bit0(Uu)) ).

% ATP.lambda_34
tff(fact_8212_ATP_Olambda__35,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_pp(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),Uu)) ).

% ATP.lambda_35
tff(fact_8213_ATP_Olambda__36,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ps(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,abs_abs(int),Uu)) ).

% ATP.lambda_36
tff(fact_8214_ATP_Olambda__37,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_oc(nat,fun(nat,product_prod(nat,nat))),Uu) = aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_37
tff(fact_8215_ATP_Olambda__38,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: int] : aa(int,nat,aTP_Lamp_ue(int,nat),Uu) = type_len0_len_of(A,type2(A)) ) ).

% ATP.lambda_38
tff(fact_8216_ATP_Olambda__39,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [Uu: fun(int,A)] : aa(fun(int,A),nat,aTP_Lamp_va(fun(int,A),nat),Uu) = type_len0_len_of(B,type2(B)) ) ).

% ATP.lambda_39
tff(fact_8217_ATP_Olambda__40,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [Uu: fun(A,int)] : aa(fun(A,int),nat,aTP_Lamp_ux(fun(A,int),nat),Uu) = type_len0_len_of(B,type2(B)) ) ).

% ATP.lambda_40
tff(fact_8218_ATP_Olambda__41,axiom,
    ! [Uu: nat] : aa(nat,option(num),aTP_Lamp_wc(nat,option(num)),Uu) = some(num,one2) ).

% ATP.lambda_41
tff(fact_8219_ATP_Olambda__42,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_vs(num,fun(nat,option(num)),Uu),Uua) = aa(num,option(num),aa(fun(num,option(num)),fun(num,option(num)),aa(fun(num,option(num)),fun(fun(num,option(num)),fun(num,option(num))),aa(option(num),fun(fun(num,option(num)),fun(fun(num,option(num)),fun(num,option(num)))),case_num(option(num)),some(num,one2)),aTP_Lamp_vq(nat,fun(num,option(num)),Uua)),aTP_Lamp_vr(nat,fun(num,option(num)),Uua)),Uu) ).

% ATP.lambda_42
tff(fact_8220_ATP_Olambda__43,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_of(A,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_43
tff(fact_8221_ATP_Olambda__44,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_km(nat,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_44
tff(fact_8222_ATP_Olambda__45,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_od(A,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_45
tff(fact_8223_ATP_Olambda__46,axiom,
    ! [Uu: fun(nat,real),Uua: nat] :
      aa(nat,real,aTP_Lamp_go(fun(nat,real),fun(nat,real),Uu),Uua) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),zero_zero(real),aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_46
tff(fact_8224_ATP_Olambda__47,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: nat,Uua: int] : aa(int,int,aa(nat,fun(int,int),aTP_Lamp_vf(nat,fun(int,int)),Uu),Uua) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Uu),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),Uu),type_len0_len_of(A,type2(A)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uua))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Uu),type_len0_len_of(A,type2(A)))),Uua)) ) ).

% ATP.lambda_47
tff(fact_8225_ATP_Olambda__48,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_uk(int,fun(int,int)),Uu),Uua) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(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),Uu),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),Uu),aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uua))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Uu),aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))),Uua)) ) ).

% ATP.lambda_48
tff(fact_8226_ATP_Olambda__49,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] :
      aa(nat,option(A),aTP_Lamp_pm(list(A),fun(nat,option(A)),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),aa(list(A),nat,size_size(list(A)),Uu)),some(A,aa(nat,A,nth(A,Uu),Uua)),none(A)) ).

% ATP.lambda_49
tff(fact_8227_ATP_Olambda__50,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: nat,Uua: int] : aa(int,int,aa(nat,fun(int,int),aTP_Lamp_vg(nat,fun(int,int)),Uu),Uua) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Uu),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),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Uu),type_len0_len_of(A,type2(A))))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uua))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Uu),type_len0_len_of(A,type2(A))))),Uua)) ) ).

% ATP.lambda_50
tff(fact_8228_ATP_Olambda__51,axiom,
    ! [Uu: int,Uua: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_pn(int,fun(int,product_prod(int,int))),Uu),Uua) = $ite(Uu = zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,Uu)),Uua)),aa(int,int,abs_abs(int),Uu))) ).

% ATP.lambda_51
tff(fact_8229_ATP_Olambda__52,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_iw(nat,fun(nat,A),Uu),Uua) = $ite(~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_52
tff(fact_8230_ATP_Olambda__53,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_tw(int,fun(int,int)),Uu),Uua) = aa(int,int,bit_concat_bit(type_len0_len_of(B,type2(B)),Uua),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uu)) ) ).

% ATP.lambda_53
tff(fact_8231_ATP_Olambda__54,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(nat,A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aik(fun(nat,A),fun(A,$o),Uu),Uua)
        <=> filterlim(nat,A,Uu,topolo7230453075368039082e_nhds(A,Uua),at_top(nat)) ) ) ).

% ATP.lambda_54
tff(fact_8232_ATP_Olambda__55,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Uu: option(A),Uua: A] : aa(A,option(A),aTP_Lamp_ny(option(A),fun(A,option(A)),Uu),Uua) = case_option(option(A),A,none(A),aTP_Lamp_nx(A,fun(A,option(A)),Uua),Uu) ) ).

% ATP.lambda_55
tff(fact_8233_ATP_Olambda__56,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_rg(option(A),fun(A,$o),Uu),Uua)
        <=> case_option($o,A,$true,aTP_Lamp_rf(A,fun(A,$o),Uua),Uu) ) ) ).

% ATP.lambda_56
tff(fact_8234_ATP_Olambda__57,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_vt(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_vs(num,fun(nat,option(num)),Uua),Uu) ).

% ATP.lambda_57
tff(fact_8235_ATP_Olambda__58,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_re(option(A),fun(A,$o),Uu),Uua)
        <=> case_option($o,A,$false,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_58
tff(fact_8236_ATP_Olambda__59,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_vq(nat,fun(num,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_ri(num,option(num)),bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_59
tff(fact_8237_ATP_Olambda__60,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_vw(num,fun(nat,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_ri(num,option(num)),bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_60
tff(fact_8238_ATP_Olambda__61,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_nh(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ng(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_61
tff(fact_8239_ATP_Olambda__62,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_nf(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ne(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_62
tff(fact_8240_ATP_Olambda__63,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_uh(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uu)),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uua)) ) ).

% ATP.lambda_63
tff(fact_8241_ATP_Olambda__64,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_ui(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uu)),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uua)) ) ).

% ATP.lambda_64
tff(fact_8242_ATP_Olambda__65,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: int,Uua: int] :
          ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_uf(int,fun(int,$o)),Uu),Uua)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uu)),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uua)) ) ) ).

% ATP.lambda_65
tff(fact_8243_ATP_Olambda__66,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: int,Uua: int] :
          ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ug(int,fun(int,$o)),Uu),Uua)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uu)),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uua)) ) ) ).

% ATP.lambda_66
tff(fact_8244_ATP_Olambda__67,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_cd(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Uu),one_one(real))),aa(nat,nat,suc,Uua))) ).

% ATP.lambda_67
tff(fact_8245_ATP_Olambda__68,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ir(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(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,aa(real,fun(nat,real),power_power(real),Uu),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_68
tff(fact_8246_ATP_Olambda__69,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_iq(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(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,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ).

% ATP.lambda_69
tff(fact_8247_ATP_Olambda__70,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_kf(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),Uua))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_70
tff(fact_8248_ATP_Olambda__71,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_lk(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_71
tff(fact_8249_ATP_Olambda__72,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_pa(real,fun(real,$o),Uu),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))))
        & ( aa(real,real,sin(real),Uua) = Uu ) ) ) ).

% ATP.lambda_72
tff(fact_8250_ATP_Olambda__73,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_ou(real,fun(real,$o),Uu),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) = Uu ) ) ) ).

% ATP.lambda_73
tff(fact_8251_ATP_Olambda__74,axiom,
    ! [Uu: nat,Uua: nat] :
      aa(nat,a,aa(nat,fun(nat,a),aTP_Lamp_ob(nat,fun(nat,a)),Uu),Uua) = $let(
        m2: a,
        m2:= aa(a,a,aa(a,fun(a,a),times_times(a),aa(num,a,numeral_numeral(a),bit0(one2))),aa(nat,a,semiring_1_of_nat(a),Uu)),
        $ite(Uua = zero_zero(nat),m2,aa(a,a,aa(a,fun(a,a),plus_plus(a),m2),one_one(a))) ) ).

% ATP.lambda_74
tff(fact_8252_ATP_Olambda__75,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa(real,$o,aTP_Lamp_pf(complex,fun(real,$o),Uu),Uua)
    <=> ( ( sgn_sgn(complex,Uu) = 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_75
tff(fact_8253_ATP_Olambda__76,axiom,
    ! [Uu: real,Uua: int] :
      ( aa(int,$o,aTP_Lamp_pd(real,fun(int,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Uua)),Uu)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uu),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_76
tff(fact_8254_ATP_Olambda__77,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_pe(rat,fun(int,$o),Uu),Uua)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu)
        & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Uu),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_77
tff(fact_8255_ATP_Olambda__78,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_bf(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),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,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: int,Uua: int] :
          ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ub(int,fun(int,$o)),Uu),Uua)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uu)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uua)) ) ) ).

% ATP.lambda_79
tff(fact_8257_ATP_Olambda__80,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_tz(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uu)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uua)) ) ).

% ATP.lambda_80
tff(fact_8258_ATP_Olambda__81,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_ua(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uu)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uua)) ) ).

% ATP.lambda_81
tff(fact_8259_ATP_Olambda__82,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: int,Uua: int] :
          ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_tx(int,fun(int,$o)),Uu),Uua)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uu)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uua)) ) ) ).

% ATP.lambda_82
tff(fact_8260_ATP_Olambda__83,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: int,Uua: int] :
          ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ty(int,fun(int,$o)),Uu),Uua)
        <=> aa(int,$o,dvd_dvd(int,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uu)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uua)) ) ) ).

% ATP.lambda_83
tff(fact_8261_ATP_Olambda__84,axiom,
    ! [C: $tType] :
      ( type_len(C)
     => ! [Uu: int,Uua: int] :
          ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_tv(int,fun(int,$o)),Uu),Uua)
        <=> ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(C,type2(C))),Uu) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(C,type2(C))),Uua) ) ) ) ).

% ATP.lambda_84
tff(fact_8262_ATP_Olambda__85,axiom,
    ! [B: $tType] :
      ( type_len(B)
     => ! [Uu: int,Uua: int] :
          ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_tm(int,fun(int,$o)),Uu),Uua)
        <=> ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(B,type2(B))),Uu) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(B,type2(B))),Uua) ) ) ) ).

% ATP.lambda_85
tff(fact_8263_ATP_Olambda__86,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: int,Uua: int] :
          ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_tu(int,fun(int,$o)),Uu),Uua)
        <=> ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uu) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uua) ) ) ) ).

% ATP.lambda_86
tff(fact_8264_ATP_Olambda__87,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_xx(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_87
tff(fact_8265_ATP_Olambda__88,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_88
tff(fact_8266_ATP_Olambda__89,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_kj(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_89
tff(fact_8267_ATP_Olambda__90,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lc(A,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))),Uua) ) ).

% ATP.lambda_90
tff(fact_8268_ATP_Olambda__91,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: int,Uua: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),aTP_Lamp_ul(int,fun(nat,$o)),Uu),Uua)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),Uu),Uua) ) ) ) ).

% ATP.lambda_91
tff(fact_8269_ATP_Olambda__92,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_kt(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_92
tff(fact_8270_ATP_Olambda__93,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lf(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)) ) ).

% ATP.lambda_93
tff(fact_8271_ATP_Olambda__94,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_jz(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,binomial(Uu),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% ATP.lambda_94
tff(fact_8272_ATP_Olambda__95,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_it(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_95
tff(fact_8273_ATP_Olambda__96,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_ahz(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu)),aa(real,real,exp(real),Uua)) ).

% ATP.lambda_96
tff(fact_8274_ATP_Olambda__97,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ji(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uu) ).

% ATP.lambda_97
tff(fact_8275_ATP_Olambda__98,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_jh(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uua) ).

% ATP.lambda_98
tff(fact_8276_ATP_Olambda__99,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa(complex,$o,aTP_Lamp_gc(nat,fun(complex,$o),Uu),Uua)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_99
tff(fact_8277_ATP_Olambda__100,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( aa(A,$o,aTP_Lamp_cv(nat,fun(A,$o),Uu),Uua)
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_100
tff(fact_8278_ATP_Olambda__101,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_pl(A,fun(A,$o),Uu),Uua)
        <=> ( member(A,Uua,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu) ) ) ) ).

% ATP.lambda_101
tff(fact_8279_ATP_Olambda__102,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_agx(real,fun(nat,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Uu),aa(nat,real,semiring_1_of_nat(real),Uua)))),Uua) ).

% ATP.lambda_102
tff(fact_8280_ATP_Olambda__103,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_aiv(real,fun(real,real),Uu),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),Uu),Uua)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Uua)) ).

% ATP.lambda_103
tff(fact_8281_ATP_Olambda__104,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_aia(real,fun(real,real),Uu),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),Uu),Uua)),Uua) ).

% ATP.lambda_104
tff(fact_8282_ATP_Olambda__105,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_aw(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),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_105
tff(fact_8283_ATP_Olambda__106,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_oy(real,fun(real,$o),Uu),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)
        & ( aa(real,real,cos(real),Uua) = Uu ) ) ) ).

% ATP.lambda_106
tff(fact_8284_ATP_Olambda__107,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_id(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_107
tff(fact_8285_ATP_Olambda__108,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fi(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_108
tff(fact_8286_ATP_Olambda__109,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_agd(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_109
tff(fact_8287_ATP_Olambda__110,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fa(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_110
tff(fact_8288_ATP_Olambda__111,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_qh(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),Uu,Uua),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_111
tff(fact_8289_ATP_Olambda__112,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_qf(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_112
tff(fact_8290_ATP_Olambda__113,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bx(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_113
tff(fact_8291_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cf(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_114
tff(fact_8292_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dw(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_115
tff(fact_8293_ATP_Olambda__116,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bw(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_116
tff(fact_8294_ATP_Olambda__117,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fh(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_117
tff(fact_8295_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fg(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_118
tff(fact_8296_ATP_Olambda__119,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_agc(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_119
tff(fact_8297_ATP_Olambda__120,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gg(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_120
tff(fact_8298_ATP_Olambda__121,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_je(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),Uu,Uua),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_121
tff(fact_8299_ATP_Olambda__122,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_jb(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),Uu,Uua)),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_122
tff(fact_8300_ATP_Olambda__123,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(B,A),aTP_Lamp_uv(fun(A,B),fun(A,product_prod(B,A)),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),aa(A,B,Uu,Uua)),Uua) ).

% ATP.lambda_123
tff(fact_8301_ATP_Olambda__124,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(B,real),Uua: B] : aa(B,real,aTP_Lamp_aah(fun(B,real),fun(B,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(B,real,Uu,Uua)),zero_zero(real)) ) ).

% ATP.lambda_124
tff(fact_8302_ATP_Olambda__125,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_oi(fun(A,real),fun(A,complex),Uu),Uua) = complex2(aa(A,real,Uu,Uua),zero_zero(real)) ).

% ATP.lambda_125
tff(fact_8303_ATP_Olambda__126,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ew(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_126
tff(fact_8304_ATP_Olambda__127,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_hv(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_127
tff(fact_8305_ATP_Olambda__128,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ahh(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),Uu)),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_128
tff(fact_8306_ATP_Olambda__129,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ahg(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afe(fun(nat,real),fun(nat,real),Uu)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ).

% ATP.lambda_129
tff(fact_8307_ATP_Olambda__130,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_td(list(A),fun(list(A),list(list(A))),Uu),Uua) = map(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_tc(list(A),fun(A,list(A))),Uua),Uu) ).

% ATP.lambda_130
tff(fact_8308_ATP_Olambda__131,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_aie(fun(A,B),fun(A,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua))),real_V7770717601297561774m_norm(A,Uua)) ) ).

% ATP.lambda_131
tff(fact_8309_ATP_Olambda__132,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ln(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% ATP.lambda_132
tff(fact_8310_ATP_Olambda__133,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lt(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,suc,Uua)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_133
tff(fact_8311_ATP_Olambda__134,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lq(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_134
tff(fact_8312_ATP_Olambda__135,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lp(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_135
tff(fact_8313_ATP_Olambda__136,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fp(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_136
tff(fact_8314_ATP_Olambda__137,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: num,Uua: num] : aa(num,word(A),aTP_Lamp_rh(num,fun(num,word(A)),Uu),Uua) = aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se2584673776208193580ke_bit(word(A)),aa(num,nat,numeral_numeral(nat),Uu)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,word(A),numeral_numeral(word(A)),Uua))) ) ).

% ATP.lambda_137
tff(fact_8315_ATP_Olambda__138,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_rj(num,fun(num,int),Uu),Uua) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(num,nat,numeral_numeral(nat),Uu)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,int,numeral_numeral(int),Uua))) ).

% ATP.lambda_138
tff(fact_8316_ATP_Olambda__139,axiom,
    ! [Uu: int,Uua: $o] : aa($o,int,aa(int,fun($o,int),aTP_Lamp_ve(int,fun($o,int)),Uu),(Uua)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($o,int,zero_neq_one_of_bool(int),(Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uu)) ).

% ATP.lambda_139
tff(fact_8317_ATP_Olambda__140,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_le(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,cos_coeff,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uu)),Uua)) ) ).

% ATP.lambda_140
tff(fact_8318_ATP_Olambda__141,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_sw(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Uua)),Uu)) ).

% ATP.lambda_141
tff(fact_8319_ATP_Olambda__142,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ky(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,sin_coeff,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_142
tff(fact_8320_ATP_Olambda__143,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_kz(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,cos_coeff,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_143
tff(fact_8321_ATP_Olambda__144,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ahc(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_144
tff(fact_8322_ATP_Olambda__145,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ahb(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_145
tff(fact_8323_ATP_Olambda__146,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_is(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,sin_coeff,Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_146
tff(fact_8324_ATP_Olambda__147,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_if(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,cos_coeff,Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_147
tff(fact_8325_ATP_Olambda__148,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aa(num,fun(num,int),aTP_Lamp_ut(num,fun(num,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Uu)),aa(num,int,numeral_numeral(int),Uua)) ).

% ATP.lambda_148
tff(fact_8326_ATP_Olambda__149,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_sy(list(list(A)),fun(A,list(list(A))),Uu),Uua) = map(list(A),list(A),cons(A,Uua),product_lists(A,Uu)) ).

% ATP.lambda_149
tff(fact_8327_ATP_Olambda__150,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_lg(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_150
tff(fact_8328_ATP_Olambda__151,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_sz(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu) ).

% ATP.lambda_151
tff(fact_8329_ATP_Olambda__152,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_ov(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,real,exp(real),Uua) = Uu ) ) ).

% ATP.lambda_152
tff(fact_8330_ATP_Olambda__153,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_nv(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))) ) ).

% ATP.lambda_153
tff(fact_8331_ATP_Olambda__154,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: int,Uua: nat] : aa(nat,int,aa(int,fun(nat,int),aTP_Lamp_vn(int,fun(nat,int)),Uu),Uua) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Uua),aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uu)) ) ).

% ATP.lambda_154
tff(fact_8332_ATP_Olambda__155,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_nw(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_155
tff(fact_8333_ATP_Olambda__156,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gu(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% ATP.lambda_156
tff(fact_8334_ATP_Olambda__157,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: nat,Uua: int] : aa(int,int,aa(nat,fun(int,int),aTP_Lamp_vm(nat,fun(int,int)),Uu),Uua) = aa(int,int,aa(nat,fun(int,int),bit_ri4674362597316999326ke_bit(int),Uu),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uua)) ) ).

% ATP.lambda_157
tff(fact_8335_ATP_Olambda__158,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: int,Uua: nat] : aa(nat,int,aa(int,fun(nat,int),aTP_Lamp_vk(int,fun(nat,int)),Uu),Uua) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Uua),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uu)) ) ).

% ATP.lambda_158
tff(fact_8336_ATP_Olambda__159,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ki(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(nat,nat,binomial(Uu),Uua)) ).

% ATP.lambda_159
tff(fact_8337_ATP_Olambda__160,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ox(set(A),fun(A,$o),Uu),Uua)
    <=> ( Uu = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_160
tff(fact_8338_ATP_Olambda__161,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_agz(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),Uu),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_161
tff(fact_8339_ATP_Olambda__162,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ags(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or1337092689740270186AtMost(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_162
tff(fact_8340_ATP_Olambda__163,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_us(fun(A,B),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(A,B,Uu,Uua)) ).

% ATP.lambda_163
tff(fact_8341_ATP_Olambda__164,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_rm(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Uua),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Uu),Uua))) ) ).

% ATP.lambda_164
tff(fact_8342_ATP_Olambda__165,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_agu(fun(nat,A),fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua))) ) ).

% ATP.lambda_165
tff(fact_8343_ATP_Olambda__166,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_ot(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),aa($o,int,zero_neq_one_of_bool(int),Uua != zero_zero(int))) ).

% ATP.lambda_166
tff(fact_8344_ATP_Olambda__167,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_agt(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua))))) ).

% ATP.lambda_167
tff(fact_8345_ATP_Olambda__168,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_agj(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))) ).

% ATP.lambda_168
tff(fact_8346_ATP_Olambda__169,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_agk(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_169
tff(fact_8347_ATP_Olambda__170,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_agv(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_170
tff(fact_8348_ATP_Olambda__171,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_agl(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_171
tff(fact_8349_ATP_Olambda__172,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_agw(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_172
tff(fact_8350_ATP_Olambda__173,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_afx(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_173
tff(fact_8351_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lj(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_174
tff(fact_8352_ATP_Olambda__175,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_qo(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_175
tff(fact_8353_ATP_Olambda__176,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ib(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_176
tff(fact_8354_ATP_Olambda__177,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_akj(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> eventually(A,Uu,topolo7230453075368039082e_nhds(A,Uua)) ) ) ).

% ATP.lambda_177
tff(fact_8355_ATP_Olambda__178,axiom,
    ! [Uu: int,Uua: nat] : aa(nat,int,aa(int,fun(nat,int),aTP_Lamp_vj(int,fun(nat,int)),Uu),Uua) = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Uua),Uu) ).

% ATP.lambda_178
tff(fact_8356_ATP_Olambda__179,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat,Uua: A] : aa(A,A,aTP_Lamp_aes(nat,fun(A,A),Uu),Uua) = comm_s3205402744901411588hammer(A,Uua,Uu) ) ).

% ATP.lambda_179
tff(fact_8357_ATP_Olambda__180,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: real] : aa(real,A,aTP_Lamp_ahp(A,fun(real,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),Uu) ) ).

% ATP.lambda_180
tff(fact_8358_ATP_Olambda__181,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_cz(set(A),fun(set(A),$o),Uu),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu) ) ).

% ATP.lambda_181
tff(fact_8359_ATP_Olambda__182,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_cx(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uu) ) ).

% ATP.lambda_182
tff(fact_8360_ATP_Olambda__183,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ajf(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_183
tff(fact_8361_ATP_Olambda__184,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ix(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_184
tff(fact_8362_ATP_Olambda__185,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_sv(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Uua),Uu) ).

% ATP.lambda_185
tff(fact_8363_ATP_Olambda__186,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ahr(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_186
tff(fact_8364_ATP_Olambda__187,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mg(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_187
tff(fact_8365_ATP_Olambda__188,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_tn(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_188
tff(fact_8366_ATP_Olambda__189,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_cy(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu) ) ).

% ATP.lambda_189
tff(fact_8367_ATP_Olambda__190,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ajh(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_190
tff(fact_8368_ATP_Olambda__191,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_rf(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_191
tff(fact_8369_ATP_Olambda__192,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_fu(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_192
tff(fact_8370_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aft(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu) ).

% ATP.lambda_193
tff(fact_8371_ATP_Olambda__194,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ahq(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_194
tff(fact_8372_ATP_Olambda__195,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_an(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_195
tff(fact_8373_ATP_Olambda__196,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mu(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_196
tff(fact_8374_ATP_Olambda__197,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ql(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).

% ATP.lambda_197
tff(fact_8375_ATP_Olambda__198,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mc(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_198
tff(fact_8376_ATP_Olambda__199,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_oe(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_199
tff(fact_8377_ATP_Olambda__200,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_wg(nat,fun(real,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu) ).

% ATP.lambda_200
tff(fact_8378_ATP_Olambda__201,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,code_integer,aTP_Lamp_qt(code_integer,fun(code_integer,code_integer),Uu),Uua) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),Uua),Uu) ).

% ATP.lambda_201
tff(fact_8379_ATP_Olambda__202,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_sj(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_202
tff(fact_8380_ATP_Olambda__203,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_qs(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_203
tff(fact_8381_ATP_Olambda__204,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ts(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_204
tff(fact_8382_ATP_Olambda__205,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mb(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_205
tff(fact_8383_ATP_Olambda__206,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: list(A),Uua: array(A)] : aa(array(A),assn,aTP_Lamp_rv(list(A),fun(array(A),assn),Uu),Uua) = aa(list(A),assn,snga_assn(A,Uua),Uu) ) ).

% ATP.lambda_206
tff(fact_8384_ATP_Olambda__207,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_xk(real,fun(real,real),Uu),Uua) = powr(real,Uua,Uu) ).

% ATP.lambda_207
tff(fact_8385_ATP_Olambda__208,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_cu(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,dvd_dvd(nat,Uua),Uu) ) ).

% ATP.lambda_208
tff(fact_8386_ATP_Olambda__209,axiom,
    ! [Uu: int,Uua: int] :
      ( aa(int,$o,aTP_Lamp_df(int,fun(int,$o),Uu),Uua)
    <=> aa(int,$o,dvd_dvd(int,Uua),Uu) ) ).

% ATP.lambda_209
tff(fact_8387_ATP_Olambda__210,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ch(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,dvd_dvd(A,Uua),Uu) ) ) ).

% ATP.lambda_210
tff(fact_8388_ATP_Olambda__211,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_uu(A,fun(B,product_prod(B,A))),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uu) ).

% ATP.lambda_211
tff(fact_8389_ATP_Olambda__212,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_iy(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_212
tff(fact_8390_ATP_Olambda__213,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_tc(list(A),fun(A,list(A))),Uu),Uua) = aa(list(A),list(A),cons(A,Uua),Uu) ).

% ATP.lambda_213
tff(fact_8391_ATP_Olambda__214,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_agi(real,fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),Uu) ).

% ATP.lambda_214
tff(fact_8392_ATP_Olambda__215,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_akh(set(A),fun(A,$o),Uu),Uua)
        <=> member(A,Uua,Uu) ) ) ).

% ATP.lambda_215
tff(fact_8393_ATP_Olambda__216,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_a(set(A),fun(A,$o),Uu),Uua)
    <=> member(A,Uua,Uu) ) ).

% ATP.lambda_216
tff(fact_8394_ATP_Olambda__217,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_aa(A,fun(A,$o),Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_217
tff(fact_8395_ATP_Olambda__218,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] :
      ( aa(A,$o,aTP_Lamp_dh(fun(A,nat),fun(A,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,Uu,Uua)) ) ).

% ATP.lambda_218
tff(fact_8396_ATP_Olambda__219,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_gs(fun(nat,real),fun(nat,real),Uu),Uua) = aa(nat,real,Uu,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_219
tff(fact_8397_ATP_Olambda__220,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_gr(fun(nat,real),fun(nat,real),Uu),Uua) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)) ).

% ATP.lambda_220
tff(fact_8398_ATP_Olambda__221,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(set(nat),A),Uua: nat] : aa(nat,A,aTP_Lamp_afu(fun(set(nat),A),fun(nat,A),Uu),Uua) = aa(set(nat),A,Uu,set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_221
tff(fact_8399_ATP_Olambda__222,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(set(nat),A),Uua: nat] : aa(nat,A,aTP_Lamp_afv(fun(set(nat),A),fun(nat,A),Uu),Uua) = aa(set(nat),A,Uu,set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_222
tff(fact_8400_ATP_Olambda__223,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real] : aa(real,A,aTP_Lamp_aja(fun(real,A),fun(real,A),Uu),Uua) = aa(real,A,Uu,aa(real,real,inverse_inverse(real),Uua)) ).

% ATP.lambda_223
tff(fact_8401_ATP_Olambda__224,axiom,
    ! [A: $tType,Uu: fun(int,A),Uua: nat] : aa(nat,A,aTP_Lamp_ahu(fun(int,A),fun(nat,A),Uu),Uua) = aa(int,A,Uu,aa(nat,int,semiring_1_of_nat(int),Uua)) ).

% ATP.lambda_224
tff(fact_8402_ATP_Olambda__225,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_wj(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,Uu,aa(real,real,uminus_uminus(real),Uua)) ).

% ATP.lambda_225
tff(fact_8403_ATP_Olambda__226,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real] : aa(real,A,aTP_Lamp_aix(fun(real,A),fun(real,A),Uu),Uua) = aa(real,A,Uu,aa(real,real,uminus_uminus(real),Uua)) ).

% ATP.lambda_226
tff(fact_8404_ATP_Olambda__227,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_tj(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_227
tff(fact_8405_ATP_Olambda__228,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bs(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_228
tff(fact_8406_ATP_Olambda__229,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_afq(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_229
tff(fact_8407_ATP_Olambda__230,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fy(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_230
tff(fact_8408_ATP_Olambda__231,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_hr(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_231
tff(fact_8409_ATP_Olambda__232,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_es(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_232
tff(fact_8410_ATP_Olambda__233,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_so(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_233
tff(fact_8411_ATP_Olambda__234,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_tg(A,fun(option(A),option(A))),Uu),Uua) = some(A,case_option(A,A,Uu,aa(A,fun(A,A),ord_min(A),Uu),Uua)) ) ).

% ATP.lambda_234
tff(fact_8412_ATP_Olambda__235,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_rd(A,fun(option(A),option(A))),Uu),Uua) = some(A,case_option(A,A,Uu,aa(A,fun(A,A),ord_max(A),Uu),Uua)) ) ).

% ATP.lambda_235
tff(fact_8413_ATP_Olambda__236,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_vr(nat,fun(num,option(num)),Uu),Uua) = some(num,case_option(num,num,one2,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_236
tff(fact_8414_ATP_Olambda__237,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_vv(num,fun(nat,option(num)),Uu),Uua) = some(num,case_option(num,num,one2,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_237
tff(fact_8415_ATP_Olambda__238,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,assn)),Uua: A] : aa(A,assn,aTP_Lamp_si(fun(B,fun(A,assn)),fun(A,assn),Uu),Uua) = ex_assn(B,aa(A,fun(B,assn),aTP_Lamp_sh(fun(B,fun(A,assn)),fun(A,fun(B,assn)),Uu),Uua)) ).

% ATP.lambda_238
tff(fact_8416_ATP_Olambda__239,axiom,
    ! [Uu: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_ya(fun(nat,real),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_xz(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua)) ).

% ATP.lambda_239
tff(fact_8417_ATP_Olambda__240,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_xi(fun(nat,A),fun(A,A),Uu),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua)) ) ).

% ATP.lambda_240
tff(fact_8418_ATP_Olambda__241,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_oj(nat,fun(nat,complex),Uu),Uua) = cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),aa(nat,real,semiring_1_of_nat(real),Uua))),aa(nat,real,semiring_1_of_nat(real),Uu))) ).

% ATP.lambda_241
tff(fact_8419_ATP_Olambda__242,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_zd(A,fun(A,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Uu)),Uua)),aa(A,A,inverse_inverse(A),Uu))) ) ).

% ATP.lambda_242
tff(fact_8420_ATP_Olambda__243,axiom,
    ! [Uu: fun(real,real),Uua: real] :
      ( aa(real,$o,aTP_Lamp_ajc(fun(real,real),fun(real,$o),Uu),Uua)
    <=> ( aa(real,real,Uu,Uua) != zero_zero(real) ) ) ).

% ATP.lambda_243
tff(fact_8421_ATP_Olambda__244,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_lr(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_244
tff(fact_8422_ATP_Olambda__245,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_op(nat,fun(nat,$o),Uu),Uua)
    <=> ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ) ).

% ATP.lambda_245
tff(fact_8423_ATP_Olambda__246,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ld(A,fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,sin_coeff,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uu)),Uua))) ) ).

% ATP.lambda_246
tff(fact_8424_ATP_Olambda__247,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_la(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,sin_coeff,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_247
tff(fact_8425_ATP_Olambda__248,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_lb(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,cos_coeff,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_248
tff(fact_8426_ATP_Olambda__249,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_lm(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_249
tff(fact_8427_ATP_Olambda__250,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_gw(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_250
tff(fact_8428_ATP_Olambda__251,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_agr(real,fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_251
tff(fact_8429_ATP_Olambda__252,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_wi(A,fun(A,A),Uu),Uua) = aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)) ) ).

% ATP.lambda_252
tff(fact_8430_ATP_Olambda__253,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Uu: A,Uua: A] : aa(A,option(A),aTP_Lamp_nz(A,fun(A,option(A)),Uu),Uua) = some(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu),Uua)) ) ).

% ATP.lambda_253
tff(fact_8431_ATP_Olambda__254,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Uu: A,Uua: A] : aa(A,option(A),aTP_Lamp_nx(A,fun(A,option(A)),Uu),Uua) = some(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),Uua)) ) ).

% ATP.lambda_254
tff(fact_8432_ATP_Olambda__255,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_wa(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Uu),Uua)) ).

% ATP.lambda_255
tff(fact_8433_ATP_Olambda__256,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_wb(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Uua),Uu)) ).

% ATP.lambda_256
tff(fact_8434_ATP_Olambda__257,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_vy(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uu),Uua)) ).

% ATP.lambda_257
tff(fact_8435_ATP_Olambda__258,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_vx(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uua),Uu)) ).

% ATP.lambda_258
tff(fact_8436_ATP_Olambda__259,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ba(set(A),fun(A,$o),Uu),Uua)
    <=> ~ member(A,Uua,Uu) ) ).

% ATP.lambda_259
tff(fact_8437_ATP_Olambda__260,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_akk(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_260
tff(fact_8438_ATP_Olambda__261,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_akg(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_261
tff(fact_8439_ATP_Olambda__262,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ajg(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_262
tff(fact_8440_ATP_Olambda__263,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_ci(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_263
tff(fact_8441_ATP_Olambda__264,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_jo(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_264
tff(fact_8442_ATP_Olambda__265,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_cg(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_265
tff(fact_8443_ATP_Olambda__266,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_eb(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_266
tff(fact_8444_ATP_Olambda__267,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_hn(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_267
tff(fact_8445_ATP_Olambda__268,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_aay(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_268
tff(fact_8446_ATP_Olambda__269,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_ho(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_269
tff(fact_8447_ATP_Olambda__270,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_acp(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_270
tff(fact_8448_ATP_Olambda__271,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_agf(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_271
tff(fact_8449_ATP_Olambda__272,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ze(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_272
tff(fact_8450_ATP_Olambda__273,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_xc(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_273
tff(fact_8451_ATP_Olambda__274,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_abo(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_274
tff(fact_8452_ATP_Olambda__275,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aht(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_275
tff(fact_8453_ATP_Olambda__276,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ack(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_276
tff(fact_8454_ATP_Olambda__277,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_gy(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_277
tff(fact_8455_ATP_Olambda__278,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_dq(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_278
tff(fact_8456_ATP_Olambda__279,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,real,aTP_Lamp_nl(fun(A,nat),fun(A,real),Uu),Uua) = aa(nat,real,semiring_1_of_nat(real),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_279
tff(fact_8457_ATP_Olambda__280,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,int,aTP_Lamp_fr(fun(A,nat),fun(A,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_280
tff(fact_8458_ATP_Olambda__281,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zm(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_281
tff(fact_8459_ATP_Olambda__282,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_afs(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_282
tff(fact_8460_ATP_Olambda__283,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_im(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_283
tff(fact_8461_ATP_Olambda__284,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,complex,aTP_Lamp_bl(fun(nat,real),fun(nat,complex),Uu),Uua) = aa(real,complex,real_Vector_of_real(complex),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_284
tff(fact_8462_ATP_Olambda__285,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_ft(fun(nat,real),fun(nat,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_285
tff(fact_8463_ATP_Olambda__286,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_bv(fun(nat,real),fun(nat,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_286
tff(fact_8464_ATP_Olambda__287,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_ii(fun(nat,real),fun(nat,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_287
tff(fact_8465_ATP_Olambda__288,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_dt(fun(B,real),fun(B,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(B,real,Uu,Uua)) ) ).

% ATP.lambda_288
tff(fact_8466_ATP_Olambda__289,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2191834092415804123ebra_1(A) )
     => ! [Uu: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_ha(fun(B,real),fun(B,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(B,real,Uu,Uua)) ) ).

% ATP.lambda_289
tff(fact_8467_ATP_Olambda__290,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_yv(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_290
tff(fact_8468_ATP_Olambda__291,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_aaz(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_291
tff(fact_8469_ATP_Olambda__292,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_acq(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_292
tff(fact_8470_ATP_Olambda__293,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_acr(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_293
tff(fact_8471_ATP_Olambda__294,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bt(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_294
tff(fact_8472_ATP_Olambda__295,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_pj(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_295
tff(fact_8473_ATP_Olambda__296,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ij(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_296
tff(fact_8474_ATP_Olambda__297,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ej(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_297
tff(fact_8475_ATP_Olambda__298,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_yq(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_298
tff(fact_8476_ATP_Olambda__299,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_wt(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_299
tff(fact_8477_ATP_Olambda__300,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_abe(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_300
tff(fact_8478_ATP_Olambda__301,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_add(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_301
tff(fact_8479_ATP_Olambda__302,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aiw(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,uminus_uminus(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_302
tff(fact_8480_ATP_Olambda__303,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_adc(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_303
tff(fact_8481_ATP_Olambda__304,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_hp(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_304
tff(fact_8482_ATP_Olambda__305,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_gz(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_305
tff(fact_8483_ATP_Olambda__306,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_dr(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_306
tff(fact_8484_ATP_Olambda__307,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ie(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_307
tff(fact_8485_ATP_Olambda__308,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ain(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_308
tff(fact_8486_ATP_Olambda__309,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_adm(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_309
tff(fact_8487_ATP_Olambda__310,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_afl(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arsinh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_310
tff(fact_8488_ATP_Olambda__311,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_acs(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arsinh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_311
tff(fact_8489_ATP_Olambda__312,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zy(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_312
tff(fact_8490_ATP_Olambda__313,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_afk(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_313
tff(fact_8491_ATP_Olambda__314,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_acu(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ).

% ATP.lambda_314
tff(fact_8492_ATP_Olambda__315,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_yb(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_315
tff(fact_8493_ATP_Olambda__316,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aim(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_316
tff(fact_8494_ATP_Olambda__317,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_adj(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_317
tff(fact_8495_ATP_Olambda__318,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aae(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_318
tff(fact_8496_ATP_Olambda__319,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_abu(fun(A,B),fun(A,B),Uu),Uua) = sgn_sgn(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_319
tff(fact_8497_ATP_Olambda__320,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_adl(fun(A,B),fun(A,B),Uu),Uua) = sgn_sgn(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_320
tff(fact_8498_ATP_Olambda__321,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_bu(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_321
tff(fact_8499_ATP_Olambda__322,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ds(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_322
tff(fact_8500_ATP_Olambda__323,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_hd(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_323
tff(fact_8501_ATP_Olambda__324,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_afn(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_324
tff(fact_8502_ATP_Olambda__325,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ade(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_325
tff(fact_8503_ATP_Olambda__326,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aew(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_326
tff(fact_8504_ATP_Olambda__327,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_xo(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tanh(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_327
tff(fact_8505_ATP_Olambda__328,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_adh(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_328
tff(fact_8506_ATP_Olambda__329,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_acw(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sinh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_329
tff(fact_8507_ATP_Olambda__330,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_wy(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,sinh(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_330
tff(fact_8508_ATP_Olambda__331,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_acx(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sinh(B),aa(A,B,Uu,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) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_acm(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,cosh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_332
tff(fact_8510_ATP_Olambda__333,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_wz(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cosh(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_333
tff(fact_8511_ATP_Olambda__334,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_acn(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,cosh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_334
tff(fact_8512_ATP_Olambda__335,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aaa(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,tan(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_335
tff(fact_8513_ATP_Olambda__336,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_adg(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tan(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_336
tff(fact_8514_ATP_Olambda__337,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_yz(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sin(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_337
tff(fact_8515_ATP_Olambda__338,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_abw(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sin(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_338
tff(fact_8516_ATP_Olambda__339,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_wm(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,sin(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_339
tff(fact_8517_ATP_Olambda__340,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_yx(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,exp(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_340
tff(fact_8518_ATP_Olambda__341,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_abx(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,exp(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_341
tff(fact_8519_ATP_Olambda__342,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_wk(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,exp(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_342
tff(fact_8520_ATP_Olambda__343,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_acl(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,exp(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_343
tff(fact_8521_ATP_Olambda__344,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_iu(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,exp(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_344
tff(fact_8522_ATP_Olambda__345,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_adi(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cot(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_345
tff(fact_8523_ATP_Olambda__346,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ahf(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,cos(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_346
tff(fact_8524_ATP_Olambda__347,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zi(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,cos(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_347
tff(fact_8525_ATP_Olambda__348,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_abv(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,cos(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_348
tff(fact_8526_ATP_Olambda__349,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_wl(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cos(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_349
tff(fact_8527_ATP_Olambda__350,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),Uua: A] : aa(A,assn,aTP_Lamp_sg(fun(A,fun(B,assn)),fun(A,assn),Uu),Uua) = ex_assn(B,aa(A,fun(B,assn),Uu,Uua)) ).

% ATP.lambda_350
tff(fact_8528_ATP_Olambda__351,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: real] : aa(real,real,aTP_Lamp_aad(fun(real,fun(nat,real)),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),Uu,Uua)) ).

% ATP.lambda_351
tff(fact_8529_ATP_Olambda__352,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_aex(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_352
tff(fact_8530_ATP_Olambda__353,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_eu(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_353
tff(fact_8531_ATP_Olambda__354,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zw(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_354
tff(fact_8532_ATP_Olambda__355,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_afj(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_355
tff(fact_8533_ATP_Olambda__356,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_acv(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ).

% ATP.lambda_356
tff(fact_8534_ATP_Olambda__357,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_rc(fun(A,B),fun(A,fun(set(B),set(B))),Uu),Uua) = aa(B,fun(set(B),set(B)),insert(B),aa(A,B,Uu,Uua)) ).

% ATP.lambda_357
tff(fact_8535_ATP_Olambda__358,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: B] :
      ( aa(B,$o,aTP_Lamp_me(fun(B,$o),fun(B,$o),Uu),Uua)
    <=> ~ aa(B,$o,Uu,Uua) ) ).

% ATP.lambda_358
tff(fact_8536_ATP_Olambda__359,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aaq(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ~ aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_359
tff(fact_8537_ATP_Olambda__360,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_az(fun(A,$o),fun(A,$o),Uu),Uua)
    <=> ~ aa(A,$o,Uu,Uua) ) ).

% ATP.lambda_360
tff(fact_8538_ATP_Olambda__361,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu: fun(real,A),Uua: real] : aa(real,real,aTP_Lamp_afa(fun(real,A),fun(real,real),Uu),Uua) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(A,aa(real,A,Uu,Uua))) ) ).

% ATP.lambda_361
tff(fact_8539_ATP_Olambda__362,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B)
        & ring_1(C)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,C,aTP_Lamp_aci(fun(A,B),fun(A,C),Uu),Uua) = aa(int,C,ring_1_of_int(C),archim6421214686448440834_floor(B,aa(A,B,Uu,Uua))) ) ).

% ATP.lambda_362
tff(fact_8540_ATP_Olambda__363,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B)
        & ring_1(C)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,C,aTP_Lamp_acj(fun(A,B),fun(A,C),Uu),Uua) = aa(int,C,ring_1_of_int(C),archimedean_ceiling(B,aa(A,B,Uu,Uua))) ) ).

% ATP.lambda_363
tff(fact_8541_ATP_Olambda__364,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_gp(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(nat,real,Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_364
tff(fact_8542_ATP_Olambda__365,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_gq(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(nat,real,Uu,Uub),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_365
tff(fact_8543_ATP_Olambda__366,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_nu(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),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),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_366
tff(fact_8544_ATP_Olambda__367,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_nq(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_367
tff(fact_8545_ATP_Olambda__368,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_np(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uub),aa(num,int,numeral_numeral(int),Uu))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_368
tff(fact_8546_ATP_Olambda__369,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: num,Uua: A,Uub: A] :
          aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_ns(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),aa(num,A,numeral_numeral(A),Uu))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),Uub)) ) ).

% ATP.lambda_369
tff(fact_8547_ATP_Olambda__370,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: set(nat),Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cq(set(nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(member(nat,Uub,Uu),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_370
tff(fact_8548_ATP_Olambda__371,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_ev(fun(A,B),fun(set(A),fun(A,B)),Uu),Uua),Uub) = $ite(member(A,Uub,Uua),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_371
tff(fact_8549_ATP_Olambda__372,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_hu(fun(A,B),fun(set(A),fun(A,B)),Uu),Uua),Uub) = $ite(member(A,Uub,Uua),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_372
tff(fact_8550_ATP_Olambda__373,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_du(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_373
tff(fact_8551_ATP_Olambda__374,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_hc(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_374
tff(fact_8552_ATP_Olambda__375,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat,Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bm(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_375
tff(fact_8553_ATP_Olambda__376,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_dv(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_376
tff(fact_8554_ATP_Olambda__377,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_hb(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_377
tff(fact_8555_ATP_Olambda__378,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: set(A)] :
      aa(set(A),set(A),aa(A,fun(set(A),set(A)),aTP_Lamp_rp(fun(A,$o),fun(A,fun(set(A),set(A))),Uu),Uua),Uub) = $ite(aa(A,$o,Uu,Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),Uub),Uub) ).

% ATP.lambda_378
tff(fact_8556_ATP_Olambda__379,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,$o),Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cp(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(aa(nat,$o,Uu,Uub),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_379
tff(fact_8557_ATP_Olambda__380,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_er(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_380
tff(fact_8558_ATP_Olambda__381,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_hq(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_381
tff(fact_8559_ATP_Olambda__382,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] :
      ( aa(real,$o,aa(fun(real,real),fun(real,$o),aTP_Lamp_ajd(fun(real,real),fun(fun(real,real),fun(real,$o)),Uu),Uua),Uub)
    <=> has_field_derivative(real,Uu,aa(real,real,Uua,Uub),topolo174197925503356063within(real,Uub,top_top(set(real)))) ) ).

% ATP.lambda_382
tff(fact_8560_ATP_Olambda__383,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Uu: option(A),Uua: option(A),Uub: A] : aa(A,option(A),aa(option(A),fun(A,option(A)),aTP_Lamp_oa(option(A),fun(option(A),fun(A,option(A))),Uu),Uua),Uub) = case_option(option(A),A,Uu,aTP_Lamp_nz(A,fun(A,option(A)),Uub),Uua) ) ).

% ATP.lambda_383
tff(fact_8561_ATP_Olambda__384,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: filter(B),Uua: fun(B,A),Uub: A] :
          ( aa(A,$o,aa(fun(B,A),fun(A,$o),aTP_Lamp_aij(filter(B),fun(fun(B,A),fun(A,$o)),Uu),Uua),Uub)
        <=> filterlim(B,A,Uua,topolo7230453075368039082e_nhds(A,Uub),Uu) ) ) ).

% ATP.lambda_384
tff(fact_8562_ATP_Olambda__385,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ng(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_385
tff(fact_8563_ATP_Olambda__386,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ne(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_386
tff(fact_8564_ATP_Olambda__387,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_aac(fun(real,fun(nat,real)),fun(nat,fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),Uu,Uub),Uua) ).

% ATP.lambda_387
tff(fact_8565_ATP_Olambda__388,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jf(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ) ).

% ATP.lambda_388
tff(fact_8566_ATP_Olambda__389,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jc(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ) ).

% ATP.lambda_389
tff(fact_8567_ATP_Olambda__390,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(B,fun(A,B)),Uua: A,Uub: B] : aa(B,B,aa(A,fun(B,B),aTP_Lamp_xf(fun(B,fun(A,B)),fun(A,fun(B,B)),Uu),Uua),Uub) = aa(A,B,aa(B,fun(A,B),Uu,Uub),Uua) ) ).

% ATP.lambda_390
tff(fact_8568_ATP_Olambda__391,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_hg(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_391
tff(fact_8569_ATP_Olambda__392,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_dz(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_392
tff(fact_8570_ATP_Olambda__393,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,assn)),Uua: A,Uub: B] : aa(B,assn,aa(A,fun(B,assn),aTP_Lamp_sh(fun(B,fun(A,assn)),fun(A,fun(B,assn)),Uu),Uua),Uub) = aa(A,assn,aa(B,fun(A,assn),Uu,Uub),Uua) ).

% ATP.lambda_393
tff(fact_8571_ATP_Olambda__394,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_aee(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu,Uub),Uua) ) ).

% ATP.lambda_394
tff(fact_8572_ATP_Olambda__395,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_aeg(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu,Uub),Uua) ) ).

% ATP.lambda_395
tff(fact_8573_ATP_Olambda__396,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_ep(fun(A,fun(nat,B)),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(A,fun(nat,B),Uu,Uub),Uua) ) ).

% ATP.lambda_396
tff(fact_8574_ATP_Olambda__397,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_yl(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_397
tff(fact_8575_ATP_Olambda__398,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_zq(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_398
tff(fact_8576_ATP_Olambda__399,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_aeo(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_399
tff(fact_8577_ATP_Olambda__400,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_aeq(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_400
tff(fact_8578_ATP_Olambda__401,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_abr(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_401
tff(fact_8579_ATP_Olambda__402,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,assn)),Uua: B,Uub: A] : aa(A,assn,aa(B,fun(A,assn),aTP_Lamp_sf(fun(A,fun(B,assn)),fun(B,fun(A,assn)),Uu),Uua),Uub) = aa(B,assn,aa(A,fun(B,assn),Uu,Uub),Uua) ).

% ATP.lambda_402
tff(fact_8580_ATP_Olambda__403,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_aek(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_403
tff(fact_8581_ATP_Olambda__404,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_aem(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_404
tff(fact_8582_ATP_Olambda__405,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_aei(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_405
tff(fact_8583_ATP_Olambda__406,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_hl(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_406
tff(fact_8584_ATP_Olambda__407,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_em(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_407
tff(fact_8585_ATP_Olambda__408,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ks(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_kr(A,fun(A,fun(nat,fun(nat,A))),Uu),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) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_kq(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_kp(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_409
tff(fact_8587_ATP_Olambda__410,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ko(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_kn(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_410
tff(fact_8588_ATP_Olambda__411,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_jq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_jp(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_411
tff(fact_8589_ATP_Olambda__412,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat,Uua: A,Uub: nat] :
          aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gd(nat,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(Uub = Uu,one_one(A),zero_zero(A))),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_412
tff(fact_8590_ATP_Olambda__413,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ro(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,aa(A,fun(A,$o),Uu,Uua),Uub)
        | ( Uua = Uub ) ) ) ).

% ATP.lambda_413
tff(fact_8591_ATP_Olambda__414,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jg(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jf(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_414
tff(fact_8592_ATP_Olambda__415,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jd(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jc(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_415
tff(fact_8593_ATP_Olambda__416,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_aez(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_bz(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_416
tff(fact_8594_ATP_Olambda__417,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_qd(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_qc(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_417
tff(fact_8595_ATP_Olambda__418,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_qb(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_qa(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_418
tff(fact_8596_ATP_Olambda__419,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_pz(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_py(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_419
tff(fact_8597_ATP_Olambda__420,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_px(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_pw(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_420
tff(fact_8598_ATP_Olambda__421,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_pu(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aa(int,fun(int,fun(int,$o)),aTP_Lamp_pt(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_421
tff(fact_8599_ATP_Olambda__422,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_pr(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aa(int,fun(int,fun(int,$o)),aTP_Lamp_pq(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_422
tff(fact_8600_ATP_Olambda__423,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4987421752381908075d_mult(C)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aer(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_aeq(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_423
tff(fact_8601_ATP_Olambda__424,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aep(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_aeo(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_424
tff(fact_8602_ATP_Olambda__425,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aen(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_aem(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_425
tff(fact_8603_ATP_Olambda__426,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ael(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_aek(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_426
tff(fact_8604_ATP_Olambda__427,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_zr(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_zq(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_427
tff(fact_8605_ATP_Olambda__428,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_hh(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(B,A,aa(C,fun(B,A),aTP_Lamp_hg(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub),Uua) ) ).

% ATP.lambda_428
tff(fact_8606_ATP_Olambda__429,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aej(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_aei(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_429
tff(fact_8607_ATP_Olambda__430,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(C)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_abs(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_abr(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_430
tff(fact_8608_ATP_Olambda__431,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ym(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_yl(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_431
tff(fact_8609_ATP_Olambda__432,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(nat,B)),Uub: nat] : aa(nat,B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_eq(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),Uu),Uua),Uub) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(nat,fun(A,B),aTP_Lamp_ep(fun(A,fun(nat,B)),fun(nat,fun(A,B)),Uua),Uub)),Uu) ) ).

% ATP.lambda_432
tff(fact_8610_ATP_Olambda__433,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_ea(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(C,fun(B,A),aTP_Lamp_dz(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua) ) ).

% ATP.lambda_433
tff(fact_8611_ATP_Olambda__434,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_xp(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,Uub),zero_zero(real))),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_434
tff(fact_8612_ATP_Olambda__435,axiom,
    ! [Uu: real,Uua: fun(nat,fun(real,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_xq(real,fun(fun(nat,fun(real,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uua,Uub),zero_zero(real))),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_435
tff(fact_8613_ATP_Olambda__436,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Uu: real,Uua: fun(nat,fun(A,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_io(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(A,real,aa(nat,fun(A,real),Uua,Uub),zero_zero(A))),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ) ).

% ATP.lambda_436
tff(fact_8614_ATP_Olambda__437,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_jm(fun(nat,A),fun(nat,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_iz(fun(nat,A),fun(A,fun(nat,A)),Uu),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)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aet(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu,Uua))),Uub) ) ).

% ATP.lambda_438
tff(fact_8616_ATP_Olambda__439,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aat(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu,Uua))),Uub) ) ).

% ATP.lambda_439
tff(fact_8617_ATP_Olambda__440,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dx(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uub))),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_440
tff(fact_8618_ATP_Olambda__441,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_abn(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),Uua)) ) ).

% ATP.lambda_441
tff(fact_8619_ATP_Olambda__442,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aau(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),Uua)) ) ).

% ATP.lambda_442
tff(fact_8620_ATP_Olambda__443,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_xy(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uub)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_443
tff(fact_8621_ATP_Olambda__444,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ip(real,fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,Uua,Uub)),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_444
tff(fact_8622_ATP_Olambda__445,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ih(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),aa(nat,A,Uu,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% ATP.lambda_445
tff(fact_8623_ATP_Olambda__446,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_ta(nat,fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set2(A,Uub)),set2(A,Uua)) ) ) ).

% ATP.lambda_446
tff(fact_8624_ATP_Olambda__447,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_lz(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set2(A,Uub)),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua) ) ) ).

% ATP.lambda_447
tff(fact_8625_ATP_Olambda__448,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_ly(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set2(A,Uub)),Uu)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_448
tff(fact_8626_ATP_Olambda__449,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_xl(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,diffs(A,Uu)),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_449
tff(fact_8627_ATP_Olambda__450,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_qn(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_450
tff(fact_8628_ATP_Olambda__451,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ph(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) = lattic643756798350308766er_Min(B,aa(set(A),set(B),image(A,B,Uua),Uu)) ) ) ) ) ).

% ATP.lambda_451
tff(fact_8629_ATP_Olambda__452,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ig(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_452
tff(fact_8630_ATP_Olambda__453,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_il(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_453
tff(fact_8631_ATP_Olambda__454,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_in(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uua),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uub)) ) ).

% ATP.lambda_454
tff(fact_8632_ATP_Olambda__455,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_jk(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub)) ).

% ATP.lambda_455
tff(fact_8633_ATP_Olambda__456,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_af(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        | member(A,Uub,Uua) ) ) ).

% ATP.lambda_456
tff(fact_8634_ATP_Olambda__457,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ad(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member(A,Uub,Uua) ) ) ).

% ATP.lambda_457
tff(fact_8635_ATP_Olambda__458,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ok(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uua)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_458
tff(fact_8636_ATP_Olambda__459,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: word(A),Uua: word(A),Uub: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),aTP_Lamp_ak(word(A),fun(word(A),fun(word(A),$o)),Uu),Uua),Uub)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Uu),Uub)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Uub),Uua) ) ) ) ).

% ATP.lambda_459
tff(fact_8637_ATP_Olambda__460,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_da(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_460
tff(fact_8638_ATP_Olambda__461,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ol(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_461
tff(fact_8639_ATP_Olambda__462,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_dd(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_462
tff(fact_8640_ATP_Olambda__463,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_de(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_463
tff(fact_8641_ATP_Olambda__464,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_db(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_464
tff(fact_8642_ATP_Olambda__465,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_bc(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,vEBT_vebt_member(Uu),Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uub) ) ) ).

% ATP.lambda_465
tff(fact_8643_ATP_Olambda__466,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_bb(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,vEBT_vebt_member(Uu),Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_466
tff(fact_8644_ATP_Olambda__467,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ap(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & member(A,Uub,Uua) ) ) ).

% ATP.lambda_467
tff(fact_8645_ATP_Olambda__468,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_cr(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uub),Uu) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uua),Uu) ) ) ).

% ATP.lambda_468
tff(fact_8646_ATP_Olambda__469,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_al(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_469
tff(fact_8647_ATP_Olambda__470,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_rq(fun(A,$o),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uua)
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_470
tff(fact_8648_ATP_Olambda__471,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ag(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_471
tff(fact_8649_ATP_Olambda__472,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ah(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_472
tff(fact_8650_ATP_Olambda__473,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_dn(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_473
tff(fact_8651_ATP_Olambda__474,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_rs(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_474
tff(fact_8652_ATP_Olambda__475,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_dl(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_475
tff(fact_8653_ATP_Olambda__476,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_rt(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_476
tff(fact_8654_ATP_Olambda__477,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: $o] : aa($o,assn,aa(vEBT_VEBTi,fun($o,assn),aTP_Lamp_ar(vEBT_VEBT,fun(vEBT_VEBTi,fun($o,assn)),Uu),Uua),(Uub)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu),Uua)),pure_assn((Uub) = vEBT_VEBT_minNull(Uu))) ).

% ATP.lambda_477
tff(fact_8655_ATP_Olambda__478,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option(nat)] : aa(option(nat),assn,aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_be(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu),Uua)),pure_assn(Uub = vEBT_vebt_mint(Uu))) ).

% ATP.lambda_478
tff(fact_8656_ATP_Olambda__479,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option(nat)] : aa(option(nat),assn,aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_iv(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu),Uua)),pure_assn(Uub = vEBT_vebt_maxt(Uu))) ).

% ATP.lambda_479
tff(fact_8657_ATP_Olambda__480,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: list(A),Uub: nat] : aa(nat,assn,aa(list(A),fun(nat,assn),aTP_Lamp_qx(array(A),fun(list(A),fun(nat,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(A),assn,snga_assn(A,Uu),Uua)),pure_assn(Uub = aa(list(A),nat,size_size(list(A)),Uua))) ) ).

% ATP.lambda_480
tff(fact_8658_ATP_Olambda__481,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: list(A),Uub: list(A)] : aa(list(A),assn,aa(list(A),fun(list(A),assn),aTP_Lamp_qw(array(A),fun(list(A),fun(list(A),assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(A),assn,snga_assn(A,Uu),Uua)),pure_assn(Uub = Uua)) ) ).

% ATP.lambda_481
tff(fact_8659_ATP_Olambda__482,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_am(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & ~ member(A,Uub,Uua) ) ) ).

% ATP.lambda_482
tff(fact_8660_ATP_Olambda__483,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ja(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uu) ) ).

% ATP.lambda_483
tff(fact_8661_ATP_Olambda__484,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_nd(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_484
tff(fact_8662_ATP_Olambda__485,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_aip(A,fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_485
tff(fact_8663_ATP_Olambda__486,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ni(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_486
tff(fact_8664_ATP_Olambda__487,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_my(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_487
tff(fact_8665_ATP_Olambda__488,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mw(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_488
tff(fact_8666_ATP_Olambda__489,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mx(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_489
tff(fact_8667_ATP_Olambda__490,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mv(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_490
tff(fact_8668_ATP_Olambda__491,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_pi(set(product_prod(B,A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uub),Uu) ) ).

% ATP.lambda_491
tff(fact_8669_ATP_Olambda__492,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_lv(set(product_prod(A,B)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub),Uu) ) ).

% ATP.lambda_492
tff(fact_8670_ATP_Olambda__493,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_dc(nat,fun(complex,fun(complex,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uu) = Uua ) ) ).

% ATP.lambda_493
tff(fact_8671_ATP_Olambda__494,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa(complex,$o,aa(nat,fun(complex,$o),aTP_Lamp_oq(complex,fun(nat,fun(complex,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uua) = Uu ) ) ).

% ATP.lambda_494
tff(fact_8672_ATP_Olambda__495,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_pk(A,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu),Uub)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uub),Uua) ) ) ) ).

% ATP.lambda_495
tff(fact_8673_ATP_Olambda__496,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_by(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_496
tff(fact_8674_ATP_Olambda__497,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cb(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_497
tff(fact_8675_ATP_Olambda__498,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,int),Uub: nat] : aa(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_ahe(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu,Uub)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),aa(nat,int,Uua,Uub))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) ).

% ATP.lambda_498
tff(fact_8676_ATP_Olambda__499,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_xz(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),aa(nat,nat,suc,Uub))) ).

% ATP.lambda_499
tff(fact_8677_ATP_Olambda__500,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_bi(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_500
tff(fact_8678_ATP_Olambda__501,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_jv(fun(nat,nat),fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uub)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uua),Uub)) ).

% ATP.lambda_501
tff(fact_8679_ATP_Olambda__502,axiom,
    ! [B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(nat,B),Uua: B,Uub: nat] : aa(nat,B,aa(B,fun(nat,B),aTP_Lamp_afb(fun(nat,B),fun(B,fun(nat,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uu,Uub)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Uua),Uub)) ) ).

% ATP.lambda_502
tff(fact_8680_ATP_Olambda__503,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_iz(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_503
tff(fact_8681_ATP_Olambda__504,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bz(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_504
tff(fact_8682_ATP_Olambda__505,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ca(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_505
tff(fact_8683_ATP_Olambda__506,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_506
tff(fact_8684_ATP_Olambda__507,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_jj(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_507
tff(fact_8685_ATP_Olambda__508,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_508
tff(fact_8686_ATP_Olambda__509,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_cs(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,Uu,Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_509
tff(fact_8687_ATP_Olambda__510,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yf(fun(A,real),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_510
tff(fact_8688_ATP_Olambda__511,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aba(fun(A,real),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_511
tff(fact_8689_ATP_Olambda__512,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acy(fun(A,real),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_512
tff(fact_8690_ATP_Olambda__513,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_aiq(fun(A,B),fun(fun(A,B),fun(A,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_513
tff(fact_8691_ATP_Olambda__514,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_air(fun(A,B),fun(fun(A,B),fun(A,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_514
tff(fact_8692_ATP_Olambda__515,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] :
      ( aa(A,$o,aa(fun(A,real),fun(A,$o),aTP_Lamp_ajm(fun(A,real),fun(fun(A,real),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_515
tff(fact_8693_ATP_Olambda__516,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajn(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_516
tff(fact_8694_ATP_Olambda__517,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajk(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_517
tff(fact_8695_ATP_Olambda__518,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_he(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_518
tff(fact_8696_ATP_Olambda__519,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zo(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_519
tff(fact_8697_ATP_Olambda__520,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zb(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_520
tff(fact_8698_ATP_Olambda__521,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_wp(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_521
tff(fact_8699_ATP_Olambda__522,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_abq(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_522
tff(fact_8700_ATP_Olambda__523,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahs(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_523
tff(fact_8701_ATP_Olambda__524,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_524
tff(fact_8702_ATP_Olambda__525,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aje(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,Uua,Uub)),aa(real,real,Uu,Uub)) ).

% ATP.lambda_525
tff(fact_8703_ATP_Olambda__526,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_qr(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uub)),aa(nat,nat,Uua,Uub)) ).

% ATP.lambda_526
tff(fact_8704_ATP_Olambda__527,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_527
tff(fact_8705_ATP_Olambda__528,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_hi(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_528
tff(fact_8706_ATP_Olambda__529,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yh(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_529
tff(fact_8707_ATP_Olambda__530,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_xa(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_530
tff(fact_8708_ATP_Olambda__531,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_abm(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_531
tff(fact_8709_ATP_Olambda__532,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adt(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_532
tff(fact_8710_ATP_Olambda__533,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahn(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_533
tff(fact_8711_ATP_Olambda__534,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aeb(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_534
tff(fact_8712_ATP_Olambda__535,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ady(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_535
tff(fact_8713_ATP_Olambda__536,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ads(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_536
tff(fact_8714_ATP_Olambda__537,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahv(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),aa(A,real,Uu,Uub)) ).

% ATP.lambda_537
tff(fact_8715_ATP_Olambda__538,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ff(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_538
tff(fact_8716_ATP_Olambda__539,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_age(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu,Uub)),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_539
tff(fact_8717_ATP_Olambda__540,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_540
tff(fact_8718_ATP_Olambda__541,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_eh(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_541
tff(fact_8719_ATP_Olambda__542,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yw(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_542
tff(fact_8720_ATP_Olambda__543,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_wq(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_543
tff(fact_8721_ATP_Olambda__544,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aax(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_544
tff(fact_8722_ATP_Olambda__545,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acg(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_545
tff(fact_8723_ATP_Olambda__546,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acd(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_546
tff(fact_8724_ATP_Olambda__547,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ach(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_547
tff(fact_8725_ATP_Olambda__548,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ge(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uua,Uub)),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_548
tff(fact_8726_ATP_Olambda__549,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_fs(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_549
tff(fact_8727_ATP_Olambda__550,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_fx(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_550
tff(fact_8728_ATP_Olambda__551,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acf(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_551
tff(fact_8729_ATP_Olambda__552,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_acc(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_552
tff(fact_8730_ATP_Olambda__553,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_acb(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_553
tff(fact_8731_ATP_Olambda__554,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aed(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_554
tff(fact_8732_ATP_Olambda__555,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aec(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_555
tff(fact_8733_ATP_Olambda__556,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adn(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_556
tff(fact_8734_ATP_Olambda__557,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ado(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_557
tff(fact_8735_ATP_Olambda__558,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,complex,aa(fun(A,real),fun(A,complex),aTP_Lamp_aco(fun(A,real),fun(fun(A,real),fun(A,complex)),Uu),Uua),Uub) = complex2(aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_558
tff(fact_8736_ATP_Olambda__559,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_sl(fun(A,assn),fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(A,assn,Uu,Uub)),aa(A,assn,Uua,Uub)) ).

% ATP.lambda_559
tff(fact_8737_ATP_Olambda__560,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bo(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_560
tff(fact_8738_ATP_Olambda__561,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ec(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_561
tff(fact_8739_ATP_Olambda__562,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_562
tff(fact_8740_ATP_Olambda__563,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_ws(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_563
tff(fact_8741_ATP_Olambda__564,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_abd(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_564
tff(fact_8742_ATP_Olambda__565,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahm(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_565
tff(fact_8743_ATP_Olambda__566,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ada(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_566
tff(fact_8744_ATP_Olambda__567,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ru(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_567
tff(fact_8745_ATP_Olambda__568,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_xn(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),aa(real,real,Uua,Uub)) ).

% ATP.lambda_568
tff(fact_8746_ATP_Olambda__569,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zu(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_569
tff(fact_8747_ATP_Olambda__570,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_afr(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_570
tff(fact_8748_ATP_Olambda__571,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adf(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_571
tff(fact_8749_ATP_Olambda__572,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_agm(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_572
tff(fact_8750_ATP_Olambda__573,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adk(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_573
tff(fact_8751_ATP_Olambda__574,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_aap(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_574
tff(fact_8752_ATP_Olambda__575,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_aby(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_575
tff(fact_8753_ATP_Olambda__576,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_av(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_576
tff(fact_8754_ATP_Olambda__577,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aih(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,$o,Uu,Uub)
            | aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_577
tff(fact_8755_ATP_Olambda__578,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ae(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_578
tff(fact_8756_ATP_Olambda__579,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_wf(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,$o,Uu,Uub)
        & aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_579
tff(fact_8757_ATP_Olambda__580,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aig(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,$o,Uu,Uub)
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_580
tff(fact_8758_ATP_Olambda__581,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_581
tff(fact_8759_ATP_Olambda__582,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_ajv(fun(B,A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,A,Uu,Uub) = aa(B,A,Uua,Uub) ) ) ).

% ATP.lambda_582
tff(fact_8760_ATP_Olambda__583,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_akr(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_583
tff(fact_8761_ATP_Olambda__584,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] :
          ( aa(A,$o,aa(fun(A,A),fun(A,$o),aTP_Lamp_akm(fun(A,A),fun(fun(A,A),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,A,Uu,Uub) = aa(A,A,Uua,Uub) ) ) ) ).

% ATP.lambda_584
tff(fact_8762_ATP_Olambda__585,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akn(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
      <=> aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_585
tff(fact_8763_ATP_Olambda__586,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajt(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_586
tff(fact_8764_ATP_Olambda__587,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajl(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_587
tff(fact_8765_ATP_Olambda__588,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] : aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_os(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uua,Uub))) ) ).

% ATP.lambda_588
tff(fact_8766_ATP_Olambda__589,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_xd(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Uu,Uub)),aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_589
tff(fact_8767_ATP_Olambda__590,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_sk(fun(A,assn),fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(A,assn,Uu,Uub)),ex_assn(A,Uua)) ).

% ATP.lambda_590
tff(fact_8768_ATP_Olambda__591,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,real),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_kx(fun(nat,real),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_591
tff(fact_8769_ATP_Olambda__592,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(B,real),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_kw(fun(B,real),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(B,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_592
tff(fact_8770_ATP_Olambda__593,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,real),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_yt(fun(A,real),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_593
tff(fact_8771_ATP_Olambda__594,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_hf(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_594
tff(fact_8772_ATP_Olambda__595,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_aef(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),Uua,Uub),Uu) ) ).

% ATP.lambda_595
tff(fact_8773_ATP_Olambda__596,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_dy(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(B,fun(C,A),Uu,Uub)),Uua) ) ).

% ATP.lambda_596
tff(fact_8774_ATP_Olambda__597,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: set(A),Uua: fun(B,fun(A,B)),Uub: B] : aa(B,B,aa(fun(B,fun(A,B)),fun(B,B),aTP_Lamp_xg(set(A),fun(fun(B,fun(A,B)),fun(B,B)),Uu),Uua),Uub) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(B,fun(A,B),Uua,Uub)),Uu) ) ).

% ATP.lambda_597
tff(fact_8775_ATP_Olambda__598,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_aeh(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),Uua,Uub)),Uu) ) ).

% ATP.lambda_598
tff(fact_8776_ATP_Olambda__599,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,real,aa(B,fun(A,real),aTP_Lamp_ais(fun(A,B),fun(B,fun(A,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_599
tff(fact_8777_ATP_Olambda__600,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ct(fun(nat,nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,Uu,Uub)),Uua) ) ).

% ATP.lambda_600
tff(fact_8778_ATP_Olambda__601,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aji(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_601
tff(fact_8779_ATP_Olambda__602,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_eo(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_602
tff(fact_8780_ATP_Olambda__603,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bg(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_603
tff(fact_8781_ATP_Olambda__604,axiom,
    ! [B: $tType,A: $tType] :
      ( field(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ei(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_604
tff(fact_8782_ATP_Olambda__605,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_xb(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_605
tff(fact_8783_ATP_Olambda__606,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_fb(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_606
tff(fact_8784_ATP_Olambda__607,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_adq(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_607
tff(fact_8785_ATP_Olambda__608,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_afh(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_608
tff(fact_8786_ATP_Olambda__609,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_akf(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_609
tff(fact_8787_ATP_Olambda__610,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_akd(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_610
tff(fact_8788_ATP_Olambda__611,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ajo(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_611
tff(fact_8789_ATP_Olambda__612,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bp(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_612
tff(fact_8790_ATP_Olambda__613,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ef(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_613
tff(fact_8791_ATP_Olambda__614,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_yn(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_614
tff(fact_8792_ATP_Olambda__615,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ww(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_615
tff(fact_8793_ATP_Olambda__616,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_adv(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_616
tff(fact_8794_ATP_Olambda__617,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_cw(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Uu,Uub)),Uua) ).

% ATP.lambda_617
tff(fact_8795_ATP_Olambda__618,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_aea(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_618
tff(fact_8796_ATP_Olambda__619,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_adw(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_619
tff(fact_8797_ATP_Olambda__620,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zh(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),Uu) ) ).

% ATP.lambda_620
tff(fact_8798_ATP_Olambda__621,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fw(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_621
tff(fact_8799_ATP_Olambda__622,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_afg(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_622
tff(fact_8800_ATP_Olambda__623,axiom,
    ! [B: $tType,A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_aam(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_623
tff(fact_8801_ATP_Olambda__624,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gh(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_624
tff(fact_8802_ATP_Olambda__625,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ace(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_625
tff(fact_8803_ATP_Olambda__626,axiom,
    ! [Uu: fun(real,real),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_xj(fun(real,real),fun(nat,fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,Uu,Uub)),Uua) ).

% ATP.lambda_626
tff(fact_8804_ATP_Olambda__627,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,A),Uua: nat,Uub: B] : aa(B,A,aa(nat,fun(B,A),aTP_Lamp_hj(fun(B,A),fun(nat,fun(B,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_627
tff(fact_8805_ATP_Olambda__628,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_zk(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_628
tff(fact_8806_ATP_Olambda__629,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_xe(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_629
tff(fact_8807_ATP_Olambda__630,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_aas(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_630
tff(fact_8808_ATP_Olambda__631,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_abz(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_631
tff(fact_8809_ATP_Olambda__632,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_aca(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_632
tff(fact_8810_ATP_Olambda__633,axiom,
    ! [Uu: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aiu(nat,fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,Uua,Uub)),Uu) ).

% ATP.lambda_633
tff(fact_8811_ATP_Olambda__634,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aao(nat,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uua,Uub)),Uu) ).

% ATP.lambda_634
tff(fact_8812_ATP_Olambda__635,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_sc(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(A,assn,Uu,Uub)),Uua) ).

% ATP.lambda_635
tff(fact_8813_ATP_Olambda__636,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_sd(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(A,assn,Uu,Uub)),Uua) ).

% ATP.lambda_636
tff(fact_8814_ATP_Olambda__637,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_mt(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_637
tff(fact_8815_ATP_Olambda__638,axiom,
    ! [Uu: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_xm(fun(real,real),fun(real,fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),Uua) ).

% ATP.lambda_638
tff(fact_8816_ATP_Olambda__639,axiom,
    ! [A: $tType,Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahw(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uua,Uub),Uu) ).

% ATP.lambda_639
tff(fact_8817_ATP_Olambda__640,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(B,A),Uua: set(A),Uub: B] :
          ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_ajx(fun(B,A),fun(set(A),fun(B,$o)),Uu),Uua),Uub)
        <=> member(A,aa(B,A,Uu,Uub),Uua) ) ) ).

% ATP.lambda_640
tff(fact_8818_ATP_Olambda__641,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: set(B),Uub: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_ajy(fun(A,B),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
        <=> member(B,aa(A,B,Uu,Uub),Uua) ) ) ).

% ATP.lambda_641
tff(fact_8819_ATP_Olambda__642,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_tp(set(A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> member(A,aa(B,A,Uua,Uub),Uu) ) ).

% ATP.lambda_642
tff(fact_8820_ATP_Olambda__643,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: $o,Uub: A] :
      ( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_ako(fun(A,$o),fun($o,fun(A,$o)),Uu),(Uua)),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        | (Uua) ) ) ).

% ATP.lambda_643
tff(fact_8821_ATP_Olambda__644,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ajz(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_644
tff(fact_8822_ATP_Olambda__645,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo8865339358273720382pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_akb(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_645
tff(fact_8823_ATP_Olambda__646,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ac(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_646
tff(fact_8824_ATP_Olambda__647,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_qm(nat,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub))) ) ).

% ATP.lambda_647
tff(fact_8825_ATP_Olambda__648,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: fun(nat,A),Uub: nat] : aa(nat,real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_ck(real,fun(fun(nat,A),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uua,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ) ).

% ATP.lambda_648
tff(fact_8826_ATP_Olambda__649,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_or(fun(A,$o),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu,Uub))),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_649
tff(fact_8827_ATP_Olambda__650,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_lu(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_650
tff(fact_8828_ATP_Olambda__651,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: nat,Uua: nat,Uub: word(A)] :
          ( aa(word(A),$o,aa(nat,fun(word(A),$o),aTP_Lamp_te(nat,fun(nat,fun(word(A),$o)),Uu),Uua),Uub)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Uub),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Uu)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Uua))) ) ) ).

% ATP.lambda_651
tff(fact_8829_ATP_Olambda__652,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_tf(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ) ).

% ATP.lambda_652
tff(fact_8830_ATP_Olambda__653,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_qj(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(nat,A,Uu,set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_653
tff(fact_8831_ATP_Olambda__654,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_qk(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_654
tff(fact_8832_ATP_Olambda__655,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_qi(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_655
tff(fact_8833_ATP_Olambda__656,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fm(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_656
tff(fact_8834_ATP_Olambda__657,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fk(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_657
tff(fact_8835_ATP_Olambda__658,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: array(A)] : aa(array(A),assn,aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_sp(nat,fun(fun(nat,A),fun(array(A),assn)),Uu),Uua),Uub) = aa(list(A),assn,snga_assn(A,Uub),map(nat,A,Uua,upt(zero_zero(nat),Uu))) ) ).

% ATP.lambda_658
tff(fact_8836_ATP_Olambda__659,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gn(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_659
tff(fact_8837_ATP_Olambda__660,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_agq(real,fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),Uua),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_660
tff(fact_8838_ATP_Olambda__661,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ex(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ) ).

% ATP.lambda_661
tff(fact_8839_ATP_Olambda__662,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fl(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ).

% ATP.lambda_662
tff(fact_8840_ATP_Olambda__663,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: A,Uub: array(A)] : aa(array(A),assn,aa(A,fun(array(A),assn),aTP_Lamp_ra(nat,fun(A,fun(array(A),assn)),Uu),Uua),Uub) = aa(list(A),assn,snga_assn(A,Uub),replicate(A,Uu,Uua)) ) ).

% ATP.lambda_663
tff(fact_8841_ATP_Olambda__664,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: real,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kv(real,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_664
tff(fact_8842_ATP_Olambda__665,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: real,Uub: nat] : aa(nat,A,aa(real,fun(nat,A),aTP_Lamp_ku(fun(nat,A),fun(real,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_665
tff(fact_8843_ATP_Olambda__666,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: real,Uub: A] : aa(A,B,aa(real,fun(A,B),aTP_Lamp_ys(fun(A,B),fun(real,fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_666
tff(fact_8844_ATP_Olambda__667,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: real,Uub: A] : aa(A,A,aa(real,fun(A,A),aTP_Lamp_wr(fun(A,A),fun(real,fun(A,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_667
tff(fact_8845_ATP_Olambda__668,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ajj(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_668
tff(fact_8846_ATP_Olambda__669,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_tl(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_669
tff(fact_8847_ATP_Olambda__670,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ake(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_670
tff(fact_8848_ATP_Olambda__671,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ajs(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_671
tff(fact_8849_ATP_Olambda__672,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_akc(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_672
tff(fact_8850_ATP_Olambda__673,axiom,
    ! [A: $tType,Uu: assn,Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_sa(assn,fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu),aa(A,assn,Uua,Uub)) ).

% ATP.lambda_673
tff(fact_8851_ATP_Olambda__674,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bh(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_674
tff(fact_8852_ATP_Olambda__675,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_eg(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_675
tff(fact_8853_ATP_Olambda__676,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fv(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_676
tff(fact_8854_ATP_Olambda__677,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aff(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_677
tff(fact_8855_ATP_Olambda__678,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_aan(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_678
tff(fact_8856_ATP_Olambda__679,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bq(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_679
tff(fact_8857_ATP_Olambda__680,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_yo(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_680
tff(fact_8858_ATP_Olambda__681,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_wx(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_681
tff(fact_8859_ATP_Olambda__682,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_adu(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_682
tff(fact_8860_ATP_Olambda__683,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_qv(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uua),aa(A,assn,Uu,Uub)) ).

% ATP.lambda_683
tff(fact_8861_ATP_Olambda__684,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_adz(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_684
tff(fact_8862_ATP_Olambda__685,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_adx(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_685
tff(fact_8863_ATP_Olambda__686,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_aiz(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_686
tff(fact_8864_ATP_Olambda__687,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: fun(B,nat),Uub: B] : aa(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_hs(A,fun(fun(B,nat),fun(B,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(B,nat,Uua,Uub)) ) ).

% ATP.lambda_687
tff(fact_8865_ATP_Olambda__688,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu: fun(A,nat),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_agy(fun(A,nat),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),Uua),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_688
tff(fact_8866_ATP_Olambda__689,axiom,
    ! [A: $tType,Uu: assn,Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_sb(assn,fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Uu),aa(A,assn,Uua,Uub)) ).

% ATP.lambda_689
tff(fact_8867_ATP_Olambda__690,axiom,
    ! [A: $tType,Uu: assn,Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_se(assn,fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Uu),aa(A,assn,Uua,Uub)) ).

% ATP.lambda_690
tff(fact_8868_ATP_Olambda__691,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adb(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_691
tff(fact_8869_ATP_Olambda__692,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_fc(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,dvd_dvd(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_692
tff(fact_8870_ATP_Olambda__693,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_afm(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ) ).

% ATP.lambda_693
tff(fact_8871_ATP_Olambda__694,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_acz(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ).

% ATP.lambda_694
tff(fact_8872_ATP_Olambda__695,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akq($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_695
tff(fact_8873_ATP_Olambda__696,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akp($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_696
tff(fact_8874_ATP_Olambda__697,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_uq(fun(A,B),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( Uub = aa(A,B,Uu,Uua) ) ) ).

% ATP.lambda_697
tff(fact_8875_ATP_Olambda__698,axiom,
    ! [A: $tType,Uu: assn,Uua: A,Uub: A] : aa(A,assn,aa(A,fun(A,assn),aTP_Lamp_co(assn,fun(A,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu),pure_assn(Uub = Uua)) ).

% ATP.lambda_698
tff(fact_8876_ATP_Olambda__699,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ic(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_699
tff(fact_8877_ATP_Olambda__700,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_dj(fun(A,nat),fun(A,fun(A,$o)),Uu),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,Uu,Uub)),aa(A,nat,Uu,Uub))) ) ).

% ATP.lambda_700
tff(fact_8878_ATP_Olambda__701,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dk(fun(A,nat),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),
          $ite(aa(A,$o,Uua,Uub),aa(A,nat,Uu,Uub),zero_zero(nat))) ) ).

% ATP.lambda_701
tff(fact_8879_ATP_Olambda__702,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] :
      ( aa(A,$o,aa(fun(A,nat),fun(A,$o),aTP_Lamp_di(fun(A,nat),fun(fun(A,nat),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uua,Uub))) ) ).

% ATP.lambda_702
tff(fact_8880_ATP_Olambda__703,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hw(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_703
tff(fact_8881_ATP_Olambda__704,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ga(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_704
tff(fact_8882_ATP_Olambda__705,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afy(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ) ).

% ATP.lambda_705
tff(fact_8883_ATP_Olambda__706,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afi(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uua)) ) ).

% ATP.lambda_706
tff(fact_8884_ATP_Olambda__707,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_abb(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_707
tff(fact_8885_ATP_Olambda__708,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_xh(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_708
tff(fact_8886_ATP_Olambda__709,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real,Uub: real] : aa(real,A,aa(real,fun(real,A),aTP_Lamp_aiy(fun(real,A),fun(real,fun(real,A)),Uu),Uua),Uub) = aa(real,A,Uu,aa(real,real,aa(real,fun(real,real),plus_plus(real),Uub),Uua)) ).

% ATP.lambda_709
tff(fact_8887_ATP_Olambda__710,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bj(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_710
tff(fact_8888_ATP_Olambda__711,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afo(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_711
tff(fact_8889_ATP_Olambda__712,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ht(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_712
tff(fact_8890_ATP_Olambda__713,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_et(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_713
tff(fact_8891_ATP_Olambda__714,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_aaw(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_714
tff(fact_8892_ATP_Olambda__715,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_aev(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_715
tff(fact_8893_ATP_Olambda__716,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_wh(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_716
tff(fact_8894_ATP_Olambda__717,axiom,
    ! [A: $tType,B: $tType,Uu: fun(product_prod(A,B),$o),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_pg(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(A,B),$o,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ) ).

% ATP.lambda_717
tff(fact_8895_ATP_Olambda__718,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_nr(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(product_prod(A,B),C,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).

% ATP.lambda_718
tff(fact_8896_ATP_Olambda__719,axiom,
    ! [C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: B,Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_aio(B,fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),Uub)) ) ).

% ATP.lambda_719
tff(fact_8897_ATP_Olambda__720,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_abc(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uub)) ) ).

% ATP.lambda_720
tff(fact_8898_ATP_Olambda__721,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fz(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uu)) ) ).

% ATP.lambda_721
tff(fact_8899_ATP_Olambda__722,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(real,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zg(fun(real,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,Uu,aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_722
tff(fact_8900_ATP_Olambda__723,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(real,A),Uua: fun(nat,real),Uub: nat] : aa(nat,A,aa(fun(nat,real),fun(nat,A),aTP_Lamp_ahy(fun(real,A),fun(fun(nat,real),fun(nat,A)),Uu),Uua),Uub) = aa(real,A,Uu,aa(nat,real,Uua,Uub)) ) ).

% ATP.lambda_723
tff(fact_8901_ATP_Olambda__724,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,A),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_st(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_724
tff(fact_8902_ATP_Olambda__725,axiom,
    ! [C: $tType,B: $tType,D: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(D,B),Uub: D] : aa(D,fun(C,$o),aa(fun(D,B),fun(D,fun(C,$o)),aTP_Lamp_vo(fun(B,fun(C,$o)),fun(fun(D,B),fun(D,fun(C,$o))),Uu),Uua),Uub) = aa(B,fun(C,$o),Uu,aa(D,B,Uua,Uub)) ).

% ATP.lambda_725
tff(fact_8903_ATP_Olambda__726,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(num,B),Uub: num] : aa(num,A,aa(fun(num,B),fun(num,A),aTP_Lamp_vp(fun(B,A),fun(fun(num,B),fun(num,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(num,B,Uua,Uub)) ).

% ATP.lambda_726
tff(fact_8904_ATP_Olambda__727,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(nat,B),Uub: nat] : aa(nat,A,aa(fun(nat,B),fun(nat,A),aTP_Lamp_vu(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(nat,B,Uua,Uub)) ).

% ATP.lambda_727
tff(fact_8905_ATP_Olambda__728,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,B),Uub: C] : aa(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_mh(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_728
tff(fact_8906_ATP_Olambda__729,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(A,$o),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ajp(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_729
tff(fact_8907_ATP_Olambda__730,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aga(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_730
tff(fact_8908_ATP_Olambda__731,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aho(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_731
tff(fact_8909_ATP_Olambda__732,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_yj(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_732
tff(fact_8910_ATP_Olambda__733,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ahk(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_733
tff(fact_8911_ATP_Olambda__734,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ahl(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_734
tff(fact_8912_ATP_Olambda__735,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_agb(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_735
tff(fact_8913_ATP_Olambda__736,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_abj(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_736
tff(fact_8914_ATP_Olambda__737,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_wn(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_737
tff(fact_8915_ATP_Olambda__738,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_abh(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_738
tff(fact_8916_ATP_Olambda__739,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_aju(fun(A,$o),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,Uu,aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_739
tff(fact_8917_ATP_Olambda__740,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_act(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_740
tff(fact_8918_ATP_Olambda__741,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(C,A),Uua: fun(A,B),Uub: C] : aa(C,B,aa(fun(A,B),fun(C,B),aTP_Lamp_abg(fun(C,A),fun(fun(A,B),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uua,aa(C,A,Uu,Uub)) ) ).

% ATP.lambda_741
tff(fact_8919_ATP_Olambda__742,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(A,C),Uub: B] : aa(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_su(fun(B,A),fun(fun(A,C),fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu,Uub)) ).

% ATP.lambda_742
tff(fact_8920_ATP_Olambda__743,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_yd(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_743
tff(fact_8921_ATP_Olambda__744,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aav(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_744
tff(fact_8922_ATP_Olambda__745,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_wo(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uua,aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_745
tff(fact_8923_ATP_Olambda__746,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ait(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_746
tff(fact_8924_ATP_Olambda__747,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topological_t2_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_abl(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_747
tff(fact_8925_ATP_Olambda__748,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_ajw(fun(A,B),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_748
tff(fact_8926_ATP_Olambda__749,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_om(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_749
tff(fact_8927_ATP_Olambda__750,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_on(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_750
tff(fact_8928_ATP_Olambda__751,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: C] : aa(C,B,aa(A,fun(C,B),aTP_Lamp_tq(fun(A,B),fun(A,fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,Uua) ).

% ATP.lambda_751
tff(fact_8929_ATP_Olambda__752,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,B),Uub: A] : aa(A,B,aa(fun(nat,B),fun(A,B),aTP_Lamp_afd(fun(A,B),fun(fun(nat,B),fun(A,B)),Uu),Uua),Uub) = suminf(B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_afc(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub)) ) ).

% ATP.lambda_752
tff(fact_8930_ATP_Olambda__753,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aaj(fun(nat,A),fun(A,fun(A,A)),Uu),Uua),Uub) = suminf(A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_aai(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub)) ) ).

% ATP.lambda_753
tff(fact_8931_ATP_Olambda__754,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_cj(fun(nat,A),fun(A,fun(nat,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub))) ) ).

% ATP.lambda_754
tff(fact_8932_ATP_Olambda__755,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_ia(fun(A,B),fun(fun(A,B),fun(A,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub))) ) ).

% ATP.lambda_755
tff(fact_8933_ATP_Olambda__756,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_ahd(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu,Uub)),Uua)) ).

% ATP.lambda_756
tff(fact_8934_ATP_Olambda__757,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t1_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aka(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_757
tff(fact_8935_ATP_Olambda__758,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,assn,aa(nat,fun(nat,assn),aTP_Lamp_cn(nat,fun(nat,fun(nat,assn)),Uu),Uua),Uub) = pure_assn(Uub = vEBT_VEBT_high(Uu,Uua)) ).

% ATP.lambda_758
tff(fact_8936_ATP_Olambda__759,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,assn,aa(nat,fun(nat,assn),aTP_Lamp_cm(nat,fun(nat,fun(nat,assn)),Uu),Uua),Uub) = pure_assn(Uub = vEBT_VEBT_low(Uu,Uua)) ).

% ATP.lambda_759
tff(fact_8937_ATP_Olambda__760,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_fd(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ~ aa(B,$o,dvd_dvd(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_760
tff(fact_8938_ATP_Olambda__761,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu: fun(B,real),Uua: fun(real,A),Uub: B] : aa(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_aag(fun(B,real),fun(fun(real,A),fun(B,real)),Uu),Uua),Uub) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(A,aa(real,A,Uua,aa(B,real,Uu,Uub)))) ) ).

% ATP.lambda_761
tff(fact_8939_ATP_Olambda__762,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,A))),Uua: D,Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_nc(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(D,A,aa(C,fun(D,A),aa(B,fun(C,fun(D,A)),Uu,Uub),Uuc),Uua) ).

% ATP.lambda_762
tff(fact_8940_ATP_Olambda__763,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_kr(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)
            & aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uuc) ),
            aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_763
tff(fact_8941_ATP_Olambda__764,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_kn(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)
            & ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uuc) ),
            aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_764
tff(fact_8942_ATP_Olambda__765,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_kp(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_765
tff(fact_8943_ATP_Olambda__766,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A,Uuc: B] :
      aa(B,A,aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_uo(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(member(B,Uuc,aa(set(A),set(B),image(A,B,Uu),Uua)),the_inv_into(A,B,Uua,Uu,Uuc),Uub) ).

% ATP.lambda_766
tff(fact_8944_ATP_Olambda__767,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ka(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu,zero_zero(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_767
tff(fact_8945_ATP_Olambda__768,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_kc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu,one_one(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_768
tff(fact_8946_ATP_Olambda__769,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ajb(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uuc),Uu),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_769
tff(fact_8947_ATP_Olambda__770,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_kd(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_770
tff(fact_8948_ATP_Olambda__771,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_kb(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_771
tff(fact_8949_ATP_Olambda__772,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: fun(A,B),Uuc: A] :
      aa(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_oo(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uuc,Uua),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_772
tff(fact_8950_ATP_Olambda__773,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: set(nat),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_gf(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(member(nat,Uuc,Uua),aa(nat,A,Uub,Uuc),aa(nat,A,Uu,Uuc)) ) ).

% ATP.lambda_773
tff(fact_8951_ATP_Olambda__774,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_hz(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_774
tff(fact_8952_ATP_Olambda__775,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_fe(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_775
tff(fact_8953_ATP_Olambda__776,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,option(A)),Uua: B,Uub: A,Uuc: B] :
      aa(B,option(A),aa(A,fun(B,option(A)),aa(B,fun(A,fun(B,option(A))),aTP_Lamp_po(fun(B,option(A)),fun(B,fun(A,fun(B,option(A)))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uua,some(A,Uub),aa(B,option(A),Uu,Uuc)) ).

% ATP.lambda_776
tff(fact_8954_ATP_Olambda__777,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
      aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_md(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(aa(B,$o,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).

% ATP.lambda_777
tff(fact_8955_ATP_Olambda__778,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_hy(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_778
tff(fact_8956_ATP_Olambda__779,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ez(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_779
tff(fact_8957_ATP_Olambda__780,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_aar(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uua,Uuc),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_780
tff(fact_8958_ATP_Olambda__781,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(D,C),Uub: B,Uuc: D] :
      ( aa(D,$o,aa(B,fun(D,$o),aa(fun(D,C),fun(B,fun(D,$o)),aTP_Lamp_vi(fun(B,fun(C,$o)),fun(fun(D,C),fun(B,fun(D,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(C,$o,aa(B,fun(C,$o),Uu,Uub),aa(D,C,Uua,Uuc)) ) ).

% ATP.lambda_781
tff(fact_8959_ATP_Olambda__782,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fo(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fn(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(nat,nat,suc,zero_zero(nat)))),Uuc))) ) ).

% ATP.lambda_782
tff(fact_8960_ATP_Olambda__783,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_ee(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_ed(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_783
tff(fact_8961_ATP_Olambda__784,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: nat] :
          aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_ke(nat,fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(
                Uuc = zero_zero(nat),
                aa(A,A,uminus_uminus(A),Uub),
                $ite(Uuc = Uu,one_one(A),zero_zero(A)) )),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_784
tff(fact_8962_ATP_Olambda__785,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_hm(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_hl(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc),collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_dg(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_785
tff(fact_8963_ATP_Olambda__786,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_en(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_em(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_dg(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_786
tff(fact_8964_ATP_Olambda__787,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_jx(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_jw(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uub),Uuc)) ).

% ATP.lambda_787
tff(fact_8965_ATP_Olambda__788,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_jt(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_js(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uuc)) ) ).

% ATP.lambda_788
tff(fact_8966_ATP_Olambda__789,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: nat,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_xv(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Uua)),Uuc)),zero_zero(real))),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uub),Uuc)) ).

% ATP.lambda_789
tff(fact_8967_ATP_Olambda__790,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: nat,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_xt(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)),zero_zero(real))),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uub),Uuc)) ).

% ATP.lambda_790
tff(fact_8968_ATP_Olambda__791,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_xs(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,Uuc),Uua)),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Uub),Uua)),Uuc)) ).

% ATP.lambda_791
tff(fact_8969_ATP_Olambda__792,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_xr(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,Uuc),Uub)),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Uua),Uub)),Uuc)) ).

% ATP.lambda_792
tff(fact_8970_ATP_Olambda__793,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gi(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uua)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uub)) ) ).

% ATP.lambda_793
tff(fact_8971_ATP_Olambda__794,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fn(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Uuc))) ) ).

% ATP.lambda_794
tff(fact_8972_ATP_Olambda__795,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_aha(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),aa(nat,A,Uub,Uuc)))),aa(A,A,Uu,Uua))),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_795
tff(fact_8973_ATP_Olambda__796,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: fun(product_prod(A,B),$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_nn(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( aa(A,option(B),Uu,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_796
tff(fact_8974_ATP_Olambda__797,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Uu: fun(A,B),Uua: set(B),Uub: B,Uuc: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aa(set(B),fun(B,fun(A,$o)),aTP_Lamp_ajq(fun(A,B),fun(set(B),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(B,aa(A,B,Uu,Uuc),Uua)
            & ( aa(A,B,Uu,Uuc) != Uub ) ) ) ) ).

% ATP.lambda_797
tff(fact_8975_ATP_Olambda__798,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: set(B),Uuc: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_ajr(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(B,aa(A,B,Uu,Uuc),Uub)
            & ( aa(A,B,Uu,Uuc) != Uua ) ) ) ) ).

% ATP.lambda_798
tff(fact_8976_ATP_Olambda__799,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_jy(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),comm_s3205402744901411588hammer(A,Uu,Uuc))),comm_s3205402744901411588hammer(A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_799
tff(fact_8977_ATP_Olambda__800,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_jr(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,semiring_1_of_nat(nat),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_800
tff(fact_8978_ATP_Olambda__801,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ju(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_801
tff(fact_8979_ATP_Olambda__802,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ls(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))) ) ).

% ATP.lambda_802
tff(fact_8980_ATP_Olambda__803,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_gl(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uuc)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)) ) ).

% ATP.lambda_803
tff(fact_8981_ATP_Olambda__804,axiom,
    ! [A: $tType,Uu: $o,Uua: A,Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_pc($o,fun(A,fun(A,fun(A,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( ( (Uu)
         => ( Uuc = Uua ) )
        & ( ~ (Uu)
         => ( Uuc = Uub ) ) ) ) ).

% ATP.lambda_804
tff(fact_8982_ATP_Olambda__805,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_mi(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,aa(set(B),set(A),image(B,A,Uu),Uua))
        & aa(A,$o,Uub,Uuc) ) ) ).

% ATP.lambda_805
tff(fact_8983_ATP_Olambda__806,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_ek(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_806
tff(fact_8984_ATP_Olambda__807,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,$o)),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_dg(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uuc),Uub) ) ) ).

% ATP.lambda_807
tff(fact_8985_ATP_Olambda__808,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gj(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uua)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))) ) ).

% ATP.lambda_808
tff(fact_8986_ATP_Olambda__809,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: set(A),Uua: fun(A,A),Uub: fun(A,A),Uuc: A] :
          ( aa(A,$o,aa(fun(A,A),fun(A,$o),aa(fun(A,A),fun(fun(A,A),fun(A,$o)),aTP_Lamp_akl(set(A),fun(fun(A,A),fun(fun(A,A),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
           => ( aa(A,A,Uua,Uuc) = aa(A,A,Uub,Uuc) ) ) ) ) ).

% ATP.lambda_809
tff(fact_8987_ATP_Olambda__810,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_mk(set(A),fun(fun(A,B),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,B,Uua,Uuc) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_810
tff(fact_8988_ATP_Olambda__811,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(B,A),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_sx(set(B),fun(fun(B,A),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uu)
        & ( aa(B,A,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_811
tff(fact_8989_ATP_Olambda__812,axiom,
    ! [A: $tType,C: $tType,Uu: set(A),Uua: fun(A,C),Uub: C,Uuc: A] :
      ( aa(A,$o,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_mn(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_812
tff(fact_8990_ATP_Olambda__813,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_mq(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_813
tff(fact_8991_ATP_Olambda__814,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: set(B),Uuc: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_to(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uub)
        & member(A,aa(B,A,Uua,Uuc),Uu) ) ) ).

% ATP.lambda_814
tff(fact_8992_ATP_Olambda__815,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_gk(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uuc))) ) ).

% ATP.lambda_815
tff(fact_8993_ATP_Olambda__816,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_jl(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Uu),Uuc)),aa(nat,nat,binomial(Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_816
tff(fact_8994_ATP_Olambda__817,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_pt(int,fun(int,fun(int,fun(int,$o))),Uu),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),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_817
tff(fact_8995_ATP_Olambda__818,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_pq(int,fun(int,fun(int,fun(int,$o))),Uu),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),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_818
tff(fact_8996_ATP_Olambda__819,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_pb(A,fun(B,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_819
tff(fact_8997_ATP_Olambda__820,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_mj(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uua)
        & aa(A,$o,Uub,aa(B,A,Uu,Uuc)) ) ) ).

% ATP.lambda_820
tff(fact_8998_ATP_Olambda__821,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_dm(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
            & ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != one_one(B) ) ) ) ) ).

% ATP.lambda_821
tff(fact_8999_ATP_Olambda__822,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_do(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
            & ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_822
tff(fact_9000_ATP_Olambda__823,axiom,
    ! [Uu: nat,Uua: set(nat),Uub: vEBT_VEBTi,Uuc: vEBT_VEBT] :
      aa(vEBT_VEBT,assn,aa(vEBT_VEBTi,fun(vEBT_VEBT,assn),aa(set(nat),fun(vEBT_VEBTi,fun(vEBT_VEBT,assn)),aTP_Lamp_ry(nat,fun(set(nat),fun(vEBT_VEBTi,fun(vEBT_VEBT,assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uuc),Uub)),
        pure_assn(( ( Uua = vEBT_set_vebt(Uuc) )
          & vEBT_invar_vebt(Uuc,Uu) ))) ).

% ATP.lambda_823
tff(fact_9001_ATP_Olambda__824,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: $o] : aa($o,assn,aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_aq(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),Uu),Uua),Uub),(Uuc)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu),Uua)),pure_assn((Uuc) = aa(nat,$o,vEBT_vebt_member(Uu),Uub))) ).

% ATP.lambda_824
tff(fact_9002_ATP_Olambda__825,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option(nat)] : aa(option(nat),assn,aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_ay(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu),Uua)),pure_assn(Uuc = vEBT_vebt_succ(Uu,Uub))) ).

% ATP.lambda_825
tff(fact_9003_ATP_Olambda__826,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option(nat)] : aa(option(nat),assn,aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_ax(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu),Uua)),pure_assn(Uuc = vEBT_vebt_pred(Uu,Uub))) ).

% ATP.lambda_826
tff(fact_9004_ATP_Olambda__827,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: list(A),Uub: array(A),Uuc: A] : aa(A,assn,aa(array(A),fun(A,assn),aa(list(A),fun(array(A),fun(A,assn)),aTP_Lamp_qu(nat,fun(list(A),fun(array(A),fun(A,assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(A),assn,snga_assn(A,Uub),Uua)),pure_assn(Uuc = aa(nat,A,nth(A,Uua),Uu))) ) ).

% ATP.lambda_827
tff(fact_9005_ATP_Olambda__828,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_ahi(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub))))) ) ).

% ATP.lambda_828
tff(fact_9006_ATP_Olambda__829,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_cc(fun(nat,A),fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uuc)) ) ).

% ATP.lambda_829
tff(fact_9007_ATP_Olambda__830,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: list(B),Uuc: nat] : aa(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_rw(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_830
tff(fact_9008_ATP_Olambda__831,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_aai(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc))),Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))))) ) ).

% ATP.lambda_831
tff(fact_9009_ATP_Olambda__832,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: fun(nat,B),Uub: A,Uuc: nat] : aa(nat,B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_afc(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uua,Uuc)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uuc)) ) ).

% ATP.lambda_832
tff(fact_9010_ATP_Olambda__833,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_zx(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(A,real,Uu,Uua)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% ATP.lambda_833
tff(fact_9011_ATP_Olambda__834,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_kk(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),Uub)),one_one(nat)))) ) ).

% ATP.lambda_834
tff(fact_9012_ATP_Olambda__835,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B,Uuc: A] : aa(A,A,aa(B,fun(A,A),aa(A,fun(B,fun(A,A)),aTP_Lamp_rx(fun(B,A),fun(A,fun(B,fun(A,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uuc)) ) ).

% ATP.lambda_835
tff(fact_9013_ATP_Olambda__836,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_jw(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uuc)),aa(nat,nat,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_836
tff(fact_9014_ATP_Olambda__837,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_jp(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_837
tff(fact_9015_ATP_Olambda__838,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_js(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_838
tff(fact_9016_ATP_Olambda__839,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_ed(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_839
tff(fact_9017_ATP_Olambda__840,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_hk(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(B,C,aa(A,fun(B,C),Uua,Uuc),collect(B,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_ek(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_840
tff(fact_9018_ATP_Olambda__841,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_el(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(A,fun(B,C),Uua,Uuc)),collect(B,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_ek(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_841
tff(fact_9019_ATP_Olambda__842,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_aab(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,cos(real),aa(A,real,Uu,Uua))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% ATP.lambda_842
tff(fact_9020_ATP_Olambda__843,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_zz(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uub)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% ATP.lambda_843
tff(fact_9021_ATP_Olambda__844,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_yy(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,exp(real),aa(A,real,Uu,Uub))) ) ).

% ATP.lambda_844
tff(fact_9022_ATP_Olambda__845,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_za(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,cos(real),aa(A,real,Uu,Uub))) ) ).

% ATP.lambda_845
tff(fact_9023_ATP_Olambda__846,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_zn(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua))) ) ).

% ATP.lambda_846
tff(fact_9024_ATP_Olambda__847,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_yc(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% ATP.lambda_847
tff(fact_9025_ATP_Olambda__848,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_zj(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,uminus_uminus(real),aa(real,real,sin(real),aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_848
tff(fact_9026_ATP_Olambda__849,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_aaf(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ).

% ATP.lambda_849
tff(fact_9027_ATP_Olambda__850,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_ahj(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc))),aa(A,B,Uu,Uub))),aa(A,B,Uua,Uuc)))),real_V7770717601297561774m_norm(A,Uuc)) ) ).

% ATP.lambda_850
tff(fact_9028_ATP_Olambda__851,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_aib(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub))))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub))) ) ).

% ATP.lambda_851
tff(fact_9029_ATP_Olambda__852,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: list(vEBT_VEBTi),Uuc: option(nat)] : aa(option(nat),assn,aa(list(vEBT_VEBTi),fun(option(nat),assn),aa(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn)),aTP_Lamp_lx(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn(Uuc = vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),Uu)))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_852
tff(fact_9030_ATP_Olambda__853,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: list(vEBT_VEBTi),Uuc: option(nat)] : aa(option(nat),assn,aa(list(vEBT_VEBTi),fun(option(nat),assn),aa(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn)),aTP_Lamp_lw(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn(Uuc = vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),Uu)))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_853
tff(fact_9031_ATP_Olambda__854,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: A,Uub: set(A),Uuc: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aa(A,fun(set(A),fun(A,$o)),aTP_Lamp_aki(fun(A,$o),fun(A,fun(set(A),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( ( Uuc != Uua )
           => ( member(A,Uuc,Uub)
             => aa(A,$o,Uu,Uuc) ) ) ) ) ).

% ATP.lambda_854
tff(fact_9032_ATP_Olambda__855,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: filter(A),Uuc: A] : aa(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_aif(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_yu(A,A)))))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(A,B,Uu,topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_yu(A,A))))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_yu(A,A)))))) ) ).

% ATP.lambda_855
tff(fact_9033_ATP_Olambda__856,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_aic(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uua)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uub,Uuc)),aa(A,B,Uub,Uua))),aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uua)))) ) ).

% ATP.lambda_856
tff(fact_9034_ATP_Olambda__857,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_aid(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))) ) ).

% ATP.lambda_857
tff(fact_9035_ATP_Olambda__858,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: nat] : aa(nat,list(A),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_vz(A,fun(list(A),fun(A,fun(nat,list(A)))),Uu),Uua),Uub),Uuc) = aa(list(A),list(A),cons(A,Uu),list_update(A,Uua,Uuc,Uub)) ).

% ATP.lambda_858
tff(fact_9036_ATP_Olambda__859,axiom,
    ! [B: $tType,A: $tType,Uu: assn,Uua: fun(B,fun(A,assn)),Uub: B,Uuc: A] : aa(A,assn,aa(B,fun(A,assn),aa(fun(B,fun(A,assn)),fun(B,fun(A,assn)),aTP_Lamp_og(assn,fun(fun(B,fun(A,assn)),fun(B,fun(A,assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu),aa(A,assn,aa(B,fun(A,assn),Uua,Uub),Uuc)) ).

% ATP.lambda_859
tff(fact_9037_ATP_Olambda__860,axiom,
    ! [A: $tType,B: $tType,Uu: $o,Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_nk($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_860
tff(fact_9038_ATP_Olambda__861,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_mp(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,B,Uua,collect(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_mn(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_861
tff(fact_9039_ATP_Olambda__862,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_ms(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,C,Uub,collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_mq(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_862
tff(fact_9040_ATP_Olambda__863,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_mo(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,Uua),collect(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_mn(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_863
tff(fact_9041_ATP_Olambda__864,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_mr(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,Uub),collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_mq(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_864
tff(fact_9042_ATP_Olambda__865,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun($o,fun($o,B)),Uub: $o,Uuc: $o] : aa($o,A,aa($o,fun($o,A),aa(fun($o,fun($o,B)),fun($o,fun($o,A)),aTP_Lamp_au(fun(B,A),fun(fun($o,fun($o,B)),fun($o,fun($o,A))),Uu),Uua),(Uub)),(Uuc)) = aa(B,A,Uu,aa($o,B,aa($o,fun($o,B),Uua,(Uub)),(Uuc))) ).

% ATP.lambda_865
tff(fact_9043_ATP_Olambda__866,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(B,A),Uua: fun(C,fun(D,B)),Uub: C,Uuc: D] : aa(D,A,aa(C,fun(D,A),aa(fun(C,fun(D,B)),fun(C,fun(D,A)),aTP_Lamp_no(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),Uu),Uua),Uub),Uuc) = aa(B,A,Uu,aa(D,B,aa(C,fun(D,B),Uua,Uub),Uuc)) ).

% ATP.lambda_866
tff(fact_9044_ATP_Olambda__867,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_qg(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_867
tff(fact_9045_ATP_Olambda__868,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_qe(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_868
tff(fact_9046_ATP_Olambda__869,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_hx(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_869
tff(fact_9047_ATP_Olambda__870,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ey(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_870
tff(fact_9048_ATP_Olambda__871,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_qc(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_871
tff(fact_9049_ATP_Olambda__872,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_pw(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_872
tff(fact_9050_ATP_Olambda__873,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B,Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_zf(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua))),aa(B,A,Uub,Uuc))),aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)))) ) ).

% ATP.lambda_873
tff(fact_9051_ATP_Olambda__874,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_qa(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uub)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_874
tff(fact_9052_ATP_Olambda__875,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_py(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ).

% ATP.lambda_875
tff(fact_9053_ATP_Olambda__876,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,fun(A,B)),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: C] : aa(C,B,aa(fun(C,A),fun(C,B),aa(C,fun(fun(C,A),fun(C,B)),aa(fun(C,A),fun(C,fun(fun(C,A),fun(C,B))),aTP_Lamp_yk(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,B,aa(A,fun(A,B),Uu,aa(C,A,Uua,Uub)),aa(C,A,Uuc,Uud)) ) ).

% ATP.lambda_876
tff(fact_9054_ATP_Olambda__877,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B] : aa(B,C,aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_zt(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_zs(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu),Uua),Uub),Uuc),Uud)),Uu) ) ).

% ATP.lambda_877
tff(fact_9055_ATP_Olambda__878,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_kh(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aa(A,fun(A,fun(nat,fun(nat,A))),aTP_Lamp_kg(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uua),Uub),Uuc),Uud)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_878
tff(fact_9056_ATP_Olambda__879,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: 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_qp(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn)))),Uu),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,Uu,Uua),Uub)),aa(C,assn,Uuc,Uud)) ).

% ATP.lambda_879
tff(fact_9057_ATP_Olambda__880,axiom,
    ! [Uu: nat,Uua: set(nat),Uub: vEBT_VEBTi,Uuc: nat,Uud: $o] : aa($o,assn,aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn))),aTP_Lamp_as(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn)))),Uu),Uua),Uub),Uuc),(Uud)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Uu,Uua),Uub)),pure_assn((Uud) = member(nat,Uuc,Uua))) ).

% ATP.lambda_880
tff(fact_9058_ATP_Olambda__881,axiom,
    ! [Uu: nat,Uua: fun(nat,fun(real,real)),Uub: real,Uuc: nat,Uud: real] : aa(real,real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_xu(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Uu),Uua),Uub),Uuc),Uud) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(nat,fun(real,real),Uua,Uuc),Uud)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_xt(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uua),Uuc),Uud)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uuc)))),aa(real,real,aa(real,fun(real,real),times_times(real),Uub),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uud),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uuc))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uuc)))))) ).

% ATP.lambda_881
tff(fact_9059_ATP_Olambda__882,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_kl(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_kk(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uua),Uuc),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uu))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud)) ) ).

% ATP.lambda_882
tff(fact_9060_ATP_Olambda__883,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ll(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uud)),Uua)),one_one(A))),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_883
tff(fact_9061_ATP_Olambda__884,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_lh(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uu)),Uua)),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uuc),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_884
tff(fact_9062_ATP_Olambda__885,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_li(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),Uua)),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uub)),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_885
tff(fact_9063_ATP_Olambda__886,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: A,Uuc: nat,Uud: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aa(A,fun(A,fun(nat,fun(nat,A))),aTP_Lamp_kg(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uud)),one_one(nat)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_886
tff(fact_9064_ATP_Olambda__887,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: nat,Uud: A] : aa(A,B,aa(nat,fun(A,B),aa(A,fun(nat,fun(A,B)),aa(fun(A,B),fun(A,fun(nat,fun(A,B))),aTP_Lamp_zl(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),Uuc)),aa(A,B,Uua,Uud))),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),one_one(nat)))) ) ).

% ATP.lambda_887
tff(fact_9065_ATP_Olambda__888,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: list(A),Uub: array(A),Uuc: A,Uud: array(A)] : aa(array(A),assn,aa(A,fun(array(A),assn),aa(array(A),fun(A,fun(array(A),assn)),aa(list(A),fun(array(A),fun(A,fun(array(A),assn))),aTP_Lamp_qy(nat,fun(list(A),fun(array(A),fun(A,fun(array(A),assn)))),Uu),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(A),assn,snga_assn(A,Uub),list_update(A,Uua,Uu,Uuc))),pure_assn(Uud = Uub)) ) ).

% ATP.lambda_888
tff(fact_9066_ATP_Olambda__889,axiom,
    ! [B: $tType,A: $tType,Uu: assn,Uua: fun(B,fun(A,assn)),Uub: B,Uuc: nat,Uud: list(A)] : aa(list(A),assn,aa(nat,fun(list(A),assn),aa(B,fun(nat,fun(list(A),assn)),aa(fun(B,fun(A,assn)),fun(B,fun(nat,fun(list(A),assn))),aTP_Lamp_oh(assn,fun(fun(B,fun(A,assn)),fun(B,fun(nat,fun(list(A),assn)))),Uu),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu),aa(list(A),assn,vEBT_List_list_assn(B,A,Uua,replicate(B,Uuc,Uub)),Uud)) ).

% ATP.lambda_889
tff(fact_9067_ATP_Olambda__890,axiom,
    ! [A: $tType,B: $tType,Uu: 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_qz(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),Uu),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uud),aa(B,assn,aa(A,fun(B,assn),Uu,aa(nat,A,nth(A,Uua),Uuc)),aa(nat,B,nth(B,Uub),Uuc))) ).

% ATP.lambda_890
tff(fact_9068_ATP_Olambda__891,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat] :
      aa(nat,vEBT_VEBT,aa(nat,fun(nat,vEBT_VEBT),aa(nat,fun(nat,fun(nat,vEBT_VEBT)),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT)))),aTP_Lamp_na(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uud)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uue),Uuc) ),
        vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uud),Uue)),Uu,Uua,Uub),
        $ite(
          ( ( Uuc = Uud )
          & ( Uuc = Uue ) ),
          vEBT_Node(none(product_prod(nat,nat)),Uu,Uua,Uub),
          $let(
            xn: nat,
            xn:= 
              $ite(Uuc = Uud,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Uub))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),the2(nat,vEBT_vebt_mint(Uub)))))),Uuc),
            $let(
              minn: nat,
              minn:= 
                $ite(Uuc = Uud,xn,Uud),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uua)),
                  $let(
                    newnode: vEBT_VEBT,
                    newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                    $let(
                      newlist: list(vEBT_VEBT),
                      newlist:= list_update(vEBT_VEBT,Uua,h,newnode),
                      $ite(
                        vEBT_VEBT_minNull(newnode),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= vEBT_vebt_delete(Uub,h),
                          vEBT_Node(some(product_prod(nat,nat),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                $ite(
                                  xn = Uue,
                                  $let(
                                    maxs: option(nat),
                                    maxs:= vEBT_vebt_maxt(sn),
                                    $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) ),
                                  Uue ))),Uu,newlist,sn) ),
                        vEBT_Node(some(product_prod(nat,nat),
                            aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                              $ite(xn = Uue,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Uue))),Uu,newlist,Uub) ) ) ),
                  vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uud),Uue)),Uu,Uua,Uub) ) ) ) ) ) ) ).

% ATP.lambda_891
tff(fact_9069_ATP_Olambda__892,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat] :
      aa(nat,option(nat),aa(nat,fun(nat,option(nat)),aa(nat,fun(nat,fun(nat,option(nat))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat))))),aTP_Lamp_nt(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uud),
        some(nat,Uud),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Uuc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Uuc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uua)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),h)),
                $ite(
                  ( ( maxlow != none(nat) )
                  & vEBT_VEBT_less(some(nat,l),maxlow) ),
                  aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),some(nat,h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),h),l)),
                  $let(
                    sc: option(nat),
                    sc:= vEBT_vebt_succ(Uub,h),
                    $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,some(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),the2(nat,sc))))) ) ) ),
              none(nat) ) ) ) ) ).

% ATP.lambda_892
tff(fact_9070_ATP_Olambda__893,axiom,
    ! [Uu: option(product_prod(nat,nat)),Uua: nat,Uub: list(vEBT_VEBT),Uuc: vEBT_VEBT,Uud: nat,Uue: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),vEBT_VEBT,aa(nat,fun(product_prod(nat,nat),vEBT_VEBT),aa(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT)))),aTP_Lamp_nb(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),one_one(nat)),vEBT_Node(Uu,Uua,Uub,Uuc),aa(product_prod(nat,nat),vEBT_VEBT,product_case_prod(nat,nat,vEBT_VEBT,aa(nat,fun(nat,fun(nat,vEBT_VEBT)),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT)))),aTP_Lamp_na(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))))),Uua),Uub),Uuc),Uud)),Uue)) ).

% ATP.lambda_893
tff(fact_9071_ATP_Olambda__894,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat] :
      aa(nat,vEBT_VEBT,aa(nat,fun(nat,vEBT_VEBT),aa(nat,fun(nat,fun(nat,vEBT_VEBT)),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT)))),aTP_Lamp_nj(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uud),Uud,Uuc),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uua))
            & ~ ( ( Uuc = Uud )
                | ( Uuc = Uue ) ) ),
            vEBT_Node(some(product_prod(nat,nat),
                aa(nat,product_prod(nat,nat),
                  aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uud),Uuc,Uud)),
                  aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Uue))),Uu,list_update(vEBT_VEBT,Uua,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),h)),vEBT_vebt_insert(Uub,h),Uub)),
            vEBT_Node(some(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uud),Uue)),Uu,Uua,Uub) ) ) ) ).

% ATP.lambda_894
tff(fact_9072_ATP_Olambda__895,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B,Uue: A] : aa(A,C,aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_zs(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(B,C,aa(A,fun(B,C),Uub,Uue),Uud)),groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_zq(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uu),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uue),bot_bot(set(A)))))) ) ).

% ATP.lambda_895
tff(fact_9073_ATP_Olambda__896,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_zc(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uud,Uue)))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uuc,Uub)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_896
tff(fact_9074_ATP_Olambda__897,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,real),Uud: fun(A,real),Uue: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_zv(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(A,real,Uu,Uub),aa(A,real,Uuc,Uub))),aa(real,real,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,Uu,Uub)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uue)),aa(A,real,Uuc,Uub))),aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_897
tff(fact_9075_ATP_Olambda__898,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_yg(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_898
tff(fact_9076_ATP_Olambda__899,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_yi(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_899
tff(fact_9077_ATP_Olambda__900,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_zp(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))),aa(A,B,Uud,Uue))),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_900
tff(fact_9078_ATP_Olambda__901,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] : aa(vEBT_VEBTi,A,aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,A),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A))),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),aTP_Lamp_at(fun(B,A),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,A,Uu,aa(vEBT_VEBTi,B,aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,B),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B))),Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_901
tff(fact_9079_ATP_Olambda__902,axiom,
    ! [A: $tType,B: $tType,Uu: 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_rb(fun(A,fun(B,assn)),fun(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))))),Uu),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),Uu,aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Uua),Uub)),Uue)),aa(nat,B,nth(B,aa(list(B),list(B),cons(B,Uuc),Uud)),Uue))) ).

% ATP.lambda_902
tff(fact_9080_ATP_Olambda__903,axiom,
    ! [B: $tType,A: $tType,Uu: $o,Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ss($o,fun(A,fun(B,$o)),(Uu)),Uua),Uub)
    <=> (Uu) ) ).

% ATP.lambda_903
tff(fact_9081_ATP_Olambda__904,axiom,
    ! [A: $tType,Uu: assn,Uua: A] : aa(A,assn,aTP_Lamp_rz(assn,fun(A,assn),Uu),Uua) = Uu ).

% ATP.lambda_904
tff(fact_9082_ATP_Olambda__905,axiom,
    ! [Uu: $o,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_tk($o,fun(nat,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_905
tff(fact_9083_ATP_Olambda__906,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: $o,Uua: A] :
          ( aa(A,$o,aTP_Lamp_aii($o,fun(A,$o),(Uu)),Uua)
        <=> (Uu) ) ) ).

% ATP.lambda_906
tff(fact_9084_ATP_Olambda__907,axiom,
    ! [A: $tType,Uu: $o,Uua: A] :
      ( aa(A,$o,aTP_Lamp_sr($o,fun(A,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_907
tff(fact_9085_ATP_Olambda__908,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_yp(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_908
tff(fact_9086_ATP_Olambda__909,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_adp(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_909
tff(fact_9087_ATP_Olambda__910,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_aak(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_910
tff(fact_9088_ATP_Olambda__911,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo8386298272705272623_space(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_abk(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_911
tff(fact_9089_ATP_Olambda__912,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_abt(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_912
tff(fact_9090_ATP_Olambda__913,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_mf(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_913
tff(fact_9091_ATP_Olambda__914,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ml(B,fun(A,B),Uu),Uua) = Uu ).

% ATP.lambda_914
tff(fact_9092_ATP_Olambda__915,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_bn(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_915
tff(fact_9093_ATP_Olambda__916,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_wu(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_916
tff(fact_9094_ATP_Olambda__917,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_afp(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_917
tff(fact_9095_ATP_Olambda__918,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_abi(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_918
tff(fact_9096_ATP_Olambda__919,axiom,
    ! [B: $tType,A: $tType] :
      ( ( zero(A)
        & topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_abf(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_919
tff(fact_9097_ATP_Olambda__920,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_sq(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_920
tff(fact_9098_ATP_Olambda__921,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_sn(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_921
tff(fact_9099_ATP_Olambda__922,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aTP_Lamp_mm(A,fun(B,A),Uu),Uua) = Uu ).

% ATP.lambda_922
tff(fact_9100_ATP_Olambda__923,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_nm(A,fun(B,$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_923
tff(fact_9101_ATP_Olambda__924,axiom,
    ! [Uu: complex] : aa(complex,complex,aTP_Lamp_gb(complex,complex),Uu) = Uu ).

% ATP.lambda_924
tff(fact_9102_ATP_Olambda__925,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: word(A)] : aa(word(A),word(A),aTP_Lamp_up(word(A),word(A)),Uu) = Uu ) ).

% ATP.lambda_925
tff(fact_9103_ATP_Olambda__926,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_fj(nat,nat),Uu) = Uu ).

% ATP.lambda_926
tff(fact_9104_ATP_Olambda__927,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_gt(int,int),Uu) = Uu ).

% ATP.lambda_927
tff(fact_9105_ATP_Olambda__928,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_uc(B,B),Uu) = Uu ) ).

% ATP.lambda_928
tff(fact_9106_ATP_Olambda__929,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_ud(B,B),Uu) = Uu ) ).

% ATP.lambda_929
tff(fact_9107_ATP_Olambda__930,axiom,
    ! [B: $tType,Uu: B] : aa(B,B,aTP_Lamp_vl(B,B),Uu) = Uu ).

% ATP.lambda_930
tff(fact_9108_ATP_Olambda__931,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_yu(A,A),Uu) = Uu ) ).

% ATP.lambda_931
tff(fact_9109_ATP_Olambda__932,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_aal(A,A),Uu) = Uu ) ).

% ATP.lambda_932
tff(fact_9110_ATP_Olambda__933,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_wv(A,A),Uu) = Uu ) ).

% ATP.lambda_933
tff(fact_9111_ATP_Olambda__934,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_abp(A,A),Uu) = Uu ) ).

% ATP.lambda_934
tff(fact_9112_ATP_Olambda__935,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ail(A,A),Uu) = Uu ) ).

% ATP.lambda_935
tff(fact_9113_ATP_Olambda__936,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_tr(A,A),Uu) = Uu ) ).

% ATP.lambda_936
tff(fact_9114_ATP_Olambda__937,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_aj(A,A),Uu) = Uu ) ).

% ATP.lambda_937
tff(fact_9115_ATP_Olambda__938,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_ma(A,A),Uu) = Uu ).

% ATP.lambda_938
tff(fact_9116_ATP_Olambda__939,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_wd(C,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_939
tff(fact_9117_ATP_Olambda__940,axiom,
    ! [A: $tType,D: $tType,Uu: A] : aa(A,set(D),aTP_Lamp_we(A,set(D)),Uu) = bot_bot(set(D)) ).

% ATP.lambda_940
tff(fact_9118_ATP_Olambda__941,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_bk(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_941
tff(fact_9119_ATP_Olambda__942,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_ce(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_942
tff(fact_9120_ATP_Olambda__943,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_dp(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_943
tff(fact_9121_ATP_Olambda__944,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_ye(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_944
tff(fact_9122_ATP_Olambda__945,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ai(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_945
tff(fact_9123_ATP_Olambda__946,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_rn(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_946
tff(fact_9124_ATP_Olambda__947,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_gx(B,A),Uu) = one_one(A) ) ).

% ATP.lambda_947
tff(fact_9125_ATP_Olambda__948,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_pv(B,option(A)),Uu) = none(A) ).

% ATP.lambda_948
tff(fact_9126_ATP_Olambda__949,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_bd(A,option(B)),Uu) = none(B) ).

% ATP.lambda_949
tff(fact_9127_ATP_Olambda__950,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_ow(real,$o),Uu)
    <=> $false ) ).

% ATP.lambda_950
tff(fact_9128_ATP_Olambda__951,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_th(nat,$o),Uu)
    <=> $false ) ).

% ATP.lambda_951
tff(fact_9129_ATP_Olambda__952,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_ab(A,$o),Uu)
    <=> $false ) ).

% ATP.lambda_952
tff(fact_9130_ATP_Olambda__953,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_ti(nat,$o),Uu)
    <=> $true ) ).

% ATP.lambda_953
tff(fact_9131_ATP_Olambda__954,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_rl(A,$o),Uu)
    <=> $true ) ).

% ATP.lambda_954
tff(fact_9132_ATP_Olambda__955,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_mz(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_955

% Type constructors (1278)
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_1,axiom,
    ! [A12: $tType,A13: $tType] :
      ( ( finite_finite(A12)
        & finite_finite(A13) )
     => finite_finite(product_prod(A12,A13)) ) ).

tff(tcon_Numeral__Type_Onum1___Finite__Set_Ofinite_2,axiom,
    finite_finite(numeral_num1) ).

tff(tcon_Numeral__Type_Obit1___Finite__Set_Ofinite_3,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => finite_finite(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Finite__Set_Ofinite_4,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => finite_finite(numeral_bit0(A12)) ) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder_5,axiom,
    comple5582772986160207858norder(extended_enat) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder_6,axiom,
    ! [A12: $tType] :
      ( comple5582772986160207858norder(A12)
     => comple5582772986160207858norder(option(A12)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_7,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => finite_finite(option(A12)) ) ).

tff(tcon_Enum_Ofinite__3___Complete__Lattices_Ocomplete__linorder_8,axiom,
    comple5582772986160207858norder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Finite__Set_Ofinite_9,axiom,
    finite_finite(finite_3) ).

tff(tcon_Enum_Ofinite__2___Complete__Lattices_Ocomplete__linorder_10,axiom,
    comple5582772986160207858norder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Finite__Set_Ofinite_11,axiom,
    finite_finite(finite_2) ).

tff(tcon_Word_Oword___Finite__Set_Ofinite_12,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => finite_finite(word(A12)) ) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_13,axiom,
    finite_finite($o) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_14,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => finite_finite(set(A12)) ) ).

tff(tcon_fun___Finite__Set_Ofinite_15,axiom,
    ! [A12: $tType,A13: $tType] :
      ( ( finite_finite(A12)
        & finite_finite(A13) )
     => finite_finite(fun(A12,A13)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A12: $tType,A13: $tType] :
      ( bounded_lattice(A13)
     => bounde4967611905675639751up_bot(fun(A12,A13)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A12: $tType,A13: $tType] :
      ( bounded_lattice(A13)
     => bounde4346867609351753570nf_top(fun(A12,A13)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A12: $tType,A13: $tType] :
      ( boolea8198339166811842893lgebra(A13)
     => boolea8198339166811842893lgebra(fun(A12,A13)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__top,axiom,
    ! [A12: $tType,A13: $tType] :
      ( bounded_lattice(A13)
     => bounded_lattice_top(fun(A12,A13)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
    ! [A12: $tType,A13: $tType] :
      ( bounded_lattice(A13)
     => bounded_lattice_bot(fun(A12,A13)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A12: $tType,A13: $tType] :
      ( semilattice_sup(A13)
     => semilattice_sup(fun(A12,A13)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A12: $tType,A13: $tType] :
      ( semilattice_inf(A13)
     => semilattice_inf(fun(A12,A13)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A12: $tType,A13: $tType] :
      ( distrib_lattice(A13)
     => distrib_lattice(fun(A12,A13)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice,axiom,
    ! [A12: $tType,A13: $tType] :
      ( bounded_lattice(A13)
     => bounded_lattice(fun(A12,A13)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A12: $tType,A13: $tType] :
      ( order_top(A13)
     => order_top(fun(A12,A13)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A12: $tType,A13: $tType] :
      ( order_bot(A13)
     => order_bot(fun(A12,A13)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A12: $tType,A13: $tType] :
      ( preorder(A13)
     => preorder(fun(A12,A13)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A12: $tType,A13: $tType] :
      ( lattice(A13)
     => lattice(fun(A12,A13)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A12: $tType,A13: $tType] :
      ( order(A13)
     => order(fun(A12,A13)) ) ).

tff(tcon_fun___Orderings_Otop,axiom,
    ! [A12: $tType,A13: $tType] :
      ( top(A13)
     => top(fun(A12,A13)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A12: $tType,A13: $tType] :
      ( ord(A13)
     => ord(fun(A12,A13)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A12: $tType,A13: $tType] :
      ( bot(A13)
     => bot(fun(A12,A13)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A12: $tType,A13: $tType] :
      ( uminus(A13)
     => uminus(fun(A12,A13)) ) ).

tff(tcon_fun___Lattices_Osup,axiom,
    ! [A12: $tType,A13: $tType] :
      ( semilattice_sup(A13)
     => sup(fun(A12,A13)) ) ).

tff(tcon_fun___Lattices_Oinf,axiom,
    ! [A12: $tType,A13: $tType] :
      ( semilattice_inf(A13)
     => inf(fun(A12,A13)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A12: $tType,A13: $tType] :
      ( minus(A13)
     => minus(fun(A12,A13)) ) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
    bit_un5681908812861735899ations(int) ).

tff(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
    semiri1453513574482234551roduct(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
    euclid5411537665997757685th_nat(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
    euclid8789492081693882211th_nat(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le(int) ).

tff(tcon_Int_Oint___Euclidean__Division_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___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___Groups_Olinordered__ab__semigroup__add,axiom,
    linord4140545234300271783up_add(int) ).

tff(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
    bit_ri3973907225187159222ations(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
    topolo2564578578187576103pology(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
    semiri2026040879449505780visors(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
    linord181362715937106298miring(int) ).

tff(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
    topolo4211221413907600880p_mult(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring,axiom,
    euclid5891614535332579305n_ring(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_16,axiom,
    semilattice_sup(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__inf_17,axiom,
    semilattice_inf(int) ).

tff(tcon_Int_Oint___Lattices_Odistrib__lattice_18,axiom,
    distrib_lattice(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
    semiring_1_cancel(int) ).

tff(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
    algebraic_semidom(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
    ordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(int) ).

tff(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
    semiring_parity(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
    semiring_modulo(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
    linordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
    linordered_idom(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
    semigroup_mult(int) ).

tff(tcon_Int_Oint___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_19,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_20,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_21,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_22,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_23,axiom,
    uminus(int) ).

tff(tcon_Int_Oint___Rings_Oring__1,axiom,
    ring_1(int) ).

tff(tcon_Int_Oint___Rings_Oabs__if,axiom,
    abs_if(int) ).

tff(tcon_Int_Oint___Lattices_Osup_24,axiom,
    sup(int) ).

tff(tcon_Int_Oint___Lattices_Oinf_25,axiom,
    inf(int) ).

tff(tcon_Int_Oint___Groups_Ominus_26,axiom,
    minus(int) ).

tff(tcon_Int_Oint___Power_Opower,axiom,
    power(int) ).

tff(tcon_Int_Oint___Num_Onumeral,axiom,
    numeral(int) ).

tff(tcon_Int_Oint___Groups_Ozero,axiom,
    zero(int) ).

tff(tcon_Int_Oint___Groups_Oplus,axiom,
    plus(int) ).

tff(tcon_Int_Oint___Rings_Oring,axiom,
    ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom,axiom,
    idom(int) ).

tff(tcon_Int_Oint___Groups_Oone,axiom,
    one(int) ).

tff(tcon_Int_Oint___Rings_Odvd,axiom,
    dvd(int) ).

tff(tcon_Int_Oint___Heap_Oheap,axiom,
    heap(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_27,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_28,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_29,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_30,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_31,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_32,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_33,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_34,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_35,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_36,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_37,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_38,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_39,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_40,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_41,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_42,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_43,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_44,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_45,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_46,axiom,
    topolo8865339358273720382pology(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_47,axiom,
    topolo4987421752381908075d_mult(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_48,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_49,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_50,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_51,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_52,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_53,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_54,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_55,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_56,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__mult_57,axiom,
    topolo1898628316856586783d_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_58,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_59,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_60,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_61,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_62,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_63,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_64,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot1__space_65,axiom,
    topological_t1_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_66,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_67,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_68,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_69,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_70,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_71,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_72,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_73,axiom,
    distrib_lattice(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_74,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_75,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_76,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_77,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_78,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_79,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_80,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_81,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_82,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_83,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_84,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_85,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_86,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_87,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_88,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_89,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_90,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_91,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_92,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_93,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_94,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_95,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_96,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_97,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_98,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_99,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_100,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_101,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_102,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_103,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_104,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_105,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_106,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_107,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Orderings_Obot_108,axiom,
    bot(nat) ).

tff(tcon_Nat_Onat___Lattices_Osup_109,axiom,
    sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Oinf_110,axiom,
    inf(nat) ).

tff(tcon_Nat_Onat___Groups_Ominus_111,axiom,
    minus(nat) ).

tff(tcon_Nat_Onat___Power_Opower_112,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_113,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_114,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_115,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_116,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_117,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Heap_Oheap_118,axiom,
    heap(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_119,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_120,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_121,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_122,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_123,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_124,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_125,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_126,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_127,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_128,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_129,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_130,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_131,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_132,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_133,axiom,
    ordere8940638589300402666id_add(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
    archim462609752435547400_field(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_134,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_135,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_136,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_137,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_138,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_139,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_140,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_141,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_142,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_143,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_144,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_145,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_146,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_147,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_148,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_149,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_150,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_151,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_152,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_153,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_154,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_155,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_156,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_157,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_158,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_159,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_160,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_161,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_162,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_163,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_164,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_165,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_166,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_167,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_168,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_169,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_170,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_171,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_172,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_173,axiom,
    semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield__abs__sgn,axiom,
    field_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_174,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_175,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_176,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_177,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_178,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_179,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_180,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_181,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_182,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_183,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_184,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_185,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_186,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_187,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_188,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_189,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_190,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_191,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_192,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_193,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_194,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_195,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_196,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_197,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_198,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_199,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_200,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_201,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_202,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Lattices_Osup_203,axiom,
    sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Oinf_204,axiom,
    inf(rat) ).

tff(tcon_Rat_Orat___Groups_Ominus_205,axiom,
    minus(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_206,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_207,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_208,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_209,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_210,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_211,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_212,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_213,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_214,axiom,
    ! [A12: $tType] : bounde4967611905675639751up_bot(set(A12)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_215,axiom,
    ! [A12: $tType] : bounde4346867609351753570nf_top(set(A12)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_216,axiom,
    ! [A12: $tType] : boolea8198339166811842893lgebra(set(A12)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_217,axiom,
    ! [A12: $tType] : bounded_lattice_top(set(A12)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_218,axiom,
    ! [A12: $tType] : bounded_lattice_bot(set(A12)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_219,axiom,
    ! [A12: $tType] : semilattice_sup(set(A12)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_220,axiom,
    ! [A12: $tType] : semilattice_inf(set(A12)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_221,axiom,
    ! [A12: $tType] : distrib_lattice(set(A12)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_222,axiom,
    ! [A12: $tType] : bounded_lattice(set(A12)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_223,axiom,
    ! [A12: $tType] : order_top(set(A12)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_224,axiom,
    ! [A12: $tType] : order_bot(set(A12)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_225,axiom,
    ! [A12: $tType] : preorder(set(A12)) ).

tff(tcon_Set_Oset___Lattices_Olattice_226,axiom,
    ! [A12: $tType] : lattice(set(A12)) ).

tff(tcon_Set_Oset___Orderings_Oorder_227,axiom,
    ! [A12: $tType] : order(set(A12)) ).

tff(tcon_Set_Oset___Orderings_Otop_228,axiom,
    ! [A12: $tType] : top(set(A12)) ).

tff(tcon_Set_Oset___Orderings_Oord_229,axiom,
    ! [A12: $tType] : ord(set(A12)) ).

tff(tcon_Set_Oset___Orderings_Obot_230,axiom,
    ! [A12: $tType] : bot(set(A12)) ).

tff(tcon_Set_Oset___Groups_Ouminus_231,axiom,
    ! [A12: $tType] : uminus(set(A12)) ).

tff(tcon_Set_Oset___Lattices_Osup_232,axiom,
    ! [A12: $tType] : sup(set(A12)) ).

tff(tcon_Set_Oset___Lattices_Oinf_233,axiom,
    ! [A12: $tType] : inf(set(A12)) ).

tff(tcon_Set_Oset___Groups_Ominus_234,axiom,
    ! [A12: $tType] : minus(set(A12)) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_235,axiom,
    topolo4958980785337419405_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_236,axiom,
    topolo1944317154257567458pology($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_237,axiom,
    topolo8865339358273720382pology($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_238,axiom,
    bounde4967611905675639751up_bot($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_239,axiom,
    bounde4346867609351753570nf_top($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_240,axiom,
    topolo2564578578187576103pology($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_241,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_242,axiom,
    bounded_lattice_top($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_243,axiom,
    bounded_lattice_bot($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_244,axiom,
    topological_t2_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot1__space_245,axiom,
    topological_t1_space($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_246,axiom,
    semilattice_sup($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_247,axiom,
    semilattice_inf($o) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_248,axiom,
    distrib_lattice($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_249,axiom,
    bounded_lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_250,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_251,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_252,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_253,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Lattices_Olattice_254,axiom,
    lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_255,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Otop_256,axiom,
    top($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_257,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Orderings_Obot_258,axiom,
    bot($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_259,axiom,
    uminus($o) ).

tff(tcon_HOL_Obool___Lattices_Osup_260,axiom,
    sup($o) ).

tff(tcon_HOL_Obool___Lattices_Oinf_261,axiom,
    inf($o) ).

tff(tcon_HOL_Obool___Groups_Ominus_262,axiom,
    minus($o) ).

tff(tcon_HOL_Obool___Heap_Oheap_263,axiom,
    heap($o) ).

tff(tcon_List_Olist___Heap_Oheap_264,axiom,
    ! [A12: $tType] :
      ( heap(A12)
     => heap(list(A12)) ) ).

tff(tcon_List_Olist___Nat_Osize_265,axiom,
    ! [A12: $tType] : size(list(A12)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_266,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_267,axiom,
    semiri1453513574482234551roduct(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_268,axiom,
    ordere1937475149494474687imp_le(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_269,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_270,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_271,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_272,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_273,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_274,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_275,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_276,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_277,axiom,
    topolo1944317154257567458pology(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
    real_V3459762299906320749_field(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
    real_V5047593784448816457lgebra(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_278,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_279,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_280,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_281,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_282,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_283,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_284,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_285,axiom,
    linord181362715937106298miring(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra__1,axiom,
    real_V2191834092415804123ebra_1(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__semigroup__mult_286,axiom,
    topolo4211221413907600880p_mult(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
    topolo8386298272705272623_space(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_287,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_288,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_289,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_290,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_291,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_292,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_293,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_294,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_295,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_296,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_297,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_298,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_299,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
    topolo1633459387980952147up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_300,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot1__space_301,axiom,
    topological_t1_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_302,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_303,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_304,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_305,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_306,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_307,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_308,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_309,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_310,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__inf_311,axiom,
    semilattice_inf(real) ).

tff(tcon_Real_Oreal___Lattices_Odistrib__lattice_312,axiom,
    distrib_lattice(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_313,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_314,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_315,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_316,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_317,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_318,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_319,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_320,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_321,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_322,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_323,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_324,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_325,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_326,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_327,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_328,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_329,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__abs__sgn_330,axiom,
    field_abs_sgn(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_331,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_332,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_333,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_334,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_335,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_336,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_337,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_338,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_339,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_340,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_341,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_342,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_343,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_344,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_345,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_346,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_347,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_348,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_349,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_350,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_351,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_352,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_353,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_354,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_355,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_356,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_357,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_358,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_359,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_360,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_361,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_362,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Lattices_Osup_363,axiom,
    sup(real) ).

tff(tcon_Real_Oreal___Lattices_Oinf_364,axiom,
    inf(real) ).

tff(tcon_Real_Oreal___Groups_Ominus_365,axiom,
    minus(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_366,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_367,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_368,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_369,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_370,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_371,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_372,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_373,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_374,axiom,
    dvd(real) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bit__operations_375,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => bit_se359711467146920520ations(word(A12)) ) ).

tff(tcon_Word_Oword___Bit__Comprehension_Obit__comprehension_376,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => bit_bi6583157726757044596ension(word(A12)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Oring__bit__operations_377,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => bit_ri3973907225187159222ations(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__ab__semigroup__add_378,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => cancel2418104881723323429up_add(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__comm__monoid__add_379,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => cancel1802427076303600483id_add(word(A12)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1__cancel_380,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => comm_s4317794764714335236cancel(word(A12)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bits_381,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => bit_semiring_bits(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__semigroup__add_382,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => cancel_semigroup_add(word(A12)) ) ).

tff(tcon_Word_Oword___Least__significant__bit_Olsb_383,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => least_6119777620449941438nt_lsb(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__mult_384,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => ab_semigroup_mult(word(A12)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1__cancel_385,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => semiring_1_cancel(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__mult_386,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => comm_monoid_mult(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__add_387,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => ab_semigroup_add(word(A12)) ) ).

tff(tcon_Word_Oword___Parity_Osemiring__parity_388,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => semiring_parity(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__add_389,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => comm_monoid_add(word(A12)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__modulo_390,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => semiring_modulo(word(A12)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1_391,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => comm_semiring_1(word(A12)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__0_392,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => comm_semiring_0(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Osemigroup__mult_393,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => semigroup_mult(word(A12)) ) ).

tff(tcon_Word_Oword___Num_Osemiring__numeral_394,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => semiring_numeral(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Osemigroup__add_395,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => semigroup_add(word(A12)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring_396,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => comm_semiring(word(A12)) ) ).

tff(tcon_Word_Oword___Orderings_Owellorder_397,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => wellorder(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Oab__group__add_398,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => ab_group_add(word(A12)) ) ).

tff(tcon_Word_Oword___Rings_Ozero__neq__one_399,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => zero_neq_one(word(A12)) ) ).

tff(tcon_Word_Oword___Parity_Oring__parity_400,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => ring_parity(word(A12)) ) ).

tff(tcon_Word_Oword___Orderings_Opreorder_401,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => preorder(word(A12)) ) ).

tff(tcon_Word_Oword___Orderings_Olinorder_402,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => linorder(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__mult_403,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => monoid_mult(word(A12)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring__1_404,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => comm_ring_1(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__add_405,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => monoid_add(word(A12)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1_406,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => semiring_1(word(A12)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__0_407,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => semiring_0(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Ogroup__add_408,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => group_add(word(A12)) ) ).

tff(tcon_Word_Oword___Rings_Omult__zero_409,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => mult_zero(word(A12)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring_410,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => comm_ring(word(A12)) ) ).

tff(tcon_Word_Oword___Orderings_Oorder_411,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => order(word(A12)) ) ).

tff(tcon_Word_Oword___Num_Oneg__numeral_412,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => neg_numeral(word(A12)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring_413,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => semiring(word(A12)) ) ).

tff(tcon_Word_Oword___Orderings_Oord_414,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => ord(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Ouminus_415,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => uminus(word(A12)) ) ).

tff(tcon_Word_Oword___Rings_Oring__1_416,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => ring_1(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Ominus_417,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => minus(word(A12)) ) ).

tff(tcon_Word_Oword___Power_Opower_418,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => power(word(A12)) ) ).

tff(tcon_Word_Oword___Num_Onumeral_419,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => numeral(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Ozero_420,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => zero(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Oplus_421,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => plus(word(A12)) ) ).

tff(tcon_Word_Oword___Rings_Oring_422,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => ring(word(A12)) ) ).

tff(tcon_Word_Oword___Groups_Oone_423,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => one(word(A12)) ) ).

tff(tcon_Word_Oword___Rings_Odvd_424,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => dvd(word(A12)) ) ).

tff(tcon_Word_Oword___Nat_Osize_425,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => size(word(A12)) ) ).

tff(tcon_Heap_Oarray___Heap_Oheap_426,axiom,
    ! [A12: $tType] : heap(array(A12)) ).

tff(tcon_Heap_Oarray___Nat_Osize_427,axiom,
    ! [A12: $tType] : size(array(A12)) ).

tff(tcon_Enum_Ofinite__2___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_428,axiom,
    condit6923001295902523014norder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_429,axiom,
    semiri1453513574482234551roduct(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Ounique__euclidean__semiring_430,axiom,
    euclid3128863361964157862miring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring__cancel_431,axiom,
    euclid4440199948858584721cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors__cancel_432,axiom,
    semiri6575147826004484403cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring_433,axiom,
    euclid3725896446679973847miring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Obounded__semilattice__sup__bot_434,axiom,
    bounde4967611905675639751up_bot(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Obounded__semilattice__inf__top_435,axiom,
    bounde4346867609351753570nf_top(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1__no__zero__divisors_436,axiom,
    semiri2026040879449505780visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors_437,axiom,
    semiri3467727345109120633visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__ab__semigroup__add_438,axiom,
    cancel2418104881723323429up_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring__1__no__zero__divisors_439,axiom,
    ring_15535105094025558882visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__comm__monoid__add_440,axiom,
    cancel1802427076303600483id_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1__cancel_441,axiom,
    comm_s4317794764714335236cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Obounded__lattice__top_442,axiom,
    bounded_lattice_top(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Obounded__lattice__bot_443,axiom,
    bounded_lattice_bot(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__semigroup__add_444,axiom,
    cancel_semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Osemilattice__sup_445,axiom,
    semilattice_sup(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Osemilattice__inf_446,axiom,
    semilattice_inf(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Odistrib__lattice_447,axiom,
    distrib_lattice(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Obounded__lattice_448,axiom,
    bounded_lattice(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__mult_449,axiom,
    ab_semigroup_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1__cancel_450,axiom,
    semiring_1_cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oalgebraic__semidom_451,axiom,
    algebraic_semidom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__mult_452,axiom,
    comm_monoid_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__add_453,axiom,
    ab_semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__add_454,axiom,
    comm_monoid_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__modulo_455,axiom,
    semiring_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1_456,axiom,
    comm_semiring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__0_457,axiom,
    comm_semiring_0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Osemigroup__mult_458,axiom,
    semigroup_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom__modulo_459,axiom,
    semidom_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom__divide_460,axiom,
    semidom_divide(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Osemiring__numeral_461,axiom,
    semiring_numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Osemigroup__add_462,axiom,
    semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Odivision__ring_463,axiom,
    division_ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring_464,axiom,
    comm_semiring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Owellorder_465,axiom,
    wellorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder__top_466,axiom,
    order_top(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder__bot_467,axiom,
    order_bot(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__group__add_468,axiom,
    ab_group_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ozero__neq__one_469,axiom,
    zero_neq_one(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__abs__sgn_470,axiom,
    idom_abs_sgn(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Opreorder_471,axiom,
    preorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Olinorder_472,axiom,
    linorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Omonoid__mult_473,axiom,
    monoid_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__modulo_474,axiom,
    idom_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__divide_475,axiom,
    idom_divide(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__ring__1_476,axiom,
    comm_ring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Omonoid__add_477,axiom,
    monoid_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Type__Length_Olen0,axiom,
    type_len0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1_478,axiom,
    semiring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__0_479,axiom,
    semiring_0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Olattice_480,axiom,
    lattice(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ogroup__add_481,axiom,
    group_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Type__Length_Olen,axiom,
    type_len(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Omult__zero_482,axiom,
    mult_zero(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__ring_483,axiom,
    comm_ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder_484,axiom,
    order(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Oneg__numeral_485,axiom,
    neg_numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring_486,axiom,
    semiring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Oinverse_487,axiom,
    inverse(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Otop_488,axiom,
    top(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oord_489,axiom,
    ord(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Obot_490,axiom,
    bot(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ouminus_491,axiom,
    uminus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring__1_492,axiom,
    ring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Osup_493,axiom,
    sup(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Oinf_494,axiom,
    inf(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ominus_495,axiom,
    minus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Ofield_496,axiom,
    field(finite_2) ).

tff(tcon_Enum_Ofinite__2___Power_Opower_497,axiom,
    power(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Onumeral_498,axiom,
    numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ozero_499,axiom,
    zero(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oplus_500,axiom,
    plus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring_501,axiom,
    ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom_502,axiom,
    idom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oone_503,axiom,
    one(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Odvd_504,axiom,
    dvd(finite_2) ).

tff(tcon_Enum_Ofinite__3___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_505,axiom,
    condit6923001295902523014norder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_506,axiom,
    semiri1453513574482234551roduct(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Ounique__euclidean__semiring_507,axiom,
    euclid3128863361964157862miring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring__cancel_508,axiom,
    euclid4440199948858584721cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors__cancel_509,axiom,
    semiri6575147826004484403cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring_510,axiom,
    euclid3725896446679973847miring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Obounded__semilattice__sup__bot_511,axiom,
    bounde4967611905675639751up_bot(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Obounded__semilattice__inf__top_512,axiom,
    bounde4346867609351753570nf_top(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1__no__zero__divisors_513,axiom,
    semiri2026040879449505780visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors_514,axiom,
    semiri3467727345109120633visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__ab__semigroup__add_515,axiom,
    cancel2418104881723323429up_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring__1__no__zero__divisors_516,axiom,
    ring_15535105094025558882visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__comm__monoid__add_517,axiom,
    cancel1802427076303600483id_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1__cancel_518,axiom,
    comm_s4317794764714335236cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Obounded__lattice__top_519,axiom,
    bounded_lattice_top(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Obounded__lattice__bot_520,axiom,
    bounded_lattice_bot(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__semigroup__add_521,axiom,
    cancel_semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Osemilattice__sup_522,axiom,
    semilattice_sup(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Osemilattice__inf_523,axiom,
    semilattice_inf(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Odistrib__lattice_524,axiom,
    distrib_lattice(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Obounded__lattice_525,axiom,
    bounded_lattice(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__mult_526,axiom,
    ab_semigroup_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1__cancel_527,axiom,
    semiring_1_cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oalgebraic__semidom_528,axiom,
    algebraic_semidom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__mult_529,axiom,
    comm_monoid_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__add_530,axiom,
    ab_semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__add_531,axiom,
    comm_monoid_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__modulo_532,axiom,
    semiring_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1_533,axiom,
    comm_semiring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__0_534,axiom,
    comm_semiring_0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Osemigroup__mult_535,axiom,
    semigroup_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom__modulo_536,axiom,
    semidom_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom__divide_537,axiom,
    semidom_divide(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Osemiring__numeral_538,axiom,
    semiring_numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Osemigroup__add_539,axiom,
    semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Odivision__ring_540,axiom,
    division_ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring_541,axiom,
    comm_semiring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Owellorder_542,axiom,
    wellorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder__top_543,axiom,
    order_top(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder__bot_544,axiom,
    order_bot(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__group__add_545,axiom,
    ab_group_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ozero__neq__one_546,axiom,
    zero_neq_one(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__abs__sgn_547,axiom,
    idom_abs_sgn(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Opreorder_548,axiom,
    preorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Olinorder_549,axiom,
    linorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Omonoid__mult_550,axiom,
    monoid_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__modulo_551,axiom,
    idom_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__divide_552,axiom,
    idom_divide(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__ring__1_553,axiom,
    comm_ring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Omonoid__add_554,axiom,
    monoid_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Type__Length_Olen0_555,axiom,
    type_len0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1_556,axiom,
    semiring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__0_557,axiom,
    semiring_0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Olattice_558,axiom,
    lattice(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ogroup__add_559,axiom,
    group_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Type__Length_Olen_560,axiom,
    type_len(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Omult__zero_561,axiom,
    mult_zero(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__ring_562,axiom,
    comm_ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder_563,axiom,
    order(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Oneg__numeral_564,axiom,
    neg_numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring_565,axiom,
    semiring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Oinverse_566,axiom,
    inverse(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Otop_567,axiom,
    top(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oord_568,axiom,
    ord(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Obot_569,axiom,
    bot(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ouminus_570,axiom,
    uminus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring__1_571,axiom,
    ring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Osup_572,axiom,
    sup(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Oinf_573,axiom,
    inf(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ominus_574,axiom,
    minus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Ofield_575,axiom,
    field(finite_3) ).

tff(tcon_Enum_Ofinite__3___Power_Opower_576,axiom,
    power(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Onumeral_577,axiom,
    numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ozero_578,axiom,
    zero(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oplus_579,axiom,
    plus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring_580,axiom,
    ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom_581,axiom,
    idom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oone_582,axiom,
    one(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Odvd_583,axiom,
    dvd(finite_3) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_584,axiom,
    ! [A12: $tType] : bounde4967611905675639751up_bot(filter(A12)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_585,axiom,
    ! [A12: $tType] : bounde4346867609351753570nf_top(filter(A12)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_586,axiom,
    ! [A12: $tType] : bounded_lattice_top(filter(A12)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_587,axiom,
    ! [A12: $tType] : bounded_lattice_bot(filter(A12)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_588,axiom,
    ! [A12: $tType] : semilattice_sup(filter(A12)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_589,axiom,
    ! [A12: $tType] : semilattice_inf(filter(A12)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_590,axiom,
    ! [A12: $tType] : distrib_lattice(filter(A12)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_591,axiom,
    ! [A12: $tType] : bounded_lattice(filter(A12)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_592,axiom,
    ! [A12: $tType] : order_top(filter(A12)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_593,axiom,
    ! [A12: $tType] : order_bot(filter(A12)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_594,axiom,
    ! [A12: $tType] : preorder(filter(A12)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_595,axiom,
    ! [A12: $tType] : lattice(filter(A12)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_596,axiom,
    ! [A12: $tType] : order(filter(A12)) ).

tff(tcon_Filter_Ofilter___Orderings_Otop_597,axiom,
    ! [A12: $tType] : top(filter(A12)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_598,axiom,
    ! [A12: $tType] : ord(filter(A12)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_599,axiom,
    ! [A12: $tType] : bot(filter(A12)) ).

tff(tcon_Filter_Ofilter___Lattices_Osup_600,axiom,
    ! [A12: $tType] : sup(filter(A12)) ).

tff(tcon_Filter_Ofilter___Lattices_Oinf_601,axiom,
    ! [A12: $tType] : inf(filter(A12)) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_602,axiom,
    ! [A12: $tType] :
      ( comple5582772986160207858norder(A12)
     => condit6923001295902523014norder(option(A12)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_603,axiom,
    ! [A12: $tType] :
      ( lattice(A12)
     => bounde4967611905675639751up_bot(option(A12)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_604,axiom,
    ! [A12: $tType] :
      ( bounded_lattice_top(A12)
     => bounde4346867609351753570nf_top(option(A12)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice__top_605,axiom,
    ! [A12: $tType] :
      ( bounded_lattice_top(A12)
     => bounded_lattice_top(option(A12)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_606,axiom,
    ! [A12: $tType] :
      ( lattice(A12)
     => bounded_lattice_bot(option(A12)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__sup_607,axiom,
    ! [A12: $tType] :
      ( semilattice_sup(A12)
     => semilattice_sup(option(A12)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__inf_608,axiom,
    ! [A12: $tType] :
      ( semilattice_inf(A12)
     => semilattice_inf(option(A12)) ) ).

tff(tcon_Option_Ooption___Lattices_Odistrib__lattice_609,axiom,
    ! [A12: $tType] :
      ( distrib_lattice(A12)
     => distrib_lattice(option(A12)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice_610,axiom,
    ! [A12: $tType] :
      ( bounded_lattice_top(A12)
     => bounded_lattice(option(A12)) ) ).

tff(tcon_Option_Ooption___Orderings_Owellorder_611,axiom,
    ! [A12: $tType] :
      ( wellorder(A12)
     => wellorder(option(A12)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__top_612,axiom,
    ! [A12: $tType] :
      ( order_top(A12)
     => order_top(option(A12)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__bot_613,axiom,
    ! [A12: $tType] :
      ( order(A12)
     => order_bot(option(A12)) ) ).

tff(tcon_Option_Ooption___Orderings_Opreorder_614,axiom,
    ! [A12: $tType] :
      ( preorder(A12)
     => preorder(option(A12)) ) ).

tff(tcon_Option_Ooption___Orderings_Olinorder_615,axiom,
    ! [A12: $tType] :
      ( linorder(A12)
     => linorder(option(A12)) ) ).

tff(tcon_Option_Ooption___Lattices_Olattice_616,axiom,
    ! [A12: $tType] :
      ( lattice(A12)
     => lattice(option(A12)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder_617,axiom,
    ! [A12: $tType] :
      ( order(A12)
     => order(option(A12)) ) ).

tff(tcon_Option_Ooption___Orderings_Otop_618,axiom,
    ! [A12: $tType] :
      ( order_top(A12)
     => top(option(A12)) ) ).

tff(tcon_Option_Ooption___Orderings_Oord_619,axiom,
    ! [A12: $tType] :
      ( preorder(A12)
     => ord(option(A12)) ) ).

tff(tcon_Option_Ooption___Orderings_Obot_620,axiom,
    ! [A12: $tType] :
      ( order(A12)
     => bot(option(A12)) ) ).

tff(tcon_Option_Ooption___Lattices_Osup_621,axiom,
    ! [A12: $tType] :
      ( sup(A12)
     => sup(option(A12)) ) ).

tff(tcon_Option_Ooption___Lattices_Oinf_622,axiom,
    ! [A12: $tType] :
      ( inf(A12)
     => inf(option(A12)) ) ).

tff(tcon_Option_Ooption___Heap_Oheap_623,axiom,
    ! [A12: $tType] :
      ( heap(A12)
     => heap(option(A12)) ) ).

tff(tcon_Option_Ooption___Nat_Osize_624,axiom,
    ! [A12: $tType] : size(option(A12)) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bit__operations_625,axiom,
    bit_se359711467146920520ations(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Comprehension_Obit__comprehension_626,axiom,
    bit_bi6583157726757044596ension(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Oring__bit__operations_627,axiom,
    bit_ri3973907225187159222ations(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__ab__semigroup__add_628,axiom,
    cancel2418104881723323429up_add(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__comm__monoid__add_629,axiom,
    cancel1802427076303600483id_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1__cancel_630,axiom,
    comm_s4317794764714335236cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bits_631,axiom,
    bit_semiring_bits(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__semigroup__add_632,axiom,
    cancel_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Least__significant__bit_Olsb_633,axiom,
    least_6119777620449941438nt_lsb(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__mult_634,axiom,
    ab_semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1__cancel_635,axiom,
    semiring_1_cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__mult_636,axiom,
    comm_monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__add_637,axiom,
    ab_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Parity_Osemiring__parity_638,axiom,
    semiring_parity(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__add_639,axiom,
    comm_monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__modulo_640,axiom,
    semiring_modulo(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1_641,axiom,
    comm_semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__0_642,axiom,
    comm_semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Osemigroup__mult_643,axiom,
    semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Osemiring__numeral_644,axiom,
    semiring_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Osemigroup__add_645,axiom,
    semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring_646,axiom,
    comm_semiring(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__group__add_647,axiom,
    ab_group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ozero__neq__one_648,axiom,
    zero_neq_one(uint32) ).

tff(tcon_Uint32_Ouint32___Parity_Oring__parity_649,axiom,
    ring_parity(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Opreorder_650,axiom,
    preorder(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Olinorder_651,axiom,
    linorder(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__mult_652,axiom,
    monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring__1_653,axiom,
    comm_ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__add_654,axiom,
    monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1_655,axiom,
    semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__0_656,axiom,
    semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ogroup__add_657,axiom,
    group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Omult__zero_658,axiom,
    mult_zero(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring_659,axiom,
    comm_ring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oorder_660,axiom,
    order(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Oneg__numeral_661,axiom,
    neg_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring_662,axiom,
    semiring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oord_663,axiom,
    ord(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ouminus_664,axiom,
    uminus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring__1_665,axiom,
    ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ominus_666,axiom,
    minus(uint32) ).

tff(tcon_Uint32_Ouint32___Power_Opower_667,axiom,
    power(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Onumeral_668,axiom,
    numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ozero_669,axiom,
    zero(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oplus_670,axiom,
    plus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring_671,axiom,
    ring(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oone_672,axiom,
    one(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Odvd_673,axiom,
    dvd(uint32) ).

tff(tcon_Uint32_Ouint32___Nat_Osize_674,axiom,
    size(uint32) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_675,axiom,
    bounde4967611905675639751up_bot(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__inf__top_676,axiom,
    bounde4346867609351753570nf_top(assn) ).

tff(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_677,axiom,
    boolea8198339166811842893lgebra(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_678,axiom,
    bounded_lattice_top(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__bot_679,axiom,
    bounded_lattice_bot(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_680,axiom,
    semilattice_sup(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_681,axiom,
    semilattice_inf(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Odistrib__lattice_682,axiom,
    distrib_lattice(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice_683,axiom,
    bounded_lattice(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_684,axiom,
    ab_semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_685,axiom,
    comm_monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Osemigroup__mult_686,axiom,
    semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__top_687,axiom,
    order_top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__bot_688,axiom,
    order_bot(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Opreorder_689,axiom,
    preorder(assn) ).

tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_690,axiom,
    monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Olattice_691,axiom,
    lattice(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder_692,axiom,
    order(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Otop_693,axiom,
    top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oord_694,axiom,
    ord(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Obot_695,axiom,
    bot(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ouminus_696,axiom,
    uminus(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osup_697,axiom,
    sup(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Oinf_698,axiom,
    inf(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ominus_699,axiom,
    minus(assn) ).

tff(tcon_Assertions_Oassn___Power_Opower_700,axiom,
    power(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oone_701,axiom,
    one(assn) ).

tff(tcon_Assertions_Oassn___Rings_Odvd_702,axiom,
    dvd(assn) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_703,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_704,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_705,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_706,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_707,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_708,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_709,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_710,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_711,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_712,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_713,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_714,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_715,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_716,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_717,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_718,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_719,axiom,
    real_V6157519004096292374lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_720,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_721,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_722,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_723,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_724,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_725,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_726,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_727,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_728,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__group__add_729,axiom,
    topolo1633459387980952147up_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_730,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_731,axiom,
    topological_t1_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_732,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_733,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_734,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_735,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_736,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_737,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_738,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_739,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_740,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_741,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_742,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_743,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_744,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_745,axiom,
    field_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_746,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_747,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_748,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_749,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_750,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_751,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_752,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_753,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_754,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_755,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_756,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_757,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_758,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_759,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_760,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_761,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_762,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_763,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_764,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_765,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_766,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_767,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ominus_768,axiom,
    minus(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_769,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_770,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_771,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_772,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_773,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_774,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_775,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_776,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_777,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_778,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_779,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_780,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_781,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_782,axiom,
    bounde4967611905675639751up_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_783,axiom,
    bounde4346867609351753570nf_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_784,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_785,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_786,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_787,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_788,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__top_789,axiom,
    bounded_lattice_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_790,axiom,
    bounded_lattice_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_791,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_792,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_793,axiom,
    semilattice_inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_794,axiom,
    distrib_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_795,axiom,
    bounded_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_796,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_797,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_798,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_799,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_800,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_801,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_802,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_803,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_804,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_805,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_806,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_807,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_808,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_809,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_810,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_811,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_812,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_813,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_814,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_815,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_816,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_817,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_818,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_819,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_820,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_821,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_822,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Otop_823,axiom,
    top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_824,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Obot_825,axiom,
    bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osup_826,axiom,
    sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Oinf_827,axiom,
    inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ominus_828,axiom,
    minus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_829,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_830,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_831,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_832,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_833,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_834,axiom,
    dvd(extended_enat) ).

tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_835,axiom,
    ! [A12: $tType] :
      ( preorder(A12)
     => ordere6658533253407199908up_add(multiset(A12)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_836,axiom,
    ! [A12: $tType] : cancel2418104881723323429up_add(multiset(A12)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_837,axiom,
    ! [A12: $tType] : cancel1802427076303600483id_add(multiset(A12)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_838,axiom,
    ! [A12: $tType] : cancel_semigroup_add(multiset(A12)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_839,axiom,
    ! [A12: $tType] : comm_monoid_diff(multiset(A12)) ).

tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_840,axiom,
    ! [A12: $tType] : ab_semigroup_add(multiset(A12)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_841,axiom,
    ! [A12: $tType] : comm_monoid_add(multiset(A12)) ).

tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_842,axiom,
    ! [A12: $tType] : semigroup_add(multiset(A12)) ).

tff(tcon_Multiset_Omultiset___Orderings_Opreorder_843,axiom,
    ! [A12: $tType] :
      ( preorder(A12)
     => preorder(multiset(A12)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_844,axiom,
    ! [A12: $tType] : monoid_add(multiset(A12)) ).

tff(tcon_Multiset_Omultiset___Orderings_Oorder_845,axiom,
    ! [A12: $tType] :
      ( preorder(A12)
     => order(multiset(A12)) ) ).

tff(tcon_Multiset_Omultiset___Orderings_Oord_846,axiom,
    ! [A12: $tType] :
      ( preorder(A12)
     => ord(multiset(A12)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ominus_847,axiom,
    ! [A12: $tType] : minus(multiset(A12)) ).

tff(tcon_Multiset_Omultiset___Groups_Ozero_848,axiom,
    ! [A12: $tType] : zero(multiset(A12)) ).

tff(tcon_Multiset_Omultiset___Groups_Oplus_849,axiom,
    ! [A12: $tType] : plus(multiset(A12)) ).

tff(tcon_Multiset_Omultiset___Nat_Osize_850,axiom,
    ! [A12: $tType] : size(multiset(A12)) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__ab__semigroup__add_851,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => cancel2418104881723323429up_add(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__comm__monoid__add_852,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => cancel1802427076303600483id_add(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1__cancel_853,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => comm_s4317794764714335236cancel(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__semigroup__add_854,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => cancel_semigroup_add(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__mult_855,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => ab_semigroup_mult(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1__cancel_856,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => semiring_1_cancel(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__mult_857,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => comm_monoid_mult(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__add_858,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => ab_semigroup_add(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__add_859,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => comm_monoid_add(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1_860,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => comm_semiring_1(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__0_861,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => comm_semiring_0(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Osemigroup__mult_862,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => semigroup_mult(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Osemiring__numeral_863,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => semiring_numeral(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Osemigroup__add_864,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => semigroup_add(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring_865,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => comm_semiring(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Owellorder_866,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => wellorder(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__group__add_867,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => ab_group_add(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ozero__neq__one_868,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => zero_neq_one(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Opreorder_869,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => preorder(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Olinorder_870,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => linorder(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__mult_871,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => monoid_mult(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring__1_872,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => comm_ring_1(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__add_873,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => monoid_add(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen0_874,axiom,
    ! [A12: $tType] :
      ( type_len0(A12)
     => type_len0(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1_875,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => semiring_1(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__0_876,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => semiring_0(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ogroup__add_877,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => group_add(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen_878,axiom,
    ! [A12: $tType] :
      ( type_len(A12)
     => type_len(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Omult__zero_879,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => mult_zero(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring_880,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => comm_ring(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oorder_881,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => order(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Oneg__numeral_882,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => neg_numeral(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring_883,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => semiring(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oord_884,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => ord(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ouminus_885,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => uminus(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring__1_886,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => ring_1(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ominus_887,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => minus(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Power_Opower_888,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => power(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Onumeral_889,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => numeral(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ozero_890,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => zero(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oplus_891,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => plus(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring_892,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => ring(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oone_893,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => one(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Odvd_894,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => dvd(numeral_bit0(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__ab__semigroup__add_895,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => cancel2418104881723323429up_add(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__comm__monoid__add_896,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => cancel1802427076303600483id_add(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1__cancel_897,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => comm_s4317794764714335236cancel(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__semigroup__add_898,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => cancel_semigroup_add(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__mult_899,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => ab_semigroup_mult(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1__cancel_900,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => semiring_1_cancel(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__mult_901,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => comm_monoid_mult(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__add_902,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => ab_semigroup_add(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__add_903,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => comm_monoid_add(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1_904,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => comm_semiring_1(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__0_905,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => comm_semiring_0(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Osemigroup__mult_906,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => semigroup_mult(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Osemiring__numeral_907,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => semiring_numeral(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Osemigroup__add_908,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => semigroup_add(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring_909,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => comm_semiring(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Owellorder_910,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => wellorder(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__group__add_911,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => ab_group_add(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ozero__neq__one_912,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => zero_neq_one(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Opreorder_913,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => preorder(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Olinorder_914,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => linorder(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__mult_915,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => monoid_mult(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring__1_916,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => comm_ring_1(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__add_917,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => monoid_add(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen0_918,axiom,
    ! [A12: $tType] :
      ( type_len0(A12)
     => type_len0(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1_919,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => semiring_1(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__0_920,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => semiring_0(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ogroup__add_921,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => group_add(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen_922,axiom,
    ! [A12: $tType] :
      ( type_len0(A12)
     => type_len(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Omult__zero_923,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => mult_zero(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring_924,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => comm_ring(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oorder_925,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => order(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Oneg__numeral_926,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => neg_numeral(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring_927,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => semiring(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oord_928,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => ord(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ouminus_929,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => uminus(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring__1_930,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => ring_1(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ominus_931,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => minus(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Power_Opower_932,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => power(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Onumeral_933,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => numeral(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ozero_934,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => zero(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oplus_935,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => plus(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring_936,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => ring(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oone_937,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => one(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Odvd_938,axiom,
    ! [A12: $tType] :
      ( finite_finite(A12)
     => dvd(numeral_bit1(A12)) ) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__ab__semigroup__add_939,axiom,
    cancel2418104881723323429up_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__comm__monoid__add_940,axiom,
    cancel1802427076303600483id_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__semigroup__add_941,axiom,
    cancel_semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__mult_942,axiom,
    ab_semigroup_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__mult_943,axiom,
    comm_monoid_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__add_944,axiom,
    ab_semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__add_945,axiom,
    comm_monoid_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring__0_946,axiom,
    comm_semiring_0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Osemigroup__mult_947,axiom,
    semigroup_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Osemigroup__add_948,axiom,
    semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring_949,axiom,
    comm_semiring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Owellorder_950,axiom,
    wellorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__group__add_951,axiom,
    ab_group_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Opreorder_952,axiom,
    preorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Olinorder_953,axiom,
    linorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Omonoid__mult_954,axiom,
    monoid_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Omonoid__add_955,axiom,
    monoid_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Type__Length_Olen0_956,axiom,
    type_len0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Osemiring__0_957,axiom,
    semiring_0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ogroup__add_958,axiom,
    group_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Type__Length_Olen_959,axiom,
    type_len(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Omult__zero_960,axiom,
    mult_zero(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__ring_961,axiom,
    comm_ring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Oorder_962,axiom,
    order(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Osemiring_963,axiom,
    semiring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Oord_964,axiom,
    ord(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ouminus_965,axiom,
    uminus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ominus_966,axiom,
    minus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Power_Opower_967,axiom,
    power(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Num_Onumeral_968,axiom,
    numeral(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ozero_969,axiom,
    zero(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oplus_970,axiom,
    plus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Oring_971,axiom,
    ring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oone_972,axiom,
    one(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Odvd_973,axiom,
    dvd(numeral_num1) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_974,axiom,
    ! [A12: $tType,A13: $tType] :
      ( ( topolo4958980785337419405_space(A12)
        & topolo4958980785337419405_space(A13) )
     => topolo4958980785337419405_space(product_prod(A12,A13)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_975,axiom,
    ! [A12: $tType,A13: $tType] :
      ( ( topological_t2_space(A12)
        & topological_t2_space(A13) )
     => topological_t2_space(product_prod(A12,A13)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_976,axiom,
    ! [A12: $tType,A13: $tType] :
      ( ( topological_t1_space(A12)
        & topological_t1_space(A13) )
     => topological_t1_space(product_prod(A12,A13)) ) ).

tff(tcon_Product__Type_Oprod___Heap_Oheap_977,axiom,
    ! [A12: $tType,A13: $tType] :
      ( ( heap(A12)
        & heap(A13) )
     => heap(product_prod(A12,A13)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_978,axiom,
    ! [A12: $tType,A13: $tType] : size(product_prod(A12,A13)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_979,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_980,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_981,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_982,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_983,axiom,
    bounded_lattice_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_984,axiom,
    bounded_lattice_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_985,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_986,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_987,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_988,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_989,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_990,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_991,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_992,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_993,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_994,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_995,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Otop_996,axiom,
    top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_997,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Obot_998,axiom,
    bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_999,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osup_1000,axiom,
    sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Oinf_1001,axiom,
    inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ominus_1002,axiom,
    minus(product_unit) ).

tff(tcon_Product__Type_Ounit___Heap_Oheap_1003,axiom,
    heap(product_unit) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_1004,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_1005,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_1006,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_1007,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_1008,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_1009,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_1010,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_1011,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_1012,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_1013,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_1014,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_1015,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_1016,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_1017,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_1018,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_1019,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_1020,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_1021,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_1022,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_1023,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_1024,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_1025,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_1026,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_1027,axiom,
    euclid5891614535332579305n_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_1028,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_1029,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_1030,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_1031,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_1032,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_1033,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_1034,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_1035,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_1036,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_1037,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_1038,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_1039,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_1040,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_1041,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_1042,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_1043,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_1044,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Least__significant__bit_Olsb_1045,axiom,
    least_6119777620449941438nt_lsb(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_1046,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_1047,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_1048,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_1049,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_1050,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_1051,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_1052,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_1053,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_1054,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_1055,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_1056,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_1057,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_1058,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_1059,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_1060,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_1061,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_1062,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_1063,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_1064,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_1065,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_1066,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_1067,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_1068,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_1069,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_1070,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_1071,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_1072,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_1073,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_1074,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_1075,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_1076,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_1077,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_1078,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_1079,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_1080,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_1081,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_1082,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_1083,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_1084,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_1085,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_1086,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_1087,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_1088,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_1089,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_1090,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_1091,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_1092,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_1093,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_1094,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_1095,axiom,
    minus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_1096,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_1097,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_1098,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_1099,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_1100,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_1101,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_1102,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_1103,axiom,
    dvd(code_integer) ).

tff(tcon_Heap__Time__Monad_OHeap___Nat_Osize_1104,axiom,
    ! [A12: $tType] : size(heap_Time_Heap(A12)) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_1105,axiom,
    size(vEBT_VEBT) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Heap_Oheap_1106,axiom,
    heap(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Nat_Osize_1107,axiom,
    size(vEBT_VEBTi) ).

% Helper facts (3)
tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X11: A,Y: A] :
      ( ( X11 != Y )
      | aa(A,$o,aa(A,fun(A,$o),fequal(A),X11),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X11: A,Y: A] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),fequal(A),X11),Y)
      | ( X11 = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
      = ( ? [X9: A] : aa(A,$o,P,X9) ) ) ).

% Free types (1)
tff(tfree_0,hypothesis,
    semiring_1(a) ).

% Conjectures (2)
tff(conj_0,hypothesis,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),x),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),n)) ).

tff(conj_1,conjecture,
    time_htt(vEBT_VEBTi,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(n,s),ti),vEBT_vebt_inserti(ti,x),vEBT_Intf_vebt_assn(n,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),s),aa(set(nat),set(nat),aa(nat,fun(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),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,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))))))) ).