%------------------------------------------------------------------------------
% File     : ITP225_4 : TPTP v9.0.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Member 00300_011794
% 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    : 0065_VEBT_Member_00300_011794 [Des22]

% Status   : Theorem
% Rating   : 0.67 v9.0.0, 1.00 v8.1.0
% Syntax   : Number of formulae    : 11462 (4198 unt;1594 typ;   0 def)
%            Number of atoms       : 18244 (7896 equ)
%            Maximal formula atoms :   47 (   1 avg)
%            Number of connectives : 17589 (1653   ~; 286   |;1859   &)
%                                         (1950 <=>;11841  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   38 (   2 avg)
%            Number of FOOLs       : 1226 ( 903 fml; 323 var)
%            Number of X terms     :  578 (   0  []; 508 ite;  70 let)
%            Number of types       :   15 (  14 usr)
%            Number of type conns  : 1365 (1095   >; 270   *;   0   +;   0  <<)
%            Number of predicates  :  266 ( 263 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1333 (1333 usr; 107 con; 0-8 aty)
%            Number of variables   : 30055 (27011   !; 675   ?;30055   :)
%                                         (2369  !>;   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
%            2022-02-17 18:47:24.801
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
tff(ty_t_VEBT__Definitions_OVEBT,type,
    vEBT_VEBT: $tType ).

tff(ty_t_Code__Numeral_Onatural,type,
    code_natural: $tType ).

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

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

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

tff(ty_t_Old__Datatype_Onode,type,
    old_node: ( $tType * $tType ) > $tType ).

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

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

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

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

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

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

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

tff(ty_t_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 (1571)
tff(sy_cl_Lattices_Obounded__lattice,type,
    bounded_lattice: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_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_Real__Vector__Spaces_Odist__norm,type,
    real_V6936659425649961206t_norm: 
      !>[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_GCD_Osemiring__gcd__mult__normalize,type,
    semiri6843258321239162965malize: 
      !>[A: $tType] : $o ).

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

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

tff(sy_cl_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_Osgn__div__norm,type,
    real_V6567297691418259687v_norm: 
      !>[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_Rings_Osemidom__divide__unit__factor,type,
    semido2269285787275462019factor: 
      !>[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_Ocomplete__space,type,
    real_V8037385150606011577_space: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Oreal__algebra__1,type,
    real_V2191834092415804123ebra_1: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__ok____,type,
    aTP_Lamp_ok: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ol____,type,
    aTP_Lamp_ol: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ot____,type,
    aTP_Lamp_ot: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ou____,type,
    aTP_Lamp_ou: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ov____,type,
    aTP_Lamp_ov: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ow____,type,
    aTP_Lamp_ow: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ox____,type,
    aTP_Lamp_ox: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oy____,type,
    aTP_Lamp_oy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__uo____,type,
    aTP_Lamp_uo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set(product_prod(nat,nat)) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set(product_prod(nat,nat)) ).

tff(sy_c_BNF__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__Relation_Owo__rel,type,
    bNF_Wellorder_wo_rel: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
    basic_BNF_size_prod: 
      !>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) * product_prod(A,B) ) > nat ) ).

tff(sy_c_Basic__BNF__LFPs_Osum_Osize__sum,type,
    basic_BNF_size_sum: 
      !>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) * sum_sum(A,B) ) > nat ) ).

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

tff(sy_c_Binomial_Obinomial,type,
    binomial: nat > fun(nat,nat) ).

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

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun(product_prod(int,int),fun(product_prod(int,int),$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 * int ) > int ).

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : ( A > fun(nat,$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__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
    bit_un7362597486090784418nd_num: ( num * num ) > option(num) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
    bit_un2480387367778600638or_num: ( num * num ) > option(num) ).

tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
    boolea2506097494486148201lgebra: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A ) > $o ) ).

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

tff(sy_c_Code__Numeral_ONat,type,
    code_Nat: fun(code_integer,code_natural) ).

tff(sy_c_Code__Numeral_OSuc,type,
    code_Suc: fun(code_natural,code_natural) ).

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

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

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

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

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

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

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

tff(sy_c_Code__Numeral_Ointeger__of__natural,type,
    code_i5400310926305786745atural: fun(code_natural,code_integer) ).

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

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

tff(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
    code_nat_of_natural: fun(code_natural,nat) ).

tff(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
    code_natural_of_nat: fun(nat,code_natural) ).

tff(sy_c_Code__Numeral_Onatural__of__integer,type,
    code_n4118661773612635043nteger: fun(code_integer,code_natural) ).

tff(sy_c_Code__Numeral_Onegative,type,
    code_negative: fun(num,code_integer) ).

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

tff(sy_c_Code__Numeral_Opcr__integer,type,
    code_pcr_integer: fun(int,fun(code_integer,$o)) ).

tff(sy_c_Code__Numeral_Opcr__natural,type,
    code_pcr_natural: fun(nat,fun(code_natural,$o)) ).

tff(sy_c_Code__Numeral_Osub,type,
    code_sub: fun(num,fun(num,code_integer)) ).

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

tff(sy_c_Code__Target__Nat_ONat,type,
    code_Target_Nat: fun(code_integer,nat) ).

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

tff(sy_c_Complete__Lattices_OInf__class_OInf,type,
    complete_Inf_Inf: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Complete__Lattices_OSup__class_OSup,type,
    complete_Sup_Sup: 
      !>[A: $tType] : fun(set(A),A) ).

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

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

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

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

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

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

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

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

tff(sy_c_Complex_Orcis,type,
    rcis: ( real * real ) > complex ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
    condit941137186595557371_above: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
    condit1013018076250108175_below: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd,type,
    condit622319405099724424ng_bdd: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Countable_Onth__item__rel,type,
    nth_item_rel: fun(nat,fun(nat,$o)) ).

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

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

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

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

tff(sy_c_Deriv_Ohas__vector__derivative,type,
    has_ve8173657378732805170vative: 
      !>[B: $tType] : ( ( fun(real,B) * B * filter(real) ) > $o ) ).

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

tff(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: ( int * int ) > int ).

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

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

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

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

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

tff(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
    euclid6346220572633701492n_size: 
      !>[A: $tType] : fun(A,nat) ).

tff(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
    euclid7384307370059645450egment: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Extended__Nat_OeSuc,type,
    extended_eSuc: extended_enat > extended_enat ).

tff(sy_c_Extended__Nat_Oenat,type,
    extended_enat2: nat > extended_enat ).

tff(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
    extended_case_enat: 
      !>[T: $tType] : ( ( fun(nat,T) * T * extended_enat ) > T ) ).

tff(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
    extend4730790105801354508finity: 
      !>[A: $tType] : A ).

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

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

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

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

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

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

tff(sy_c_Filter_Ofiltercomap,type,
    filtercomap: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) ) > filter(A) ) ).

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

tff(sy_c_Filter_Ofiltermap,type,
    filtermap: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(A) ) > filter(B) ) ).

tff(sy_c_Filter_Oprincipal,type,
    principal: 
      !>[A: $tType] : ( set(A) > filter(A) ) ).

tff(sy_c_Filter_Oprod__filter,type,
    prod_filter: 
      !>[A: $tType,B: $tType] : ( ( filter(A) * filter(B) ) > filter(product_prod(A,B)) ) ).

tff(sy_c_Finite__Set_Ocard,type,
    finite_card: 
      !>[B: $tType] : fun(set(B),nat) ).

tff(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
    finite4664212375090638736ute_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
    finite673082921795544331dem_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

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

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

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

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

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

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

tff(sy_c_Fun_Ofun__upd,type,
    fun_upd: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A * B ) > fun(A,B) ) ).

tff(sy_c_Fun_Oid,type,
    id: 
      !>[A: $tType] : fun(A,A) ).

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

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

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

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

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

tff(sy_c_Fun__Def_Opair__leq,type,
    fun_pair_leq: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).

tff(sy_c_Fun__Def_Opair__less,type,
    fun_pair_less: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).

tff(sy_c_GCD_OGcd__class_OGcd,type,
    gcd_Gcd: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_GCD_OGcd__class_OLcm,type,
    gcd_Lcm: 
      !>[A: $tType] : ( set(A) > A ) ).

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

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

tff(sy_c_GCD_Obounded__quasi__semilattice,type,
    bounde8507323023520639062attice: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $o ) ).

tff(sy_c_GCD_Obounded__quasi__semilattice__set,type,
    bounde6485984586167503788ce_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $o ) ).

tff(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
    bounde2362111253966948842tice_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A ) > fun(set(A),A) ) ).

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

tff(sy_c_GCD_Ogcd__class_Olcm,type,
    gcd_lcm: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_GCD_Ogcd__nat__rel,type,
    gcd_nat_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

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

tff(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
    semiring_gcd_Gcd_fin: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
    semiring_gcd_Lcm_fin: 
      !>[A: $tType] : fun(set(A),A) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
    groups7311177749621191930dd_sum: 
      !>[B: $tType,A: $tType] : fun(fun(B,A),fun(set(B),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
    groups1027152243600224163dd_sum: 
      !>[C: $tType,A: $tType] : fun(fun(C,A),fun(set(C),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
    groups7121269368397514597t_prod: 
      !>[B: $tType,A: $tType] : fun(fun(B,A),fun(set(B),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
    groups1962203154675924110t_prod: 
      !>[C: $tType,A: $tType] : fun(fun(C,A),fun(set(C),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__set,type,
    groups778175481326437816id_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

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

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

tff(sy_c_Groups__List_Ocomm__monoid__list,type,
    groups1828464146339083142d_list: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups__List_Ocomm__monoid__list__set,type,
    groups4802862169904069756st_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
    groups4207007520872428315er_sum: 
      !>[B: $tType,A: $tType] : fun(fun(B,A),fun(A,fun(list(B),A))) ).

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

tff(sy_c_Groups__List_Omonoid__list_OF,type,
    groups_monoid_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(list(A),A) ) ).

tff(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
    groups5270119922927024881d_list: 
      !>[A: $tType] : fun(list(A),A) ).

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

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

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

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

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

tff(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
    complete_lattice_gfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
    complete_lattice_lfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
    infini527867602293511546merate: 
      !>[A: $tType] : ( ( set(A) * nat ) > A ) ).

tff(sy_c_Int_OAbs__Integ,type,
    abs_Integ: fun(product_prod(nat,nat),int) ).

tff(sy_c_Int_ONeg,type,
    neg: num > int ).

tff(sy_c_Int_OPos,type,
    pos: fun(num,int) ).

tff(sy_c_Int_ORep__Integ,type,
    rep_Integ: fun(int,product_prod(nat,nat)) ).

tff(sy_c_Int_Ocr__int,type,
    cr_int: fun(product_prod(nat,nat),fun(int,$o)) ).

tff(sy_c_Int_Odup,type,
    dup: int > int ).

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

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

tff(sy_c_Int_Ointrel,type,
    intrel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

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

tff(sy_c_Int_Opcr__int,type,
    pcr_int: fun(product_prod(nat,nat),fun(int,$o)) ).

tff(sy_c_Int_Opower__int,type,
    power_int: 
      !>[A: $tType] : ( ( A * int ) > A ) ).

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

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

tff(sy_c_Int_Osub,type,
    sub: ( num * num ) > int ).

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

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

tff(sy_c_Lattices_Osemilattice__neutr__order,type,
    semila1105856199041335345_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

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

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

tff(sy_c_Lattices__Big_Olinorder__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__max,type,
    lattices_ord_arg_max: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,$o) ) > B ) ).

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

tff(sy_c_Lattices__Big_Oord__class_Ois__arg__max,type,
    lattic501386751176901750rg_max: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,$o) * B ) > $o ) ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
    lattic7752659483105999362nf_fin: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
    lattic5214292709420241887eutr_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * set(A) ) > A ) ).

tff(sy_c_Lattices__Big_Osemilattice__order__set,type,
    lattic4895041142388067077er_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
    lattic5882676163264333800up_fin: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Lifting_OQuotient,type,
    quotient: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,B) * fun(B,A) * fun(A,fun(B,$o)) ) > $o ) ).

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

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

tff(sy_c_Limits_Oat__infinity,type,
    at_infinity: 
      !>[A: $tType] : filter(A) ).

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

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

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

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

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

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

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

tff(sy_c_List_Odistinct,type,
    distinct: 
      !>[A: $tType] : ( list(A) > $o ) ).

tff(sy_c_List_Odistinct__adj,type,
    distinct_adj: 
      !>[A: $tType] : ( list(A) > $o ) ).

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

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

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

tff(sy_c_List_Oextract,type,
    extract: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > option(product_prod(list(A),product_prod(A,list(A)))) ) ).

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

tff(sy_c_List_Ofind,type,
    find: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > option(A) ) ).

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

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

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

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

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

tff(sy_c_List_Olast,type,
    last: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Olenlex,type,
    lenlex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olex,type,
    lex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olexn,type,
    lexn: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * nat ) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olexord,type,
    lexord: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

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

tff(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
    linord329482645794927042rt_key: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).

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

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

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

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

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

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

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

tff(sy_c_List_Olist_Olist__all2,type,
    list_all2: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(list(A),fun(list(B),$o)) ) ).

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

tff(sy_c_List_Olist_Orec__list,type,
    rec_list: 
      !>[C: $tType,A: $tType] : ( ( C * fun(A,fun(list(A),fun(C,C))) * list(A) ) > C ) ).

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

tff(sy_c_List_Olist_Osize__list,type,
    size_list: 
      !>[A: $tType] : ( ( fun(A,nat) * list(A) ) > nat ) ).

tff(sy_c_List_Olist_Otl,type,
    tl: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

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

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

tff(sy_c_List_Olistrel,type,
    listrel: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(list(A),list(B))) ) ).

tff(sy_c_List_Olistrel1,type,
    listrel1: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

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

tff(sy_c_List_Omeasures,type,
    measures: 
      !>[A: $tType] : ( list(fun(A,nat)) > set(product_prod(A,A)) ) ).

tff(sy_c_List_Omin__list,type,
    min_list: 
      !>[A: $tType] : ( list(A) > A ) ).

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

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

tff(sy_c_List_Onths,type,
    nths: 
      !>[A: $tType] : ( ( list(A) * set(nat) ) > list(A) ) ).

tff(sy_c_List_Oord__class_Olexordp,type,
    ord_lexordp: 
      !>[A: $tType] : fun(list(A),fun(list(A),$o)) ).

tff(sy_c_List_Oord__class_Olexordp__eq,type,
    ord_lexordp_eq: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > $o ) ).

tff(sy_c_List_Opartition,type,
    partition: 
      !>[A: $tType] : ( fun(A,$o) > fun(list(A),product_prod(list(A),list(A))) ) ).

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

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

tff(sy_c_List_Oremdups,type,
    remdups: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oremdups__adj,type,
    remdups_adj: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

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

tff(sy_c_Map_Odom,type,
    dom: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(A) ) ).

tff(sy_c_Map_Ograph,type,
    graph: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).

tff(sy_c_Map_Omap__of,type,
    map_of: 
      !>[A: $tType,B: $tType] : ( list(product_prod(A,B)) > fun(A,option(B)) ) ).

tff(sy_c_Map_Omap__upds,type,
    map_upds: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) * list(B) ) > fun(A,option(B)) ) ).

tff(sy_c_Map_Oran,type,
    ran: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).

tff(sy_c_Map_Orestrict__map,type,
    restrict_map: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > fun(A,option(B)) ) ).

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

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

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

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

tff(sy_c_Nat_Onat_Opred,type,
    pred: nat > nat ).

tff(sy_c_Nat_Oold_Onat_Orec__nat,type,
    rec_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) ) > fun(nat,T) ) ).

tff(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
    rec_set_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) * nat ) > fun(T,$o) ) ).

tff(sy_c_Nat_Osemiring__1__class_ONats,type,
    semiring_1_Nats: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
    semiring_1_of_nat: 
      !>[A: $tType] : fun(nat,A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
    semiri8178284476397505188at_aux: 
      !>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).

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

tff(sy_c_Nat__Bijection_Oint__decode,type,
    nat_int_decode: fun(nat,int) ).

tff(sy_c_Nat__Bijection_Oint__encode,type,
    nat_int_encode: fun(int,nat) ).

tff(sy_c_Nat__Bijection_Olist__decode,type,
    nat_list_decode: fun(nat,list(nat)) ).

tff(sy_c_Nat__Bijection_Olist__decode__rel,type,
    nat_list_decode_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: fun(list(nat),nat) ).

tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: fun(list(nat),fun(list(nat),$o)) ).

tff(sy_c_Nat__Bijection_Oprod__decode,type,
    nat_prod_decode: fun(nat,product_prod(nat,nat)) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > fun(nat,product_prod(nat,nat)) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: fun(product_prod(nat,nat),nat) ).

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

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun(set(nat),nat) ).

tff(sy_c_Nat__Bijection_Osum__decode,type,
    nat_sum_decode: fun(nat,sum_sum(nat,nat)) ).

tff(sy_c_Nat__Bijection_Osum__encode,type,
    nat_sum_encode: fun(sum_sum(nat,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_Onat__of__num,type,
    nat_of_num: fun(num,nat) ).

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

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

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

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

tff(sy_c_Num_Oneg__numeral__class_Osub,type,
    neg_numeral_sub: 
      !>[A: $tType] : ( ( num * num ) > A ) ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: num > num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun(num,num) ).

tff(sy_c_Num_Onum_OOne,type,
    one2: num ).

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

tff(sy_c_Num_Osqr,type,
    sqr: num > num ).

tff(sy_c_Old__Datatype_OAtom,type,
    old_Atom: 
      !>[A: $tType,B: $tType] : ( sum_sum(A,nat) > set(old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_OIn0,type,
    old_In0: 
      !>[A: $tType,B: $tType] : ( set(old_node(A,B)) > set(old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_OIn1,type,
    old_In1: 
      !>[A: $tType,B: $tType] : ( set(old_node(A,B)) > set(old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_OLeaf,type,
    old_Leaf: 
      !>[A: $tType,B: $tType] : ( A > set(old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_ONode,type,
    old_Node: 
      !>[B: $tType,A: $tType] : set(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))) ).

tff(sy_c_Old__Datatype_ONumb,type,
    old_Numb: 
      !>[A: $tType,B: $tType] : ( nat > set(old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_OPush,type,
    old_Push: 
      !>[B: $tType] : ( ( sum_sum(B,nat) * fun(nat,sum_sum(B,nat)) * nat ) > sum_sum(B,nat) ) ).

tff(sy_c_Old__Datatype_OPush__Node,type,
    old_Push_Node: 
      !>[B: $tType,A: $tType] : ( sum_sum(B,nat) > fun(old_node(A,B),old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_OScons,type,
    old_Scons: 
      !>[A: $tType,B: $tType] : ( ( set(old_node(A,B)) * set(old_node(A,B)) ) > set(old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_Ondepth,type,
    old_ndepth: 
      !>[A: $tType,B: $tType] : ( old_node(A,B) > nat ) ).

tff(sy_c_Old__Datatype_Onode_OAbs__Node,type,
    old_Abs_Node: 
      !>[B: $tType,A: $tType] : ( product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)) > old_node(A,B) ) ).

tff(sy_c_Old__Datatype_Onode_ORep__Node,type,
    old_Rep_Node: 
      !>[A: $tType,B: $tType] : ( old_node(A,B) > product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)) ) ).

tff(sy_c_Old__Datatype_Ontrunc,type,
    old_ntrunc: 
      !>[A: $tType,B: $tType] : ( ( nat * set(old_node(A,B)) ) > set(old_node(A,B)) ) ).

tff(sy_c_Option_Ooption_ONone,type,
    none: 
      !>[A: $tType] : option(A) ).

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

tff(sy_c_Option_Ooption_Ocase__option,type,
    case_option: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,B) * option(A) ) > B ) ).

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

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

tff(sy_c_Order__Relation_OunderS,type,
    order_underS: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).

tff(sy_c_Order__Relation_Owell__order__on,type,
    order_well_order_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

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

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

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

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

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

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

tff(sy_c_Orderings_Oorder__class_Oantimono,type,
    order_antimono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oorder__class_Omono,type,
    order_mono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
    order_strict_mono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

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

tff(sy_c_Orderings_Otop__class_Otop,type,
    top_top: 
      !>[A: $tType] : A ).

tff(sy_c_Power_Opower_Opower,type,
    power2: 
      !>[A: $tType] : ( ( A * fun(A,fun(A,A)) * A * nat ) > A ) ).

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

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

tff(sy_c_Product__Type_OSigma,type,
    product_Sigma: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,set(B)) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Product__Type_Oapsnd,type,
    product_apsnd: 
      !>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(product_prod(A,B),product_prod(A,C))) ).

tff(sy_c_Product__Type_Oprod_Ocase__prod,type,
    product_case_prod: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(product_prod(A,B),C) ) ).

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

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

tff(sy_c_Product__Type_Oproduct,type,
    product_product: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Product__Type_Oscomp,type,
    product_scomp: 
      !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,product_prod(B,C)) * fun(B,fun(C,D)) ) > fun(A,D) ) ).

tff(sy_c_Pure_Otype,type,
    type2: 
      !>[A: $tType] : itself(A) ).

tff(sy_c_Quotient_OQuotient3,type,
    quotient3: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,B) * fun(B,A) ) > $o ) ).

tff(sy_c_Random_Oinc__shift,type,
    inc_shift: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Oiterate,type,
    iterate: 
      !>[B: $tType,A: $tType] : ( ( code_natural * fun(B,fun(A,product_prod(B,A))) ) > fun(B,fun(A,product_prod(B,A))) ) ).

tff(sy_c_Random_Oiterate__rel,type,
    iterate_rel: 
      !>[B: $tType,A: $tType] : fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),$o)) ).

tff(sy_c_Random_Olog,type,
    log: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Olog__rel,type,
    log_rel: fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),$o)) ).

tff(sy_c_Random_Ominus__shift,type,
    minus_shift: ( code_natural * code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Onext,type,
    next: fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).

tff(sy_c_Random_Opick,type,
    pick: 
      !>[A: $tType] : ( ( list(product_prod(code_natural,A)) * code_natural ) > A ) ).

tff(sy_c_Random_Orange,type,
    range: code_natural > fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).

tff(sy_c_Random_Oselect,type,
    select: 
      !>[A: $tType] : ( list(A) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Random_Oselect__weight,type,
    select_weight: 
      !>[A: $tType] : ( list(product_prod(code_natural,A)) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: fun(product_prod(int,int),rat) ).

tff(sy_c_Rat_OFract,type,
    fract: fun(int,fun(int,rat)) ).

tff(sy_c_Rat_OFrct,type,
    frct: product_prod(int,int) > rat ).

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun(rat,product_prod(int,int)) ).

tff(sy_c_Rat_Ofield__char__0__class_ORats,type,
    field_char_0_Rats: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
    field_char_0_of_rat: 
      !>[A: $tType] : fun(rat,A) ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

tff(sy_c_Rat_Oof__int,type,
    of_int: fun(int,rat) ).

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun(product_prod(int,int),fun(rat,$o)) ).

tff(sy_c_Rat_Opositive,type,
    positive: fun(rat,$o) ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod(int,int) ).

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_Real_ORatreal,type,
    ratreal: fun(rat,real) ).

tff(sy_c_Real_OReal,type,
    real2: fun(fun(nat,rat),real) ).

tff(sy_c_Real_Ocauchy,type,
    cauchy: fun(nat,rat) > $o ).

tff(sy_c_Real_Ocr__real,type,
    cr_real: fun(fun(nat,rat),fun(real,$o)) ).

tff(sy_c_Real_Opcr__real,type,
    pcr_real: fun(fun(nat,rat),fun(real,$o)) ).

tff(sy_c_Real_Opositive,type,
    positive2: fun(real,$o) ).

tff(sy_c_Real_Orealrel,type,
    realrel: fun(fun(nat,rat),fun(fun(nat,rat),$o)) ).

tff(sy_c_Real_Orep__real,type,
    rep_real: fun(real,fun(nat,rat)) ).

tff(sy_c_Real_Ovanishes,type,
    vanishes: fun(nat,rat) > $o ).

tff(sy_c_Real__Vector__Spaces_OReals,type,
    real_Vector_Reals: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Real__Vector__Spaces_Obounded__bilinear,type,
    real_V2442710119149674383linear: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > $o ) ).

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_Oconstruct,type,
    real_V4425403222259421789struct: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * A ) > B ) ).

tff(sy_c_Real__Vector__Spaces_Odependent,type,
    real_V358717886546972837endent: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Odim,type,
    real_Vector_dim: 
      !>[A: $tType] : ( set(A) > nat ) ).

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
    real_V557655796197034286t_dist: 
      !>[A: $tType] : ( ( A * A ) > real ) ).

tff(sy_c_Real__Vector__Spaces_Oextend__basis,type,
    real_V4986007116245087402_basis: 
      !>[A: $tType] : ( set(A) > set(A) ) ).

tff(sy_c_Real__Vector__Spaces_Olinear,type,
    real_Vector_linear: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

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] : ( real > A ) ).

tff(sy_c_Real__Vector__Spaces_Orepresentation,type,
    real_V7696804695334737415tation: 
      !>[A: $tType] : ( ( set(A) * A * A ) > real ) ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
    real_V8093663219630862766scaleR: 
      !>[A: $tType] : ( ( real * A ) > A ) ).

tff(sy_c_Real__Vector__Spaces_Ospan,type,
    real_Vector_span: 
      !>[A: $tType] : ( set(A) > set(A) ) ).

tff(sy_c_Real__Vector__Spaces_Osubspace,type,
    real_Vector_subspace: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Relation_OField,type,
    field2: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(A) ) ).

tff(sy_c_Relation_OId,type,
    id2: 
      !>[A: $tType] : set(product_prod(A,A)) ).

tff(sy_c_Relation_OId__on,type,
    id_on: 
      !>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).

tff(sy_c_Relation_Orelcomp,type,
    relcomp: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,B)) * set(product_prod(B,C)) ) > set(product_prod(A,C)) ) ).

tff(sy_c_Relation_Otransp,type,
    transp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
    algebr8660921524188924756oprime: 
      !>[A: $tType] : ( ( A * A ) > $o ) ).

tff(sy_c_Rings_Odivide__class_Odivide,type,
    divide_divide: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Rings_Odvd__class_Odvd,type,
    dvd_dvd: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_Rings_Omodulo__class_Omodulo,type,
    modulo_modulo: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
    normal6383669964737779283malize: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
    unit_f5069060285200089521factor: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
    zero_neq_one_of_bool: 
      !>[A: $tType] : fun($o,A) ).

tff(sy_c_Series_Osuminf,type,
    suminf: 
      !>[A: $tType] : ( fun(nat,A) > A ) ).

tff(sy_c_Series_Osummable,type,
    summable: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Series_Osums,type,
    sums: 
      !>[A: $tType] : ( ( fun(nat,A) * A ) > $o ) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : ( fun(A,$o) > set(A) ) ).

tff(sy_c_Set_OPow,type,
    pow2: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Set_Odisjnt,type,
    disjnt: 
      !>[A: $tType] : ( ( set(A) * set(A) ) > $o ) ).

tff(sy_c_Set_Ofilter,type,
    filter3: 
      !>[A: $tType] : ( ( fun(A,$o) * set(A) ) > set(A) ) ).

tff(sy_c_Set_Oimage,type,
    image: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > set(B) ) ).

tff(sy_c_Set_Oinsert,type,
    insert2: 
      !>[A: $tType] : fun(A,fun(set(A),set(A))) ).

tff(sy_c_Set_Ois__singleton,type,
    is_singleton: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Set_Oremove,type,
    remove: 
      !>[A: $tType] : fun(A,fun(set(A),set(A))) ).

tff(sy_c_Set_Othe__elem,type,
    the_elem: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Set_Ovimage,type,
    vimage: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(B) ) > set(A) ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
    set_fo6178422350223883121st_nat: 
      !>[A: $tType] : ( ( fun(nat,fun(A,A)) * nat * nat * A ) > A ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
    set_fo1817059534552279752at_rel: 
      !>[A: $tType] : fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o)) ).

tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
    set_ord_atLeast: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
    set_or1337092689740270186AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
    set_or7035219750837199246ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatMost,type,
    set_ord_atMost: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
    set_ord_greaterThan: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
    set_or3652927894154168847AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
    set_or5935395276787703475ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
    set_ord_lessThan: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_String_OCode_Oabort,type,
    abort: 
      !>[A: $tType] : ( ( literal * fun(product_unit,A) ) > A ) ).

tff(sy_c_String_OLiteral,type,
    literal2: ( $o * $o * $o * $o * $o * $o * $o * literal ) > literal ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( $o * $o * $o * $o * $o * $o * $o * $o ) > char ).

tff(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
    comm_s6883823935334413003f_char: 
      !>[A: $tType] : fun(char,A) ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
    unique5772411509450598832har_of: 
      !>[A: $tType] : fun(A,char) ).

tff(sy_c_Sum__Type_OInl,type,
    sum_Inl: 
      !>[A: $tType,B: $tType] : fun(A,sum_sum(A,B)) ).

tff(sy_c_Sum__Type_OInr,type,
    sum_Inr: 
      !>[B: $tType,A: $tType] : fun(B,sum_sum(A,B)) ).

tff(sy_c_Sum__Type_OPlus,type,
    sum_Plus: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(sum_sum(A,B)) ) ).

tff(sy_c_Sum__Type_Osum_Ocase__sum,type,
    sum_case_sum: 
      !>[A: $tType,C: $tType,B: $tType] : ( ( fun(A,C) * fun(B,C) * sum_sum(A,B) ) > C ) ).

tff(sy_c_Topological__Spaces_Ocontinuous,type,
    topolo3448309680560233919inuous: 
      !>[A: $tType,B: $tType] : ( ( filter(A) * fun(A,B) ) > $o ) ).

tff(sy_c_Topological__Spaces_Ocontinuous__on,type,
    topolo81223032696312382ous_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > $o ) ).

tff(sy_c_Topological__Spaces_Omonoseq,type,
    topological_monoseq: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
    topolo1002775350975398744n_open: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ot2__space__class_OLim,type,
    topolo3827282254853284352ce_Lim: 
      !>[F: $tType,A: $tType] : ( ( filter(F) * fun(F,A) ) > A ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
    topolo174197925503356063within: 
      !>[A: $tType] : ( ( A * set(A) ) > filter(A) ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent,type,
    topolo6863149650580417670ergent: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

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] : ( A > A ) ).

tff(sy_c_Transcendental_Ocot,type,
    cot: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Odiffs,type,
    diffs: 
      !>[A: $tType] : ( fun(nat,A) > fun(nat,A) ) ).

tff(sy_c_Transcendental_Oexp,type,
    exp: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Oln__class_Oln,type,
    ln_ln: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Olog,type,
    log2: 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_Opowr__real,type,
    powr_real: ( real * real ) > real ).

tff(sy_c_Transcendental_Osin,type,
    sin: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: nat > real ).

tff(sy_c_Transcendental_Osinh,type,
    sinh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Otan,type,
    tan: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Otanh,type,
    tanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transitive__Closure_Ontrancl,type,
    transitive_ntrancl: 
      !>[A: $tType] : ( ( nat * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Otrancl,type,
    transitive_trancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).

tff(sy_c_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_Ocase__VEBT,type,
    vEBT_case_VEBT: 
      !>[A: $tType] : fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,A))) ).

tff(sy_c_VEBT__Definitions_OVEBT_Orec__VEBT,type,
    vEBT_rec_VEBT: 
      !>[A: $tType] : fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,A))) ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > fun(nat,$o) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
    vEBT_VEBT_elim_dead: ( vEBT_VEBT * extended_enat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
    vEBT_V312737461966249ad_rel: fun(product_prod(vEBT_VEBT,extended_enat),fun(product_prod(vEBT_VEBT,extended_enat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list(vEBT_VEBT) * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: ( nat * nat * nat ) > nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: vEBT_VEBT > $o ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > fun(nat,$o) ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > $o ) ).

tff(sy_c_Wellfounded_Oless__than,type,
    less_than: set(product_prod(nat,nat)) ).

tff(sy_c_Wellfounded_Omeasure,type,
    measure: 
      !>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Omlex__prod,type,
    mlex_prod: 
      !>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set(product_prod(nat,nat)) ).

tff(sy_c_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_deg____,type,
    deg: nat ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList: list(vEBT_VEBT) ).

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (8969)
tff(fact_0_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_1_neq0__conv,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% neq0_conv
tff(fact_2_less__nat__zero__code,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% less_nat_zero_code
tff(fact_3_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Nb)
        <=> ( Nb = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_4__C3_Ohyps_C_I3_J,axiom,
    m = aa(nat,nat,suc,na) ).

% "3.hyps"(3)
tff(fact_5_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_6_gr0I,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% gr0I
tff(fact_7_not__gr0,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
    <=> ( Nb = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_8_not__less0,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% not_less0
tff(fact_9_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% less_zeroE
tff(fact_10_gr__implies__not0,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
     => ( Nb != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_11_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X2) = aa(nat,nat,suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_12_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_13_Suc__less__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ma)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ).

% Suc_less_eq
tff(fact_14_Suc__mono,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ma)),aa(nat,nat,suc,Nb)) ) ).

% Suc_mono
tff(fact_15_lessI,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Nb)) ).

% lessI
tff(fact_16_zero__less__Suc,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,Nb)) ).

% zero_less_Suc
tff(fact_17_less__Suc0,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( Nb = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_18_Suc__inject,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa(nat,nat,suc,Xb) = aa(nat,nat,suc,Y) )
     => ( Xb = Y ) ) ).

% Suc_inject
tff(fact_19_n__not__Suc__n,axiom,
    ! [Nb: nat] : ( Nb != aa(nat,nat,suc,Nb) ) ).

% n_not_Suc_n
tff(fact_20_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => ? [M: nat] : ( Nb = aa(nat,nat,suc,M) ) ) ).

% not0_implies_Suc
tff(fact_21_Zero__not__Suc,axiom,
    ! [Ma: nat] : ( zero_zero(nat) != aa(nat,nat,suc,Ma) ) ).

% Zero_not_Suc
tff(fact_22_Zero__neq__Suc,axiom,
    ! [Ma: nat] : ( zero_zero(nat) != aa(nat,nat,suc,Ma) ) ).

% Zero_neq_Suc
tff(fact_23_Suc__neq__Zero,axiom,
    ! [Ma: nat] : ( aa(nat,nat,suc,Ma) != zero_zero(nat) ) ).

% Suc_neq_Zero
tff(fact_24_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_25_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),Ma: nat,Nb: nat] :
      ( ! [X: nat] : aa(nat,$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,Ma),Nb) ) ) ) ).

% diff_induct
tff(fact_26_nat__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$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,Nb) ) ) ).

% nat_induct
tff(fact_27_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : ( Y != aa(nat,nat,suc,Nat3) ) ) ).

% old.nat.exhaust
tff(fact_28_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa(nat,nat,suc,X2) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_29_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : ( zero_zero(nat) != aa(nat,nat,suc,Nat2) ) ).

% old.nat.distinct(1)
tff(fact_30_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : ( aa(nat,nat,suc,Nat2) != zero_zero(nat) ) ).

% old.nat.distinct(2)
tff(fact_31_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : ( zero_zero(nat) != aa(nat,nat,suc,X2) ) ).

% nat.distinct(1)
tff(fact_32_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,Ma: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Ma))
      <=> ( Nb = Ma ) ) ) ).

% not_less_less_Suc_eq
tff(fact_33_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I2: nat] :
            ( ( J = aa(nat,nat,suc,I2) )
           => aa(nat,$o,P,I2) )
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
             => ( aa(nat,$o,P,aa(nat,nat,suc,I2))
               => aa(nat,$o,P,I2) ) )
         => aa(nat,$o,P,I) ) ) ) ).

% strict_inc_induct
tff(fact_34_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,I2),aa(nat,nat,suc,I2))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),K2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),P,I2),J2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),P,J2),K2)
                   => aa(nat,$o,aa(nat,fun(nat,$o),P,I2),K2) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_35_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_36_Suc__less__SucD,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ma)),aa(nat,nat,suc,Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ).

% Suc_less_SucD
tff(fact_37_less__antisym,axiom,
    ! [Nb: nat,Ma: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Ma))
       => ( Ma = Nb ) ) ) ).

% less_antisym
tff(fact_38_Suc__less__eq2,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),Ma)
    <=> ? [M2: nat] :
          ( ( Ma = aa(nat,nat,suc,M2) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M2) ) ) ).

% Suc_less_eq2
tff(fact_39_All__less__Suc,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Nb))
         => aa(nat,$o,P,I3) )
    <=> ( aa(nat,$o,P,Nb)
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
           => aa(nat,$o,P,I3) ) ) ) ).

% All_less_Suc
tff(fact_40_not__less__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Ma)) ) ).

% not_less_eq
tff(fact_41_less__Suc__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,suc,Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
        | ( Ma = Nb ) ) ) ).

% less_Suc_eq
tff(fact_42_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_43_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_44_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_45_ext,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 ) ) ).

% ext
tff(fact_46_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Nb))
          & aa(nat,$o,P,I3) )
    <=> ( aa(nat,$o,P,Nb)
        | ? [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
            & aa(nat,$o,P,I3) ) ) ) ).

% Ex_less_Suc
tff(fact_47_less__SucI,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,suc,Nb)) ) ).

% less_SucI
tff(fact_48_less__SucE,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,suc,Nb))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
       => ( Ma = Nb ) ) ) ).

% less_SucE
tff(fact_49_Suc__lessI,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
     => ( ( aa(nat,nat,suc,Ma) != Nb )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ma)),Nb) ) ) ).

% Suc_lessI
tff(fact_50_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_51_Suc__lessD,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ma)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ).

% Suc_lessD
tff(fact_52_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K)
     => ( ( K != aa(nat,nat,suc,I) )
       => ~ ! [J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_53_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,Ma: nat] :
          ( ! [N: nat] : aa(A,$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,$o),ord_less(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,Ma))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_54_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,N2: nat] :
          ( ! [N: nat] : aa(A,$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,$o),ord_less(nat),Nb),N2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,N2)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_55_less__Suc__eq__0__disj,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,suc,Nb))
    <=> ( ( Ma = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( Ma = aa(nat,nat,suc,J3) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_56_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ? [M: nat] : ( Nb = aa(nat,nat,suc,M) ) ) ).

% gr0_implies_Suc
tff(fact_57_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Nb))
         => aa(nat,$o,P,I3) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
           => aa(nat,$o,P,aa(nat,nat,suc,I3)) ) ) ) ).

% All_less_Suc2
tff(fact_58_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
    <=> ? [M3: nat] : ( Nb = aa(nat,nat,suc,M3) ) ) ).

% gr0_conv_Suc
tff(fact_59_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Nb))
          & aa(nat,$o,P,I3) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        | ? [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
            & aa(nat,$o,P,aa(nat,nat,suc,I3)) ) ) ) ).

% Ex_less_Suc2
tff(fact_60_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xb: A] :
          ( ( zero_zero(A) = Xb )
        <=> ( Xb = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_61_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X: A] :
              ( ! [Y4: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y4)),aa(A,B,F2,X))
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct_rule
tff(fact_62_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X: A] :
              ( ! [Y4: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y4)),aa(A,B,F2,X))
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct
tff(fact_63_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,$o),V: fun(A,nat),Xb: A] :
      ( ! [X: A] :
          ( ~ aa(A,$o,P,X)
         => ? [Y4: A] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V,Y4)),aa(A,nat,V,X))
              & ~ aa(A,$o,P,Y4) ) )
     => aa(A,$o,P,Xb) ) ).

% infinite_descent_measure
tff(fact_64_linorder__neqE__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xb) ) ) ).

% linorder_neqE_nat
tff(fact_65_infinite__descent,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa(nat,$o,P,N)
         => ? [M4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N)
              & ~ aa(nat,$o,P,M4) ) )
     => aa(nat,$o,P,Nb) ) ).

% infinite_descent
tff(fact_66_nat__less__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ! [M4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N)
             => aa(nat,$o,P,M4) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Nb) ) ).

% nat_less_induct
tff(fact_67_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Nb) ).

% less_irrefl_nat
tff(fact_68_less__not__refl3,axiom,
    ! [S: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S),Ta)
     => ( S != Ta ) ) ).

% less_not_refl3
tff(fact_69_less__not__refl2,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma)
     => ( Ma != Nb ) ) ).

% less_not_refl2
tff(fact_70_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Nb) ).

% less_not_refl
tff(fact_71_nat__neq__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( Ma != Nb )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma) ) ) ).

% nat_neq_iff
tff(fact_72_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Nb)
        <=> ( Nb != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_73_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Ma: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ma),Nb)
         => ( Nb != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_74_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Nb),zero_zero(A)) ) ).

% not_less_zero
tff(fact_75_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( ( Nb != zero_zero(A) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Nb) ) ) ).

% gr_zeroI
tff(fact_76_infinite__descent0__measure,axiom,
    ! [A: $tType,V: fun(A,nat),P: fun(A,$o),Xb: A] :
      ( ! [X: A] :
          ( ( aa(A,nat,V,X) = zero_zero(nat) )
         => aa(A,$o,P,X) )
     => ( ! [X: A] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,V,X))
           => ( ~ aa(A,$o,P,X)
             => ? [Y4: A] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V,Y4)),aa(A,nat,V,X))
                  & ~ aa(A,$o,P,Y4) ) ) )
       => aa(A,$o,P,Xb) ) ) ).

% infinite_descent0_measure
tff(fact_77_infinite__descent0,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( ~ aa(nat,$o,P,N)
             => ? [M4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N)
                  & ~ aa(nat,$o,P,M4) ) ) )
       => aa(nat,$o,P,Nb) ) ) ).

% infinite_descent0
tff(fact_78__C3_Ohyps_C_I4_J,axiom,
    deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),na),m) ).

% "3.hyps"(4)
tff(fact_79_exists__least__lemma,axiom,
    ! [P: fun(nat,$o)] :
      ( ~ aa(nat,$o,P,zero_zero(nat))
     => ( ? [X_1: nat] : aa(nat,$o,P,X_1)
       => ? [N: nat] :
            ( ~ aa(nat,$o,P,N)
            & aa(nat,$o,P,aa(nat,nat,suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_80_dependent__nat__choice,axiom,
    ! [A: $tType,P: fun(nat,fun(A,$o)),Q: fun(nat,fun(A,fun(A,$o)))] :
      ( ? [X_1: A] : aa(A,$o,aa(nat,fun(A,$o),P,zero_zero(nat)),X_1)
     => ( ! [X: A,N: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N),X)
           => ? [Y4: A] :
                ( aa(A,$o,aa(nat,fun(A,$o),P,aa(nat,nat,suc,N)),Y4)
                & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N),X),Y4) ) )
       => ? [F3: fun(nat,A)] :
          ! [N3: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N3),aa(nat,A,F3,N3))
            & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N3),aa(nat,A,F3,N3)),aa(nat,A,F3,aa(nat,nat,suc,N3))) ) ) ) ).

% dependent_nat_choice
tff(fact_81_vebt__buildup_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero(nat) )
     => ( ( Xb != aa(nat,nat,suc,zero_zero(nat)) )
       => ~ ! [Va: nat] : ( Xb != aa(nat,nat,suc,aa(nat,nat,suc,Va)) ) ) ) ).

% vebt_buildup.cases
tff(fact_82_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero(nat) )
     => ~ ! [N: nat] : ( Xb != aa(nat,nat,suc,N) ) ) ).

% list_decode.cases
tff(fact_83_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D1: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D1)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D2)
           => ? [E: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E),D1)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E),D2) ) ) ) ) ).

% field_lbound_gt_zero
tff(fact_84_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_85__C3_Ohyps_C_I1_J,axiom,
    vEBT_invar_vebt(summary,m) ).

% "3.hyps"(1)
tff(fact_86_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% deg_not_0
tff(fact_87_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% of_nat_0_less_iff
tff(fact_88_Suc__pred,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))) = Nb ) ) ).

% Suc_pred
tff(fact_89_even__odd__cases,axiom,
    ! [Xb: nat] :
      ( ! [N: nat] : ( Xb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N) )
     => ~ ! [N: nat] : ( Xb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,suc,N)) ) ) ).

% even_odd_cases
tff(fact_90_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_0_not
tff(fact_91_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_92_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_93_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_94_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Ma: nat,Nb: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Ma) = aa(nat,A,semiring_1_of_nat(A),Nb) )
        <=> ( Ma = Nb ) ) ) ).

% of_nat_eq_iff
tff(fact_95_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_96_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_97_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_98_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_99_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_100_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_101_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y) = zero_zero(A) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_102_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A,Y: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_103_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_104_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_105_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_106_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_107_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_108_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_109_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_110_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_111_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_112_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_113_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_114_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_115_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_116_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_117_add__Suc__right,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)) ) ).

% add_Suc_right
tff(fact_118_add__is__0,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb) = zero_zero(nat) )
    <=> ( ( Ma = zero_zero(nat) )
        & ( Nb = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_119_Nat_Oadd__0__right,axiom,
    ! [Ma: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),zero_zero(nat)) = Ma ) ).

% Nat.add_0_right
tff(fact_120_nat__add__left__cancel__less,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_121_Suc__diff__diff,axiom,
    ! [Ma: nat,Nb: 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,Ma)),Nb)),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),Ma),Nb)),K) ) ).

% Suc_diff_diff
tff(fact_122_diff__Suc__Suc,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Ma)),aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb) ) ).

% diff_Suc_Suc
tff(fact_123_diff__0__eq__0,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),Nb) = zero_zero(nat) ) ).

% diff_0_eq_0
tff(fact_124_diff__self__eq__0,axiom,
    ! [Ma: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Ma) = zero_zero(nat) ) ).

% diff_self_eq_0
tff(fact_125_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_126_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_127_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_128_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_129_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_130_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_131_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_132_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_133_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_134_of__nat__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),zero_zero(nat)) = zero_zero(A) ) ) ).

% of_nat_0
tff(fact_135_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),Nb) )
        <=> ( zero_zero(nat) = Nb ) ) ) ).

% of_nat_0_eq_iff
tff(fact_136_of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Ma: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Ma) = zero_zero(A) )
        <=> ( Ma = zero_zero(nat) ) ) ) ).

% of_nat_eq_0_iff
tff(fact_137_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Ma: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ) ).

% of_nat_less_iff
tff(fact_138_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Ma: nat,Nb: nat] : ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ).

% of_nat_add
tff(fact_139_add__gr__0,axiom,
    ! [Ma: nat,Nb: 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),Ma),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% add_gr_0
tff(fact_140_zero__less__diff,axiom,
    ! [Nb: nat,Ma: 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),Nb),Ma))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ).

% zero_less_diff
tff(fact_141__C3_Ohyps_C_I5_J,axiom,
    ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(summary),X_1) ).

% "3.hyps"(5)
tff(fact_142_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_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)) ) ) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_143_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_144_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_145_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_146_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_147_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_148_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_149_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_150_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_151_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_152_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_153_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_154_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A,D3: 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),D3) )
         => ( ( A2 = B2 )
          <=> ( C2 = D3 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_155_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_156_add_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) ) ) ).

% add.commute
tff(fact_157_add_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)) ) ) ).

% add.left_commute
tff(fact_158_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_159_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_160_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_161_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_162_diff__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) ) ) ).

% diff_right_commute
tff(fact_163_Nat_Odiff__cancel,axiom,
    ! [K: nat,Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb) ) ).

% Nat.diff_cancel
tff(fact_164_diff__cancel2,axiom,
    ! [Ma: nat,K: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb) ) ).

% diff_cancel2
tff(fact_165_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_166_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_167_diff__add__inverse,axiom,
    ! [Nb: nat,Ma: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Ma)),Nb) = Ma ) ).

% diff_add_inverse
tff(fact_168_diff__add__inverse2,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)),Nb) = Ma ) ).

% diff_add_inverse2
tff(fact_169_add__diff__add,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,C2: A,B2: A,D3: 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),D3)) = 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),D3)) ) ) ).

% add_diff_add
tff(fact_170_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_171_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_172_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_173_diff__add__0,axiom,
    ! [Nb: nat,Ma: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Ma)) = zero_zero(nat) ) ).

% diff_add_0
tff(fact_174_add__diff__inverse__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb)) = Ma ) ) ).

% add_diff_inverse_nat
tff(fact_175_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_176_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% reals_Archimedean2
tff(fact_177_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,$o,P,zero_zero(nat)) )
        & ! [D4: nat] :
            ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D4) )
           => aa(nat,$o,P,D4) ) ) ) ).

% nat_diff_split
tff(fact_178_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,$o,P,zero_zero(nat)) )
          | ? [D4: nat] :
              ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D4) )
              & ~ aa(nat,$o,P,D4) ) ) ) ).

% nat_diff_split_asm
tff(fact_179_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_180_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),D3),C2)
           => aa(A,$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),D3)) ) ) ) ).

% diff_strict_mono
tff(fact_181_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3) ) ) ) ).

% diff_eq_diff_less
tff(fact_182_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_183_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_184_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_185_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_186_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_187_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_188_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_189_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_190_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => aa(A,$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),D3)) ) ) ) ).

% add_strict_mono
tff(fact_191_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_192_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_193_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_194_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_195_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_196_minus__nat_Odiff__0,axiom,
    ! [Ma: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),zero_zero(nat)) = Ma ) ).

% minus_nat.diff_0
tff(fact_197_diffs0__imp__equal,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb) = zero_zero(nat) )
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma) = zero_zero(nat) )
       => ( Ma = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_198_diff__less__mono2,axiom,
    ! [Ma: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),Ma)) ) ) ).

% diff_less_mono2
tff(fact_199_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Nb: 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),Nb)),K) ) ).

% less_imp_diff_less
tff(fact_200_add__Suc__shift,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Ma)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),aa(nat,nat,suc,Nb)) ) ).

% add_Suc_shift
tff(fact_201_add__Suc,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Ma)),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)) ) ).

% add_Suc
tff(fact_202_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_203_plus__nat_Oadd__0,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),Nb) = Nb ) ).

% plus_nat.add_0
tff(fact_204_add__eq__self__zero,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb) = Ma )
     => ( Nb = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_205_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_206_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_207_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_208_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_209_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_210_trans__less__add1,axiom,
    ! [I: nat,J: nat,Ma: 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),Ma)) ) ).

% trans_less_add1
tff(fact_211_trans__less__add2,axiom,
    ! [I: nat,J: nat,Ma: 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),Ma),J)) ) ).

% trans_less_add2
tff(fact_212_less__add__eq__less,axiom,
    ! [K: nat,L: nat,Ma: nat,Nb: 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),Ma),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ) ).

% less_add_eq_less
tff(fact_213_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Ma: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Ma)),zero_zero(A)) ) ).

% of_nat_less_0_iff
tff(fact_214_of__nat__neq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Nb)) != zero_zero(A) ) ) ).

% of_nat_neq_0
tff(fact_215_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Ma: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ).

% less_imp_of_nat_less
tff(fact_216_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Ma: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,A,semiring_1_of_nat(A),Nb))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ) ).

% of_nat_less_imp_less
tff(fact_217_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_218_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_219_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_220_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)
         => ~ ! [C3: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3) )
               => ( C3 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_221_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_222_diff__less__Suc,axiom,
    ! [Ma: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb)),aa(nat,nat,suc,Ma)) ).

% diff_less_Suc
tff(fact_223_Suc__diff__Suc,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),aa(nat,nat,suc,Nb))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_224_diff__less,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb)),Ma) ) ) ).

% diff_less
tff(fact_225_one__is__add,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb) )
    <=> ( ( ( Ma = aa(nat,nat,suc,zero_zero(nat)) )
          & ( Nb = zero_zero(nat) ) )
        | ( ( Ma = zero_zero(nat) )
          & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_226_add__is__1,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( Ma = aa(nat,nat,suc,zero_zero(nat)) )
          & ( Nb = zero_zero(nat) ) )
        | ( ( Ma = zero_zero(nat) )
          & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_227_less__imp__Suc__add,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
     => ? [K2: nat] : ( Nb = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K2)) ) ) ).

% less_imp_Suc_add
tff(fact_228_less__iff__Suc__add,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
    <=> ? [K3: nat] : ( Nb = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K3)) ) ) ).

% less_iff_Suc_add
tff(fact_229_less__add__Suc2,axiom,
    ! [I: nat,Ma: 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),Ma),I))) ).

% less_add_Suc2
tff(fact_230_less__add__Suc1,axiom,
    ! [I: nat,Ma: 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),Ma))) ).

% less_add_Suc1
tff(fact_231_less__natE,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
     => ~ ! [Q2: nat] : ( Nb != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Q2)) ) ) ).

% less_natE
tff(fact_232_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_233_diff__Suc__less,axiom,
    ! [Nb: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,I))),Nb) ) ).

% diff_Suc_less
tff(fact_234_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => vEBT_invar_vebt(vEBT_vebt_buildup(Nb),Nb) ) ).

% buildup_gives_valid
tff(fact_235_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_236_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_237_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A)) ) ) ) ).

% add_less_zeroD
tff(fact_238_minus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( minus(A)
     => ! [A3: fun(B,A),B3: fun(B,A),Xb: 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),Xb) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,A3,Xb)),aa(B,A,B3,Xb)) ) ) ).

% minus_apply
tff(fact_239_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D3: nat] :
      ( vEBT_VEBT_valid(Ta,D3)
     => vEBT_invar_vebt(Ta,D3) ) ).

% valid_eq2
tff(fact_240_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D3: nat] :
      ( vEBT_invar_vebt(Ta,D3)
     => vEBT_VEBT_valid(Ta,D3) ) ).

% valid_eq1
tff(fact_241_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D3: nat] :
      ( vEBT_VEBT_valid(Ta,D3)
    <=> vEBT_invar_vebt(Ta,D3) ) ).

% valid_eq
tff(fact_242_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),A2: nat,B2: nat] :
      ( ! [A4: nat,B4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),P,A4),B4)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),P,B4),A4) )
     => ( ! [A4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,A4),zero_zero(nat))
       => ( ! [A4: nat,B4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,A4),B4)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,A4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B4)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,A2),B2) ) ) ) ).

% Euclid_induct
tff(fact_243_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_244_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_245_triangle__Suc,axiom,
    ! [Nb: nat] : ( nat_triangle(aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(Nb)),aa(nat,nat,suc,Nb)) ) ).

% triangle_Suc
tff(fact_246_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_247_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_248_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_249_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_250_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M: nat,N: nat] : ( Z != 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),N)) ) ).

% int_diff_cases
tff(fact_251_int__int__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa(nat,int,semiring_1_of_nat(int),Ma) = aa(nat,int,semiring_1_of_nat(int),Nb) )
    <=> ( Ma = Nb ) ) ).

% int_int_eq
tff(fact_252_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_253_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z)
    <=> ? [N4: 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,N4))) ) ) ).

% zless_iff_Suc_zadd
tff(fact_254_zadd__int__left,axiom,
    ! [Ma: nat,Nb: nat,Z: int] : ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Ma)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),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),Ma),Nb))),Z) ) ).

% zadd_int_left
tff(fact_255_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_256_fun__diff__def,axiom,
    ! [B: $tType,A: $tType] :
      ( minus(B)
     => ! [A3: fun(A,B),B3: fun(A,B),X3: 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),X3) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A3,X3)),aa(A,B,B3,X3)) ) ) ).

% fun_diff_def
tff(fact_257_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Ta)
     => ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),X_12) ) ).

% not_min_Null_member
tff(fact_258__C3_OIH_C_I2_J,axiom,
    ! [Xb: nat] :
      ( aa(nat,$o,vEBT_V8194947554948674370ptions(summary),Xb)
     => aa(nat,$o,vEBT_vebt_member(summary),Xb) ) ).

% "3.IH"(2)
tff(fact_259_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xb: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb),Xb) ).

% buildup_nothing_in_leaf
tff(fact_260_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_261_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_262_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_263_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xb: nat] : ~ vEBT_VEBT_membermima(vEBT_vebt_buildup(Nb),Xb) ).

% buildup_nothing_in_min_max
tff(fact_264_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_265_int__plus,axiom,
    ! [Nb: nat,Ma: nat] : ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Ma)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),aa(nat,int,semiring_1_of_nat(int),Ma)) ) ).

% int_plus
tff(fact_266_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_267_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_268_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_minNull(Ta)
     => ~ aa(nat,$o,vEBT_vebt_member(Ta),Xb) ) ).

% min_Null_member
tff(fact_269_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xb)
    <=> ( vEBT_V5719532721284313246member(Ta,Xb)
        | vEBT_VEBT_membermima(Ta,Xb) ) ) ).

% both_member_options_def
tff(fact_270_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tree,Nb)
     => ( aa(nat,$o,vEBT_vebt_member(Tree),Xb)
       => ( vEBT_V5719532721284313246member(Tree,Xb)
          | vEBT_VEBT_membermima(Tree,Xb) ) ) ) ).

% member_valid_both_member_options
tff(fact_271_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_272_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_273_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_274__C3_OIH_C_I1_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList))
     => ( vEBT_invar_vebt(X3,na)
        & ! [Xa: nat] :
            ( aa(nat,$o,vEBT_V8194947554948674370ptions(X3),Xa)
           => aa(nat,$o,vEBT_vebt_member(X3),Xa) ) ) ) ).

% "3.IH"(1)
tff(fact_275_Suc__diff__1,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) = Nb ) ) ).

% Suc_diff_1
tff(fact_276_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Tree,aa(nat,nat,suc,aa(nat,nat,suc,Nb)))
     => ? [Info: option(product_prod(nat,nat)),TreeList: list(vEBT_VEBT),S2: vEBT_VEBT] : ( Tree = vEBT_Node(Info,aa(nat,nat,suc,aa(nat,nat,suc,Nb)),TreeList,S2) ) ) ).

% deg_SUcn_Node
tff(fact_277_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_278_Leaf__0__not,axiom,
    ! [A2: $o,B2: $o] : ~ vEBT_invar_vebt(vEBT_Leaf((A2),(B2)),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_279_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ? [N: nat] :
              ( aa(nat,$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))),Xb) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_280_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_281_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_282_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_283_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb) ) ) ).

% zero_less_ceiling
tff(fact_284_add__eq__if,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb) = $ite(Ma = zero_zero(nat),Nb,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),Ma),one_one(nat))),Nb))) ) ).

% add_eq_if
tff(fact_285_deg__deg__n,axiom,
    ! [Info2: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info2,Dega,TreeLista,Summarya),Nb)
     => ( Dega = Nb ) ) ).

% deg_deg_n
tff(fact_286__C3_Ohyps_C_I6_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList))
     => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_1) ) ).

% "3.hyps"(6)
tff(fact_287_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
    <=> ? [A5: $o,B5: $o] : ( Ta = vEBT_Leaf((A5),(B5)) ) ) ).

% deg1Leaf
tff(fact_288_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
     => ? [A4: $o,B4: $o] : ( Ta = vEBT_Leaf((A4),(B4)) ) ) ).

% deg_1_Leaf
tff(fact_289_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( Nb = one_one(nat) )
       => ? [A4: $o,B4: $o] : ( Ta = vEBT_Leaf((A4),(B4)) ) ) ) ).

% deg_1_Leafy
tff(fact_290_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_291_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_292_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_293_uminus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( uminus(A)
     => ! [A3: fun(B,A),Xb: B] : ( aa(B,A,aa(fun(B,A),fun(B,A),uminus_uminus(fun(B,A)),A3),Xb) = aa(A,A,uminus_uminus(A),aa(B,A,A3,Xb)) ) ) ).

% uminus_apply
tff(fact_294_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_295_VEBT_Oinject_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,Y11: option(product_prod(nat,nat)),Y12: nat,Y13: list(vEBT_VEBT),Y14: vEBT_VEBT] :
      ( ( vEBT_Node(X11,X12,X13,X14) = vEBT_Node(Y11,Y12,Y13,Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_296_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leaf((X21),(X22)) = vEBT_Leaf((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X22)
        <=> (Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_297_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_298_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_299_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_300_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_301_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_302_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_303_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_304_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_305_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_306_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_307_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_308_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_309_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_310_nat__int,axiom,
    ! [Nb: nat] : ( aa(int,nat,nat2,aa(nat,int,semiring_1_of_nat(int),Nb)) = Nb ) ).

% nat_int
tff(fact_311_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_312_of__int__ceiling__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( ( aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xb)) = Xb )
        <=> ? [N4: int] : ( Xb = aa(int,A,ring_1_of_int(A),N4) ) ) ) ).

% of_int_ceiling_cancel
tff(fact_313_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_314_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_315_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_316_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_317_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_318_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_319_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_320_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_321_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_322_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_323_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_324_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_325_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_326_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_327_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Nb) = one_one(A) )
        <=> ( Nb = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_328_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),Nb) )
        <=> ( Nb = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_329_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_330_less__one,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),one_one(nat))
    <=> ( Nb = zero_zero(nat) ) ) ).

% less_one
tff(fact_331_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_332_diff__Suc__1,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Nb)),one_one(nat)) = Nb ) ).

% diff_Suc_1
tff(fact_333_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_334_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_335_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_336_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_337_negative__eq__positive,axiom,
    ! [Nb: nat,Ma: nat] :
      ( ( aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Nb)) = aa(nat,int,semiring_1_of_nat(int),Ma) )
    <=> ( ( Nb = zero_zero(nat) )
        & ( Ma = zero_zero(nat) ) ) ) ).

% negative_eq_positive
tff(fact_338_of__int__of__nat__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] : ( aa(int,A,ring_1_of_int(A),aa(nat,int,semiring_1_of_nat(int),Nb)) = aa(nat,A,semiring_1_of_nat(A),Nb) ) ) ).

% of_int_of_nat_eq
tff(fact_339_ceiling__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: nat] : ( archimedean_ceiling(A,aa(nat,A,semiring_1_of_nat(A),Nb)) = aa(nat,int,semiring_1_of_nat(int),Nb) ) ) ).

% ceiling_of_nat
tff(fact_340_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_341_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_342_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_343_of__nat__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Ma: nat] : ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Ma)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Ma)) ) ) ).

% of_nat_Suc
tff(fact_344_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_345_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_346_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xb) ) ) ).

% one_less_ceiling
tff(fact_347_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_348_ceiling__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : ( archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),one_one(int)) ) ) ).

% ceiling_add_one
tff(fact_349_negative__zless,axiom,
    ! [Nb: nat,Ma: 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,Nb)))),aa(nat,int,semiring_1_of_nat(int),Ma)) ).

% negative_zless
tff(fact_350_ceiling__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : ( archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,Xb)),one_one(int)) ) ) ).

% ceiling_diff_one
tff(fact_351_nat__zminus__int,axiom,
    ! [Nb: nat] : ( aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Nb))) = zero_zero(nat) ) ).

% nat_zminus_int
tff(fact_352_ceiling__add__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] : ( archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),Z) ) ) ).

% ceiling_add_of_int
tff(fact_353_ceiling__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] : ( archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,Xb)),Z) ) ) ).

% ceiling_diff_of_int
tff(fact_354_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_355_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_356_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option(product_prod(nat,nat)),X122: nat,X132: list(vEBT_VEBT),X142: vEBT_VEBT] : ( Y != vEBT_Node(X112,X122,X132,X142) )
     => ~ ! [X212: $o,X222: $o] : ( Y != vEBT_Leaf((X212),(X222)) ) ) ).

% VEBT.exhaust
tff(fact_357_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,X21: $o,X22: $o] : ( vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X22)) ) ).

% VEBT.distinct(1)
tff(fact_358_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [Xb: A] :
          ( ( one_one(A) = Xb )
        <=> ( Xb = one_one(A) ) ) ) ).

% one_reorient
tff(fact_359_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_360_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_361_fun__Compl__def,axiom,
    ! [B: $tType,A: $tType] :
      ( uminus(B)
     => ! [A3: fun(A,B),X3: A] : ( aa(A,B,aa(fun(A,B),fun(A,B),uminus_uminus(fun(A,B)),A3),X3) = aa(B,B,uminus_uminus(B),aa(A,B,A3,X3)) ) ) ).

% fun_Compl_def
tff(fact_362_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_363_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_364_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_365_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D3: nat] :
      ( vEBT_VEBT_valid(vEBT_Leaf((Uu),(Uv)),D3)
    <=> ( D3 = one_one(nat) ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_366_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_367_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),Xb) ) ) ).

% less_ceiling_iff
tff(fact_368_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_369_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_370_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z2: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),Xb) ) ).

% ex_of_int_less
tff(fact_371_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z2: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),Z2)) ) ).

% ex_less_of_int
tff(fact_372_vebt__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Leaf((A2),(B2))),Xb)
    <=> $ite(
          Xb = zero_zero(nat),
          (A2),
          $ite(Xb = one_one(nat),(B2),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_373_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf((A2),(B2)),Xb)
    <=> $ite(
          Xb = zero_zero(nat),
          (A2),
          $ite(Xb = one_one(nat),(B2),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_374_forall__pos__mono__1,axiom,
    ! [P: fun(real,$o),E2: real] :
      ( ! [D5: real,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D5),E)
         => ( aa(real,$o,P,D5)
           => aa(real,$o,P,E) ) )
     => ( ! [N: nat] : aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => aa(real,$o,P,E2) ) ) ) ).

% forall_pos_mono_1
tff(fact_375_real__arch__inverse,axiom,
    ! [E2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
    <=> ? [N4: nat] :
          ( ( N4 != 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),N4)))
          & 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),N4))),E2) ) ) ).

% real_arch_inverse
tff(fact_376_forall__pos__mono,axiom,
    ! [P: fun(real,$o),E2: real] :
      ( ! [D5: real,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D5),E)
         => ( aa(real,$o,P,D5)
           => aa(real,$o,P,E) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero(nat) )
           => aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N))) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => aa(real,$o,P,E2) ) ) ) ).

% forall_pos_mono
tff(fact_377_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_378_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_379_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_380_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_381_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_382_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_383_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_384_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_385_uminus__int__code_I1_J,axiom,
    aa(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).

% uminus_int_code(1)
tff(fact_386_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_387_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_388_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_389_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] : ~ vEBT_VEBT_membermima(vEBT_Leaf((Uu),(Uv)),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_390_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] : ~ vEBT_VEBT_minNull(vEBT_Leaf((Uu),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_391_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] : ~ vEBT_VEBT_minNull(vEBT_Leaf($true,(Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_392_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull(vEBT_Leaf($false,$false)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_393_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),archimedean_ceiling(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ).

% ceiling_less_cancel
tff(fact_394_nat__zero__as__int,axiom,
    zero_zero(nat) = aa(int,nat,nat2,zero_zero(int)) ).

% nat_zero_as_int
tff(fact_395_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_396_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_397_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_398_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_399_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_400_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_401_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_402_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_403_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_404_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_405_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_406_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_407_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_408_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_409_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_410_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_411_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_412_not__int__zless__negative,axiom,
    ! [Nb: nat,Ma: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Nb)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Ma))) ).

% not_int_zless_negative
tff(fact_413_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Ux: nat] : ~ vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero(nat),Uv,Uw),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_414_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_415_Suc__eq__plus1,axiom,
    ! [Nb: nat] : ( aa(nat,nat,suc,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)) ) ).

% Suc_eq_plus1
tff(fact_416_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_417_Suc__eq__plus1__left,axiom,
    ! [Nb: nat] : ( aa(nat,nat,suc,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),Nb) ) ).

% Suc_eq_plus1_left
tff(fact_418_diff__Suc__eq__diff__pred,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),one_one(nat))),Nb) ) ).

% diff_Suc_eq_diff_pred
tff(fact_419_invar__vebt_Ointros_I1_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_invar_vebt(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,zero_zero(nat))) ).

% invar_vebt.intros(1)
tff(fact_420_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_421_zless__nat__eq__int__zless,axiom,
    ! [Ma: nat,Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Ma)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_422_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_423_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(2)
tff(fact_424_int__minus,axiom,
    ! [Nb: nat,Ma: nat] : ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma)) = 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),Nb)),aa(nat,int,semiring_1_of_nat(int),Ma)))) ) ).

% int_minus
tff(fact_425_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_426_int__cases4,axiom,
    ! [Ma: int] :
      ( ! [N: nat] : ( Ma != 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)
           => ( Ma != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) ) ) ) ).

% int_cases4
tff(fact_427_nat__induct__non__zero,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$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,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_428_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ? [N: nat] : aa(A,$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)))),Xb) ) ) ).

% reals_Archimedean
tff(fact_429_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_430_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,Xb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(int,A,ring_1_of_int(A),Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Nb)),Xb) ) ) ).

% of_nat_less_of_int_iff
tff(fact_431_split__nat,axiom,
    ! [P: fun(nat,$o),I: int] :
      ( aa(nat,$o,P,aa(int,nat,nat2,I))
    <=> ( ! [N4: nat] :
            ( ( I = aa(nat,int,semiring_1_of_nat(int),N4) )
           => aa(nat,$o,P,N4) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),zero_zero(int))
         => aa(nat,$o,P,zero_zero(nat)) ) ) ) ).

% split_nat
tff(fact_432_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_433_negative__zless__0,axiom,
    ! [Nb: 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,Nb)))),zero_zero(int)) ).

% negative_zless_0
tff(fact_434_negD,axiom,
    ! [Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xb),zero_zero(int))
     => ? [N: nat] : ( Xb = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ) ).

% negD
tff(fact_435_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Ma)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_436_Suc__pred_H,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( Nb = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_437__C3_Oprems_C,axiom,
    aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(none(product_prod(nat,nat)),deg,treeList,summary)),xa) ).

% "3.prems"
tff(fact_438_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_439_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_440_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_441_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_442_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_443_inverse__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A] :
          ( ( aa(A,A,inverse_inverse(A),Xb) = one_one(A) )
        <=> ( Xb = one_one(A) ) ) ) ).

% inverse_eq_1_iff
tff(fact_444_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_445_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_446_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_447_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% compl_less_compl_iff
tff(fact_448_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,uminus_uminus(A),Xb) = aa(A,A,uminus_uminus(A),Y) )
        <=> ( Xb = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
tff(fact_449_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A] : ( aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),Xb)) = Xb ) ) ).

% boolean_algebra_class.boolean_algebra.double_compl
tff(fact_450_real__add__minus__iff,axiom,
    ! [Xb: real,A2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,uminus_uminus(real),A2)) = zero_zero(real) )
    <=> ( Xb = A2 ) ) ).

% real_add_minus_iff
tff(fact_451_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_452_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_453_nat__1,axiom,
    aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_454_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_455_real__0__less__add__iff,axiom,
    ! [Xb: 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),Xb),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),Xb)),Y) ) ).

% real_0_less_add_iff
tff(fact_456_real__add__less__0__iff,axiom,
    ! [Xb: 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),Xb),Y)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,uminus_uminus(real),Xb)) ) ).

% real_add_less_0_iff
tff(fact_457_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_458_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_459_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I)
     => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_gr_induct
tff(fact_460_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_461_nat__one__as__int,axiom,
    one_one(nat) = aa(int,nat,nat2,one_one(int)) ).

% nat_one_as_int
tff(fact_462_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)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_less_induct
tff(fact_463_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Xb: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),Xb) ).

% vebt_member.simps(2)
tff(fact_464_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : vEBT_VEBT_minNull(vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_465_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X3: A] :
        ? [Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X3) ) ).

% linordered_field_no_lb
tff(fact_466_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X3: A] :
        ? [X_12: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),X_12) ) ).

% linordered_field_no_ub
tff(fact_467_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT,Uz: nat] : ~ vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_468_int__Suc,axiom,
    ! [Nb: nat] : ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Nb)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),one_one(int)) ) ).

% int_Suc
tff(fact_469_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_470_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_471_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_472_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xb)
     => ( ( Xb != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : ( Xb != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) ) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_473_Suc__as__int,axiom,
    ! [X3: nat] : ( aa(nat,nat,suc,X3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X3)),one_one(int))) ) ).

% Suc_as_int
tff(fact_474_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,uminus_uminus(A),Xb))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_less_swap1
tff(fact_475_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Y)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Xb)),Y) ) ) ).

% compl_less_swap2
tff(fact_476_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_477_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_478_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_479_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_480_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_481_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_482_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_483_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_484_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_485_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_486_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_487_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_488_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_489_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_490_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_491_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),Xb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),one_one(A)) ) ) ) ).

% one_less_inverse_iff
tff(fact_492_Suc__if__eq,axiom,
    ! [A: $tType,F2: fun(nat,A),H: fun(nat,A),G: A,Nb: nat] :
      ( ! [N: nat] : ( aa(nat,A,F2,aa(nat,nat,suc,N)) = aa(nat,A,H,N) )
     => ( ( aa(nat,A,F2,zero_zero(nat)) = G )
       => ( aa(nat,A,F2,Nb) = $ite(Nb = zero_zero(nat),G,aa(nat,A,H,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ) ) ) ).

% Suc_if_eq
tff(fact_493_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_494_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Nb)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Nb)),one_one(A)))
           => ( archimedean_ceiling(A,Xb) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_495_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% ceiling_less_zero
tff(fact_496_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),Xb) ) ) ).

% zero_le_ceiling
tff(fact_497_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_498_Suc__le__mono,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,suc,Ma))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma) ) ).

% Suc_le_mono
tff(fact_499_le0,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Nb) ).

% le0
tff(fact_500_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_501_nat__add__left__cancel__le,axiom,
    ! [K: nat,Ma: nat,Nb: 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),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_502_diff__diff__cancel,axiom,
    ! [I: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_503_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),zero_zero(A))
        <=> ( Nb = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_504_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_505_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_506_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_507_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% compl_le_compl_iff
tff(fact_508_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Ma: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ) ).

% of_nat_le_iff
tff(fact_509_diff__is__0__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ).

% diff_is_0_eq
tff(fact_510_diff__is__0__eq_H,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_511_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_512_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_513_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_514_nat__ceiling__le__eq,axiom,
    ! [Xb: real,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,archimedean_ceiling(real,Xb))),A2)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(nat,real,semiring_1_of_nat(real),A2)) ) ).

% nat_ceiling_le_eq
tff(fact_515_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_516_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_517_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_518_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_519_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_520_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_521_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_522_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_523_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_524_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_525_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_526_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_527_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_528_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_529_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_530_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_531_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_532_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_533_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_534_negative__zle,axiom,
    ! [Nb: nat,Ma: 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),Nb))),aa(nat,int,semiring_1_of_nat(int),Ma)) ).

% negative_zle
tff(fact_535_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_536_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Ma: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Ma)),zero_zero(A))
        <=> ( Ma = zero_zero(nat) ) ) ) ).

% of_nat_le_0_iff
tff(fact_537_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_538_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_539_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_540_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_541_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_542_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_543_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_544_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_545_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_546_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_547_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_548_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A)) ) ) ).

% ceiling_le_zero
tff(fact_549_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A)) ) ) ).

% ceiling_less_one
tff(fact_550_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb) ) ) ).

% one_le_ceiling
tff(fact_551_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),one_one(A)) ) ) ).

% ceiling_le_one
tff(fact_552_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_553_minus__real__def,axiom,
    ! [Xb: real,Y: real] : ( aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,uminus_uminus(real),Y)) ) ).

% minus_real_def
tff(fact_554_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_555_verit__la__generic,axiom,
    ! [A2: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),Xb)
      | ( A2 = Xb )
      | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),A2) ) ).

% verit_la_generic
tff(fact_556_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_557_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,N2: nat] :
          ( ! [N: nat] : aa(A,$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,$o),ord_less_eq(nat),Nb),N2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,N2)) ) ) ) ).

% lift_Suc_mono_le
tff(fact_558_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,N2: nat] :
          ( ! [N: nat] : aa(A,$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,$o),ord_less_eq(nat),Nb),N2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N2)),aa(nat,A,F2,Nb)) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_559_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_560_zle__int,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Ma)),aa(nat,int,semiring_1_of_nat(int),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ).

% zle_int
tff(fact_561_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_562_nat__mono,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),Y)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,Xb)),aa(int,nat,nat2,Y)) ) ).

% nat_mono
tff(fact_563_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Y)),archimedean_ceiling(A,Xb)) ) ) ).

% ceiling_mono
tff(fact_564_nat__le__iff,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,Xb)),Nb)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),aa(nat,int,semiring_1_of_nat(int),Nb)) ) ).

% nat_le_iff
tff(fact_565_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% ceiling_le_iff
tff(fact_566_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),A2))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),A2) ) ) ).

% ceiling_le
tff(fact_567_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb) ) ).

% zero_le
tff(fact_568_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_569_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B6: A,A6: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B6),A6)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A6),B6) ) ) ).

% verit_comp_simplify1(3)
tff(fact_570_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_571_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_572_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_573_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)
        <=> ? [C4: A] : ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C4) ) ) ) ).

% le_iff_add
tff(fact_574_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_575_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)
         => ~ ! [C3: A] : ( B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3) ) ) ) ).

% less_eqE
tff(fact_576_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_577_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => aa(A,$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),D3)) ) ) ) ).

% add_mono
tff(fact_578_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_579_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_580_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_581_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D3) )
         => ( aa(A,$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),D3) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_582_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_583_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_584_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C2)
           => aa(A,$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),D3)) ) ) ) ).

% diff_mono
tff(fact_585_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_586_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_587_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_588_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),Xb)) ) ) ).

% compl_mono
tff(fact_589_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),Xb))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_le_swap1
tff(fact_590_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xb)),Y) ) ) ).

% compl_le_swap2
tff(fact_591_Suc__leD,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Ma)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ).

% Suc_leD
tff(fact_592_le__SucE,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,Nb))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
       => ( Ma = aa(nat,nat,suc,Nb) ) ) ) ).

% le_SucE
tff(fact_593_le__SucI,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,Nb)) ) ).

% le_SucI
tff(fact_594_Suc__le__D,axiom,
    ! [Nb: nat,M5: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),M5)
     => ? [M: nat] : ( M5 = aa(nat,nat,suc,M) ) ) ).

% Suc_le_D
tff(fact_595_le__Suc__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
        | ( Ma = aa(nat,nat,suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_596_Suc__n__not__le__n,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),Nb) ).

% Suc_n_not_le_n
tff(fact_597_not__less__eq__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),Ma) ) ).

% not_less_eq_eq
tff(fact_598_full__nat__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ! [M4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M4)),N)
             => aa(nat,$o,P,M4) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Nb) ) ).

% full_nat_induct
tff(fact_599_nat__induct__at__least,axiom,
    ! [Ma: nat,Nb: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => ( aa(nat,$o,P,Ma)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
             => ( aa(nat,$o,P,N)
               => aa(nat,$o,P,aa(nat,nat,suc,N)) ) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_600_transitive__stepwise__le,axiom,
    ! [Ma: nat,Nb: nat,R: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => ( ! [X: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,X),X)
       => ( ! [X: nat,Y3: nat,Z2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R,X),Y3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R,Y3),Z2)
               => aa(nat,$o,aa(nat,fun(nat,$o),R,X),Z2) ) )
         => ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,N),aa(nat,nat,suc,N))
           => aa(nat,$o,aa(nat,fun(nat,$o),R,Ma),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_601_le__0__eq,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),zero_zero(nat))
    <=> ( Nb = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_602_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_603_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_604_less__eq__nat_Osimps_I1_J,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Nb) ).

% less_eq_nat.simps(1)
tff(fact_605_int__le__real__less,axiom,
    ! [Nb: int,Ma: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),Ma)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(int,real,ring_1_of_int(real),Nb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Ma)),one_one(real))) ) ).

% int_le_real_less
tff(fact_606_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% real_arch_simple
tff(fact_607_real__add__le__0__iff,axiom,
    ! [Xb: 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),Xb),Y)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,uminus_uminus(real),Xb)) ) ).

% real_add_le_0_iff
tff(fact_608_real__0__le__add__iff,axiom,
    ! [Xb: 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),Xb),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),Xb)),Y) ) ).

% real_0_le_add_iff
tff(fact_609_less__mono__imp__le__mono,axiom,
    ! [F2: fun(nat,nat),I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,I2)),aa(nat,nat,F2,J2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,F2,I)),aa(nat,nat,F2,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_610_le__neq__implies__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => ( ( Ma != Nb )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ) ).

% le_neq_implies_less
tff(fact_611_less__or__eq__imp__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
        | ( Ma = Nb ) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ).

% less_or_eq_imp_le
tff(fact_612_le__eq__less__or__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
        | ( Ma = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_613_less__imp__le__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ).

% less_imp_le_nat
tff(fact_614_nat__less__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
        & ( Ma != Nb ) ) ) ).

% nat_less_le
tff(fact_615_less__eq__real__def,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_real_def
tff(fact_616_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_617_nat__le__iff__add,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
    <=> ? [K3: nat] : ( Nb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K3) ) ) ).

% nat_le_iff_add
tff(fact_618_trans__le__add2,axiom,
    ! [I: nat,J: nat,Ma: 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),Ma),J)) ) ).

% trans_le_add2
tff(fact_619_trans__le__add1,axiom,
    ! [I: nat,J: nat,Ma: 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),Ma)) ) ).

% trans_le_add1
tff(fact_620_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_621_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_622_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_623_add__leD2,axiom,
    ! [Ma: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ).

% add_leD2
tff(fact_624_add__leD1,axiom,
    ! [Ma: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ).

% add_leD1
tff(fact_625_le__add2,axiom,
    ! [Nb: nat,Ma: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)) ).

% le_add2
tff(fact_626_le__add1,axiom,
    ! [Nb: nat,Ma: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Ma)) ).

% le_add1
tff(fact_627_add__leE,axiom,
    ! [Ma: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K)),Nb)
     => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ) ).

% add_leE
tff(fact_628_diff__le__mono2,axiom,
    ! [Ma: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),Ma)) ) ).

% diff_le_mono2
tff(fact_629_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_630_diff__le__self,axiom,
    ! [Ma: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb)),Ma) ).

% diff_le_self
tff(fact_631_diff__le__mono,axiom,
    ! [Ma: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),L)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),L)) ) ).

% diff_le_mono
tff(fact_632_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_633_le__diff__iff,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ) ) ).

% le_diff_iff
tff(fact_634_eq__diff__iff,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K) )
        <=> ( Ma = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_635_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z2: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),Z2)) ) ).

% ex_le_of_int
tff(fact_636_int__less__real__le,axiom,
    ! [Nb: int,Ma: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),Ma)
    <=> 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),Nb)),one_one(real))),aa(int,real,ring_1_of_int(real),Ma)) ) ).

% int_less_real_le
tff(fact_637_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_638_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_639_le__nat__iff,axiom,
    ! [K: int,Nb: 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),Nb),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),Nb)),K) ) ) ).

% le_nat_iff
tff(fact_640_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),zero_zero(A)) ) ).

% not_one_le_zero
tff(fact_641_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_642_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% zero_less_one_class.zero_le_one
tff(fact_643_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_644_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_645_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_646_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_647_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_648_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_649_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$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),Xb),Y) = zero_zero(A) )
            <=> ( ( Xb = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_650_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$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),Xb),Y) = zero_zero(A) )
            <=> ( ( Xb = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_651_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_652_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_653_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_654_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => aa(A,$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),D3)) ) ) ) ).

% add_le_less_mono
tff(fact_655_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => aa(A,$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),D3)) ) ) ) ).

% add_less_le_mono
tff(fact_656_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),aa(A,A,aa(A,fun(A,A),minus_minus(A),Nb),K)) ) ) ).

% add_le_imp_le_diff
tff(fact_657_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Nb: A,J: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Nb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Nb),K)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_658_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_659_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_660_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_661_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_662_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_663_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_664_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_665_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_666_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_667_diff__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 ) ) ) ).

% diff_add
tff(fact_668_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_669_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_670_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_671_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_672_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_0_le_iff
tff(fact_673_Suc__leI,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Ma)),Nb) ) ).

% Suc_leI
tff(fact_674_Suc__le__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Ma)),Nb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ).

% Suc_le_eq
tff(fact_675_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_676_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_677_Suc__le__lessD,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Ma)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ).

% Suc_le_lessD
tff(fact_678_le__less__Suc__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Ma))
      <=> ( Nb = Ma ) ) ) ).

% le_less_Suc_eq
tff(fact_679_less__Suc__eq__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ).

% less_Suc_eq_le
tff(fact_680_less__eq__Suc__le,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),Ma) ) ).

% less_eq_Suc_le
tff(fact_681_le__imp__less__Suc,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_682_ex__least__nat__le,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
            & ! [I4: nat] :
                ( aa(nat,$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_683_mono__nat__linear__lb,axiom,
    ! [F2: fun(nat,nat),Ma: nat,K: nat] :
      ( ! [M: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,M)),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,Ma)),K)),aa(nat,nat,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K))) ) ).

% mono_nat_linear_lb
tff(fact_684_Suc__diff__le,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Ma)),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb)) ) ) ).

% Suc_diff_le
tff(fact_685_less__diff__iff,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ) ) ).

% less_diff_iff
tff(fact_686_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_687_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_688_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_689_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_690_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_691_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_692_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_693_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_694_eq__nat__nat__iff,axiom,
    ! [Z: int,Z3: 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)),Z3)
       => ( ( aa(int,nat,nat2,Z) = aa(int,nat,nat2,Z3) )
        <=> ( Z = Z3 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_695_all__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ! [X_13: nat] : aa(nat,$o,P,X_13)
    <=> ! [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_696_ex__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ? [X_13: nat] : aa(nat,$o,P,X_13)
    <=> ? [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_697_nat__le__real__less,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),Ma)),one_one(real))) ) ).

% nat_le_real_less
tff(fact_698_nat__less__real__le,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma)
    <=> 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),Nb)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),Ma)) ) ).

% nat_less_real_le
tff(fact_699_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I)
     => ( aa(int,$o,P,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_ge_induct
tff(fact_700_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z)
    <=> ? [N4: nat] : ( Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),N4)) ) ) ).

% zle_iff_zadd
tff(fact_701_int__le__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),K)
     => ( aa(int,$o,P,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_le_induct
tff(fact_702_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xb))) ) ).

% le_of_int_ceiling
tff(fact_703_real__nat__ceiling__ge,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(nat,real,semiring_1_of_nat(real),aa(int,nat,nat2,archimedean_ceiling(real,Xb)))) ).

% real_nat_ceiling_ge
tff(fact_704_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_705_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_706_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_707_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_708_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_709_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_710_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( ! [E: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% field_le_epsilon
tff(fact_711_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_712_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_713_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_714_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_715_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_716_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_717_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),Xb)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ) ).

% inverse_le_1_iff
tff(fact_718_ex__least__nat__less,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),Nb)
            & ! [I4: nat] :
                ( aa(nat,$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_719_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [Nb: nat,Ma: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ) ).

% of_nat_diff
tff(fact_720_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_721_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_722_int__zle__neg,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Nb)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Ma)))
    <=> ( ( Nb = zero_zero(nat) )
        & ( Ma = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_723_negative__zle__0,axiom,
    ! [Nb: 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),Nb))),zero_zero(int)) ).

% negative_zle_0
tff(fact_724_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_725_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_726_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_727_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_728_int__eq__iff,axiom,
    ! [Ma: nat,Z: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),Ma) = Z )
    <=> ( ( Ma = 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_729_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),archimedean_ceiling(A,Y))) ) ).

% ceiling_add_le
tff(fact_730_int__induct,axiom,
    ! [P: fun(int,$o),K: int,I: int] :
      ( aa(int,$o,P,K)
     => ( ! [I2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
           => ( aa(int,$o,P,I2)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_induct
tff(fact_731_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archimedean_ceiling(A,R2)))) ) ).

% of_nat_ceiling
tff(fact_732_dbl__inc__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xb: A] : ( neg_numeral_dbl_inc(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb)),one_one(A)) ) ) ).

% dbl_inc_def
tff(fact_733_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),Xb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),one_one(A)) ) ) ) ).

% one_le_inverse_iff
tff(fact_734_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),Xb)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xb) ) ) ) ).

% inverse_less_1_iff
tff(fact_735_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_736_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [X: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X)),Xb)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int))))
          & ! [Y4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y4)),Xb)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y4),one_one(int)))) )
             => ( Y4 = X ) ) ) ) ).

% floor_exists1
tff(fact_737_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),Xb)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int)))) ) ) ).

% floor_exists
tff(fact_738_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R2),one_one(A))) ) ).

% of_int_ceiling_le_add_one
tff(fact_739_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),one_one(A))),R2) ) ).

% of_int_ceiling_diff_one_le
tff(fact_740_not__zle__0__negative,axiom,
    ! [Nb: 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,Nb)))) ).

% not_zle_0_negative
tff(fact_741_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_742_nat__eq__iff,axiom,
    ! [W: int,Ma: nat] :
      ( ( aa(int,nat,nat2,W) = Ma )
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W),W = aa(nat,int,semiring_1_of_nat(int),Ma),Ma = zero_zero(nat)) ) ).

% nat_eq_iff
tff(fact_743_nat__eq__iff2,axiom,
    ! [Ma: nat,W: int] :
      ( ( Ma = 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),Ma),Ma = zero_zero(nat)) ) ).

% nat_eq_iff2
tff(fact_744_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_745_nat__add__distrib,axiom,
    ! [Z: int,Z3: 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)),Z3)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),Z3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z3)) ) ) ) ).

% nat_add_distrib
tff(fact_746_nat__diff__distrib,axiom,
    ! [Z3: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z3),Z)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z3)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z3)) ) ) ) ).

% nat_diff_distrib
tff(fact_747_nat__diff__distrib_H,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( 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),Xb),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Xb)),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_748_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archimedean_ceiling(A,Ta))
        <=> ! [I3: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),I3)),one_one(A))),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),aa(int,A,ring_1_of_int(A),I3)) )
             => aa(int,$o,P,I3) ) ) ) ).

% ceiling_split
tff(fact_749_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: int] :
          ( ( archimedean_ceiling(A,Xb) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),A2)),one_one(A))),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),A2)) ) ) ) ).

% ceiling_eq_iff
tff(fact_750_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),Z))
           => ( archimedean_ceiling(A,Xb) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_751_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xb))),one_one(A))),Xb)
          & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xb))) ) ) ).

% ceiling_correct
tff(fact_752_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archimedean_ceiling(A,Xb))
        <=> aa(A,$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))),Xb) ) ) ).

% le_ceiling_iff
tff(fact_753_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),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_754_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_755_nat__less__iff,axiom,
    ! [W: int,Ma: 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)),Ma)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),aa(nat,int,semiring_1_of_nat(int),Ma)) ) ) ).

% nat_less_iff
tff(fact_756_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xb: A] : ( neg_numeral_dbl_dec(A,Xb) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb)),one_one(A)) ) ) ).

% dbl_dec_def
tff(fact_757_zdiff__int__split,axiom,
    ! [P: fun(int,$o),Xb: nat,Y: nat] :
      ( aa(int,$o,P,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xb),Y)))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xb)
         => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),Xb)),aa(nat,int,semiring_1_of_nat(int),Y))) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y)
         => aa(int,$o,P,zero_zero(int)) ) ) ) ).

% zdiff_int_split
tff(fact_758_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_759_conj__le__cong,axiom,
    ! [Xb: int,X5: int,P: $o,P2: $o] :
      ( ( Xb = X5 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X5)
         => ( (P)
          <=> (P2) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
            & (P) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X5)
            & (P2) ) ) ) ) ).

% conj_le_cong
tff(fact_760_imp__le__cong,axiom,
    ! [Xb: int,X5: int,P: $o,P2: $o] :
      ( ( Xb = X5 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X5)
         => ( (P)
          <=> (P2) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
           => (P) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X5)
           => (P2) ) ) ) ) ).

% imp_le_cong
tff(fact_761_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
         => 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),I2),one_one(nat)))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Nb))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
              & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_762_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),Nb: nat] :
      ( aa(A,$o,P,K)
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
           => ? [Y4: A] :
                ( aa(A,$o,P,Y4)
                & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y4)),aa(A,nat,F2,X)) ) )
       => ? [Y3: A] :
            ( aa(A,$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)),Nb)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_763_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
            | ( Xb = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_764_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% linorder_le_less_linear
tff(fact_765_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_766_abs__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : ( aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A2)) = aa(A,A,abs_abs(A),A2) ) ) ).

% abs_abs
tff(fact_767_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_768_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_769_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_770_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_771_abs__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_0
tff(fact_772_abs__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_1
tff(fact_773_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_774_abs__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(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
tff(fact_775_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_776_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : ( aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),Nb)) = aa(nat,A,semiring_1_of_nat(A),Nb) ) ) ).

% abs_of_nat
tff(fact_777_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_778_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_779_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_780_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_781_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_782_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_783_of__int__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int] : ( aa(int,A,ring_1_of_int(A),aa(int,int,abs_abs(int),Xb)) = aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Xb)) ) ) ).

% of_int_abs
tff(fact_784_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_785_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_786_complete__real,axiom,
    ! [S3: set(real)] :
      ( ? [X3: real] : member(real,X3,S3)
     => ( ? [Z4: real] :
          ! [X: real] :
            ( member(real,X,S3)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Z4) )
       => ? [Y3: real] :
            ( ! [X3: real] :
                ( member(real,X3,S3)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),Y3) )
            & ! [Z4: real] :
                ( ! [X: real] :
                    ( member(real,X,S3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Z4) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),Z4) ) ) ) ) ).

% complete_real
tff(fact_787_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)
            & ! [Y4: nat] :
                ( aa(nat,$o,P,Y4)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y4),X) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_788_nat__le__linear,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
      | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma) ) ).

% nat_le_linear
tff(fact_789_le__antisym,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
       => ( Ma = Nb ) ) ) ).

% le_antisym
tff(fact_790_eq__imp__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( Ma = Nb )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ).

% eq_imp_le
tff(fact_791_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_792_le__refl,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Nb) ).

% le_refl
tff(fact_793_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_794_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_795_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_796_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_797_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_798_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,abs_abs(A),Xb) = aa(A,A,abs_abs(A),Y) )
        <=> ( ( Xb = Y )
            | ( Xb = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% abs_eq_iff
tff(fact_799_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_800_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_801_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_802_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_803_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_804_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_805_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_806_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_807_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_808_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_809_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_810_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_811_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_812_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [Xb: A] :
          ( ! [E: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),E) )
         => ( Xb = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_813_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_814_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,abs_abs(A),B2) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            & ( ( B2 = A2 )
              | ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_815_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
            & ( ( A2 = B2 )
              | ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_816_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X3: A] :
          ( aa(A,A,abs_abs(A),X3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),zero_zero(A)),aa(A,A,uminus_uminus(A),X3),X3) ) ) ).

% abs_if_raw
tff(fact_817_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_818_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_819_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_820_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A,C2: A,D3: A] : aa(A,$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),D3)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)))) ) ).

% abs_diff_triangle_ineq
tff(fact_821_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,A2: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A2))),R2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),R2)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R2)) ) ) ) ).

% abs_diff_le_iff
tff(fact_822_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,A2: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A2))),R2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),R2)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R2)) ) ) ) ).

% abs_diff_less_iff
tff(fact_823_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_824_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_825_minf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F4: B] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z2)
         => ( F4 = F4 ) ) ) ).

% minf(11)
tff(fact_826_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),X3) ) ) ).

% minf(7)
tff(fact_827_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Ta) ) ) ).

% minf(5)
tff(fact_828_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z2)
         => ( X3 != Ta ) ) ) ).

% minf(4)
tff(fact_829_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z2)
         => ( X3 != Ta ) ) ) ).

% minf(3)
tff(fact_830_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q3: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
             => ( aa(A,$o,P,X)
              <=> aa(A,$o,P2,X) ) )
         => ( ? [Z4: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
               => ( aa(A,$o,Q,X)
                <=> aa(A,$o,Q3,X) ) )
           => ? [Z2: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z2)
               => ( ( aa(A,$o,P,X3)
                    | aa(A,$o,Q,X3) )
                <=> ( aa(A,$o,P2,X3)
                    | aa(A,$o,Q3,X3) ) ) ) ) ) ) ).

% minf(2)
tff(fact_831_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q3: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
             => ( aa(A,$o,P,X)
              <=> aa(A,$o,P2,X) ) )
         => ( ? [Z4: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
               => ( aa(A,$o,Q,X)
                <=> aa(A,$o,Q3,X) ) )
           => ? [Z2: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z2)
               => ( ( aa(A,$o,P,X3)
                    & aa(A,$o,Q,X3) )
                <=> ( aa(A,$o,P2,X3)
                    & aa(A,$o,Q3,X3) ) ) ) ) ) ) ).

% minf(1)
tff(fact_832_pinf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F4: B] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X3)
         => ( F4 = F4 ) ) ) ).

% pinf(11)
tff(fact_833_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),X3) ) ) ).

% pinf(7)
tff(fact_834_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Ta) ) ) ).

% pinf(5)
tff(fact_835_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X3)
         => ( X3 != Ta ) ) ) ).

% pinf(4)
tff(fact_836_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X3)
         => ( X3 != Ta ) ) ) ).

% pinf(3)
tff(fact_837_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q3: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X)
             => ( aa(A,$o,P,X)
              <=> aa(A,$o,P2,X) ) )
         => ( ? [Z4: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X)
               => ( aa(A,$o,Q,X)
                <=> aa(A,$o,Q3,X) ) )
           => ? [Z2: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X3)
               => ( ( aa(A,$o,P,X3)
                    | aa(A,$o,Q,X3) )
                <=> ( aa(A,$o,P2,X3)
                    | aa(A,$o,Q3,X3) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_838_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q3: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X)
             => ( aa(A,$o,P,X)
              <=> aa(A,$o,P2,X) ) )
         => ( ? [Z4: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X)
               => ( aa(A,$o,Q,X)
                <=> aa(A,$o,Q3,X) ) )
           => ? [Z2: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X3)
               => ( ( aa(A,$o,P,X3)
                    & aa(A,$o,Q,X3) )
                <=> ( aa(A,$o,P2,X3)
                    & aa(A,$o,Q3,X3) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_839_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xb: A] :
        ? [Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),Xb) ) ).

% lt_ex
tff(fact_840_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xb: A] :
        ? [X_12: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X_12) ) ).

% gt_ex
tff(fact_841_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ? [Z2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),Y) ) ) ) ).

% dense
tff(fact_842_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( Xb != Y ) ) ) ).

% less_imp_neq
tff(fact_843_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_844_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_845_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_846_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [X: A] :
              ( ! [Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X)
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X) )
         => aa(A,$o,P,A2) ) ) ).

% less_induct
tff(fact_847_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% antisym_conv3
tff(fact_848_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( ( Xb != Y )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ) ).

% linorder_cases
tff(fact_849_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_850_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_851_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_13: A] : aa(A,$o,P,X_13)
        <=> ? [N4: A] :
              ( aa(A,$o,P,N4)
              & ! [M3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M3),N4)
                 => ~ aa(A,$o,P,M3) ) ) ) ) ).

% exists_least_iff
tff(fact_852_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A2: A,B2: A] :
          ( ! [A4: A,B4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B4)
             => aa(A,$o,aa(A,fun(A,$o),P,A4),B4) )
         => ( ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),P,A4),A4)
           => ( ! [A4: A,B4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),P,B4),A4)
                 => aa(A,$o,aa(A,fun(A,$o),P,A4),B4) )
             => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) ) ) ) ) ).

% linorder_less_wlog
tff(fact_853_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_854_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
            | ( Xb = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_855_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_856_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_857_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_858_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ) ).

% linorder_neqE
tff(fact_859_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% order_less_asym
tff(fact_860_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ) ).

% linorder_neq_iff
tff(fact_861_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_862_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z) ) ) ) ).

% order_less_trans
tff(fact_863_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_864_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_865_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Xb) ) ).

% order_less_irrefl
tff(fact_866_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_867_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_868_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% order_less_not_sym
tff(fact_869_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,P: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
           => (P) ) ) ) ).

% order_less_imp_triv
tff(fact_870_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
          | ( Xb = Y )
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% linorder_less_linear
tff(fact_871_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( Xb != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_872_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( Y != Xb ) ) ) ).

% order_less_imp_not_eq2
tff(fact_873_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% order_less_imp_not_less
tff(fact_874_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,$o,P,X)
            & ! [Y4: A] :
                ( aa(A,$o,P,Y4)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y4)),aa(A,nat,F2,X)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_875_subset__code_I1_J,axiom,
    ! [A: $tType,Xs: list(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),B3)
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => member(A,X4,B3) ) ) ).

% subset_code(1)
tff(fact_876_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] : aa(A,$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),Xb))) ) ).

% abs_add_one_gt_zero
tff(fact_877_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Nb))),Xb)
         => ( ( Nb = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ) ).

% of_int_leD
tff(fact_878_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Nb))),Xb)
         => ( ( Nb = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xb) ) ) ) ).

% of_int_lessD
tff(fact_879_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_880_nat__intermed__int__val,axiom,
    ! [Ma: nat,Nb: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),I2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb) )
         => 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,I2))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,Ma)),K)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Nb))
           => ? [I2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),I2)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
                & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_881_nat__ivt__aux,axiom,
    ! [Nb: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
         => 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,I2))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Nb))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
              & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_882_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),X3) ) ) ).

% minf(8)
tff(fact_883_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Ta) ) ) ).

% minf(6)
tff(fact_884_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),X3) ) ) ).

% pinf(8)
tff(fact_885_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Ta) ) ) ).

% pinf(6)
tff(fact_886_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ).

% leD
tff(fact_887_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% leI
tff(fact_888_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_889_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% antisym_conv1
tff(fact_890_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% antisym_conv2
tff(fact_891_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_892_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_893_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ) ).

% less_le_not_le
tff(fact_894_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ).

% not_le_imp_less
tff(fact_895_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_896_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_897_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_898_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_899_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_900_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xb)
         => ( ! [W2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),W2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W2),Xb)
                 => aa(A,$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_901_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( ! [W2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),W2)
               => ( aa(A,$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_902_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_903_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_904_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_905_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_906_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_907_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_908_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_909_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
            | ( Xb = Y ) ) ) ) ).

% order_le_less
tff(fact_910_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & ( Xb != Y ) ) ) ) ).

% order_less_le
tff(fact_911_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% linorder_not_le
tff(fact_912_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% linorder_not_less
tff(fact_913_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% order_less_imp_le
tff(fact_914_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_915_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_916_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z) ) ) ) ).

% order_le_less_trans
tff(fact_917_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z) ) ) ) ).

% order_less_le_trans
tff(fact_918_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_919_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_920_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_921_nat__descend__induct,axiom,
    ! [Nb: nat,P: fun(nat,$o),Ma: nat] :
      ( ! [K2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K2)
         => aa(nat,$o,P,K2) )
     => ( ! [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
           => ( ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),I4)
                 => aa(nat,$o,P,I4) )
             => aa(nat,$o,P,K2) ) )
       => aa(nat,$o,P,Ma) ) ) ).

% nat_descend_induct
tff(fact_922_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,P,A2)
           => ( ~ aa(A,$o,P,B2)
             => ? [C3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
                  & ! [X3: A] :
                      ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X3)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),C3) )
                     => aa(A,$o,P,X3) )
                  & ! [D6: A] :
                      ( ! [X: A] :
                          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X)
                            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),D6) )
                         => aa(A,$o,P,X) )
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D6),C3) ) ) ) ) ) ) ).

% complete_interval
tff(fact_923_incr__lemma,axiom,
    ! [D3: int,Z: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),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),Xb),Z))),one_one(int))),D3))) ) ).

% incr_lemma
tff(fact_924_decr__lemma,axiom,
    ! [D3: int,Xb: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),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),Xb),Z))),one_one(int))),D3))),Z) ) ).

% decr_lemma
tff(fact_925_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)
       => ( divide_divide(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(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_926_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] : ( aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Leaf((X21),(X22))) = zero_zero(nat) ) ).

% VEBT.size_gen(2)
tff(fact_927_bset_I6_J,axiom,
    ! [D7: int,B3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D7)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B3)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X3),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D7)),Ta) ) ) ) ).

% bset(6)
tff(fact_928_bset_I8_J,axiom,
    ! [D7: int,Ta: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D7)
     => ( member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Ta),one_one(int)),B3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),X3)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D7)) ) ) ) ) ).

% bset(8)
tff(fact_929_aset_I6_J,axiom,
    ! [D7: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D7)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int)),A3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X3),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D7)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_930_aset_I8_J,axiom,
    ! [D7: int,A3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D7)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A3)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),X3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D7)) ) ) ) ).

% aset(8)
tff(fact_931_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_932_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_933_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_934_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_935_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_936_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : ( divide_divide(A,A2,zero_zero(A)) = zero_zero(A) ) ) ).

% div_by_0
tff(fact_937_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : ( divide_divide(A,zero_zero(A),A2) = zero_zero(A) ) ) ).

% div_0
tff(fact_938_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : ( divide_divide(A,A2,zero_zero(A)) = zero_zero(A) ) ) ).

% division_ring_divide_zero
tff(fact_939_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( divide_divide(A,A2,C2) = divide_divide(A,B2,C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_940_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( divide_divide(A,C2,A2) = divide_divide(A,C2,B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_941_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( divide_divide(A,A2,B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_942_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_943_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_944_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_945_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_946_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_947_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : ( divide_divide(A,A2,one_one(A)) = A2 ) ) ).

% div_by_1
tff(fact_948_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Ma: nat,Nb: nat] : ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ).

% of_nat_mult
tff(fact_949_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_950_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_951_inverse__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : ( aa(A,A,inverse_inverse(A),divide_divide(A,A2,B2)) = divide_divide(A,B2,A2) ) ) ).

% inverse_divide
tff(fact_952_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_953_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_954_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_955_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_956_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(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_957_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(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_958_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: 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),divide_divide(A,A2,B2)) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_959_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(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)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_960_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(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)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_961_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(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)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_962_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(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)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_963_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( divide_divide(A,A2,A2) = one_one(A) ) ) ) ).

% div_self
tff(fact_964_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( divide_divide(A,A2,B2) = one_one(A) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_965_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( one_one(A) = divide_divide(A,A2,B2) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_966_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( divide_divide(A,A2,A2) = one_one(A) ) ) ) ).

% divide_self
tff(fact_967_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( divide_divide(A,A2,A2) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ) ).

% divide_self_if
tff(fact_968_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( divide_divide(A,B2,A2) = one_one(A) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_969_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( one_one(A) = divide_divide(A,B2,A2) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_970_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( divide_divide(A,one_one(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_971_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = divide_divide(A,one_one(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_972_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_973_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_974_divide__minus1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A] : ( divide_divide(A,Xb,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),Xb) ) ) ).

% divide_minus1
tff(fact_975_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_976_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,one_one(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% divide_le_0_1_iff
tff(fact_977_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,one_one(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% zero_le_divide_1_iff
tff(fact_978_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,one_one(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% zero_less_divide_1_iff
tff(fact_979_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,$o),ord_less(A),one_one(A)),divide_divide(A,B2,A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_980_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,$o),ord_less(A),one_one(A)),divide_divide(A,B2,A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_981_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,$o),ord_less(A),divide_divide(A,B2,A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_982_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,$o),ord_less(A),divide_divide(A,B2,A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_983_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,one_one(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% divide_less_0_1_iff
tff(fact_984_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = divide_divide(A,one_one(A),B2) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_985_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,B2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = divide_divide(A,one_one(A),A2) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_986_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,A2,aa(A,A,abs_abs(A),B2)))
        <=> ( aa(A,$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_987_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A2,aa(A,A,abs_abs(A),B2))),zero_zero(A))
        <=> ( aa(A,$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_988_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_989_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_990_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,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_991_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,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_992_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,$o),ord_less_eq(A),divide_divide(A,B2,A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_993_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,$o),ord_less_eq(A),divide_divide(A,B2,A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_994_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: 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_995_divide__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: 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_996_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: 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_997_mult_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)) ) ) ).

% mult.left_commute
tff(fact_998_mult_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) ) ) ).

% mult.commute
tff(fact_999_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_1000_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_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)) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_1001_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( A2 = 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_1002_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(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_1003_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 = divide_divide(A,B2,C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_1004_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) )
           => ( divide_divide(A,B2,C2) = A2 ) ) ) ) ).

% divide_eq_imp
tff(fact_1005_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = 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_1006_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: 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_1007_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( divide_divide(A,Xb,Y) = divide_divide(A,W,Z) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),W),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_1008_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)
         => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,divide_divide(A,A2,B2),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1009_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),divide_divide(A,C2,A2)),divide_divide(A,C2,B2)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1010_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),divide_divide(A,C2,A2)),divide_divide(A,C2,B2)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_1011_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),divide_divide(A,Xb,Y)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1012_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xb,Y)),Z) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1013_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,$o),ord_less(A),A2),divide_divide(A,B2,C2))
          <=> aa(A,$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_1014_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,$o),ord_less(A),divide_divide(A,B2,C2)),A2)
          <=> aa(A,$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_1015_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,$o),ord_less(A),A2),divide_divide(A,B2,C2))
          <=> aa(A,$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_1016_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,$o),ord_less(A),divide_divide(A,B2,C2)),A2)
          <=> aa(A,$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_1017_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),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_1018_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),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_1019_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xb,Z)),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_1020_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),divide_divide(A,Y,Z)) = 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),Xb),Z)),Y),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_1021_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),divide_divide(A,Xb,Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Y) ) ) ) ).

% add_num_frac
tff(fact_1022_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xb,Y)),Z) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Y) ) ) ) ).

% add_frac_num
tff(fact_1023_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xb,Y)),divide_divide(A,W,Z)) = 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),Xb),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_1024_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),divide_divide(A,B2,Z)) = $ite(Z = zero_zero(A),A2,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_1025_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),divide_divide(A,A2,Z)),B2) = $ite(Z = zero_zero(A),B2,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_1026_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Xb,Z)),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1027_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),divide_divide(A,Y,Z)) = 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),Xb),Z)),Y),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1028_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Xb,Y)),divide_divide(A,W,Z)) = 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),Xb),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_1029_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),divide_divide(A,B2,Z)) = $ite(Z = zero_zero(A),A2,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_1030_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(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_1031_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( aa(A,A,uminus_uminus(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_1032_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),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_1033_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),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_1034_add__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: 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),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ).

% add_divide_distrib
tff(fact_1035_diff__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: 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),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ).

% diff_divide_distrib
tff(fact_1036_minus__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : ( aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) = divide_divide(A,A2,aa(A,A,uminus_uminus(A),B2)) ) ) ).

% minus_divide_right
tff(fact_1037_minus__divide__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : ( divide_divide(A,aa(A,A,uminus_uminus(A),A2),aa(A,A,uminus_uminus(A),B2)) = divide_divide(A,A2,B2) ) ) ).

% minus_divide_divide
tff(fact_1038_minus__divide__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] : ( aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) = divide_divide(A,aa(A,A,uminus_uminus(A),A2),B2) ) ) ).

% minus_divide_left
tff(fact_1039_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_1040_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_1041_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_1042_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_1043_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_1044_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_1045_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_1046_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_1047_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_1048_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_1049_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_1050_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_1051_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,E2: A,B2: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),E2)),C2) ) ) ).

% combine_common_factor
tff(fact_1052_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W: A,Y: A,Xb: 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),Xb),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),Xb),Y)) )
        <=> ( ( W = Xb )
            | ( Y = Z ) ) ) ) ).

% crossproduct_eq
tff(fact_1053_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( ( ( A2 != B2 )
            & ( C2 != D3 ) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),D3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% crossproduct_noteq
tff(fact_1054_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_1055_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_1056_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_1057_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_1058_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D7: A,Q: fun(A,$o)] :
          ( ! [X: A,K2: A] :
              ( aa(A,$o,P,X)
            <=> aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D7))) )
         => ( ! [X: A,K2: A] :
                ( aa(A,$o,Q,X)
              <=> aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D7))) )
           => ! [X3: A,K4: A] :
                ( ( aa(A,$o,P,X3)
                  | aa(A,$o,Q,X3) )
              <=> ( aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D7)))
                  | aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D7))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_1059_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D7: A,Q: fun(A,$o)] :
          ( ! [X: A,K2: A] :
              ( aa(A,$o,P,X)
            <=> aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D7))) )
         => ( ! [X: A,K2: A] :
                ( aa(A,$o,Q,X)
              <=> aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D7))) )
           => ! [X3: A,K4: A] :
                ( ( aa(A,$o,P,X3)
                  & aa(A,$o,Q,X3) )
              <=> ( aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D7)))
                  & aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D7))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_1060_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_1061_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_1062_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xb: nat,Y: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Xb)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),Xb)) ) ) ).

% mult_of_nat_commute
tff(fact_1063_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_1064_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_1065_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_1066_mult__commute__imp__mult__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Y: A,Xb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Y)),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),Y)) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_1067_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: int,Y: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Xb)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(int,A,ring_1_of_int(A),Xb)) ) ) ).

% mult_of_int_commute
tff(fact_1068_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : ( aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z22)) ) ).

% int_distrib(2)
tff(fact_1069_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)),W) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W)) ) ).

% int_distrib(1)
tff(fact_1070_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : ( aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z22)) ) ).

% int_distrib(4)
tff(fact_1071_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z1),Z22)),W) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W)) ) ).

% int_distrib(3)
tff(fact_1072_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),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_1073_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),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_1074_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),divide_divide(A,C2,A2)),divide_divide(A,C2,B2)) ) ) ) ) ).

% divide_left_mono
tff(fact_1075_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,$o),ord_less_eq(A),divide_divide(A,B2,C2)),A2)
          <=> aa(A,$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_1076_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,$o),ord_less_eq(A),A2),divide_divide(A,B2,C2))
          <=> aa(A,$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_1077_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,$o),ord_less_eq(A),divide_divide(A,B2,C2)),A2)
          <=> aa(A,$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_1078_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,$o),ord_less_eq(A),A2),divide_divide(A,B2,C2))
          <=> aa(A,$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_1079_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xb,Y)),Z) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1080_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),divide_divide(A,Xb,Y)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1081_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),divide_divide(A,C2,A2)),divide_divide(A,C2,B2)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1082_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xb,Y)),divide_divide(A,W,Z))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(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),Xb),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_1083_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xb,Y)),divide_divide(A,W,Z))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(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),Xb),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_1084_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,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2)
          <=> aa(A,$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_1085_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,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
          <=> aa(A,$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_1086_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,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2)
          <=> aa(A,$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_1087_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,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
          <=> aa(A,$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_1088_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),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_1089_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),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_1090_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),divide_divide(A,A2,Z))),B2) = $ite(Z = zero_zero(A),B2,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_1091_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,Xb,Z))),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_1092_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),divide_divide(A,Xb,Z))),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_1093_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),divide_divide(A,A2,Z)),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B2),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_1094_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),divide_divide(A,A2,Z))),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B2),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_1095_ceiling__divide__eq__div,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: int,B2: int] : ( archimedean_ceiling(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),divide_divide(int,aa(int,int,uminus_uminus(int),A2),B2)) ) ) ).

% ceiling_divide_eq_div
tff(fact_1096_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D3))
        <=> ( ( ~ aa(A,$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,$o),ord_less_eq(A),B2),D3)
                & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2)
                  | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D3) ) ) )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_1097_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),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_1098_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),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_1099_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,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
          <=> aa(A,$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_1100_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,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2)
          <=> aa(A,$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_1101_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,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
          <=> aa(A,$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_1102_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,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2)
          <=> aa(A,$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_1103_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V2: A,R2: A,S: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),S)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(A,A,aa(A,fun(A,A),minus_minus(A),V2),U)),S))),V2) ) ) ) ) ).

% scaling_mono
tff(fact_1104_periodic__finite__ex,axiom,
    ! [D3: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X: int,K2: int] :
            ( aa(int,$o,P,X)
          <=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3))) )
       => ( ? [X_13: int] : aa(int,$o,P,X_13)
        <=> ? [X4: int] :
              ( member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D3))
              & aa(int,$o,P,X4) ) ) ) ) ).

% periodic_finite_ex
tff(fact_1105_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_1106_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A2,B2)),zero_zero(A))
        <=> ( ( aa(A,$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_1107_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),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ).

% divide_right_mono
tff(fact_1108_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,A2,B2))
        <=> ( ( aa(A,$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_1109_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$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)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_1110_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$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),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_1111_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$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),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_1112_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$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)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_1113_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),divide_divide(A,B2,C2)),divide_divide(A,A2,C2)) ) ) ) ).

% divide_right_mono_neg
tff(fact_1114_split__zdiv,axiom,
    ! [P: fun(int,$o),Nb: int,K: int] :
      ( aa(int,$o,P,divide_divide(int,Nb,K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,zero_zero(int)) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => aa(int,$o,P,I3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => aa(int,$o,P,I3) ) ) ) ) ).

% split_zdiv
tff(fact_1115_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q4: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R2)
         => ( divide_divide(int,A2,B2) = Q4 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1116_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q4: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
         => ( divide_divide(int,A2,B2) = Q4 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1117_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),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1118_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),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ).

% divide_strict_right_mono
tff(fact_1119_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,A2,B2))
        <=> ( ( aa(A,$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_1120_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2))
        <=> ( ( aa(A,$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_1121_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A2,B2)),zero_zero(A))
        <=> ( ( aa(A,$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_1122_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ( aa(A,$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)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_pos_pos
tff(fact_1123_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_pos_neg
tff(fact_1124_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_neg_pos
tff(fact_1125_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A))
         => ( aa(A,$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)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_neg_neg
tff(fact_1126_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( divide_divide(A,A2,B2) = one_one(A) )
          <=> ( A2 = B2 ) ) ) ) ).

% right_inverse_eq
tff(fact_1127_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) = divide_divide(A,A2,aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_1128_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,uminus_uminus(A),A2),aa(A,A,uminus_uminus(A),B2)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_1129_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_1130_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_1131_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_1132_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_1133_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_1134_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_1135_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_1136_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_1137_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_1138_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_1139_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_1140_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_1141_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_1142_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_1143_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_1144_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => ( aa(A,$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),D3)) ) ) ) ) ) ).

% mult_mono'
tff(fact_1145_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => ( aa(A,$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),D3)) ) ) ) ) ) ).

% mult_mono
tff(fact_1146_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_1147_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_1148_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_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_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_1157_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_1158_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_1159_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_1160_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_1161_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_1162_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_1163_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_1164_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_1165_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),divide_divide(A,A2,B2)) = divide_divide(A,aa(A,A,abs_abs(A),A2),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% nonzero_abs_divide
tff(fact_1166_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R2: A,A2: A,B2: A,C2: A,D3: A] :
          ( ( R2 != zero_zero(A) )
         => ( ( ( A2 = B2 )
              & ( C2 != D3 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),D3)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1167_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ma: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Ma)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Ma),Nb)) ) ) ) ).

% less_1_mult
tff(fact_1168_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),E2)),C2) = D3 ) ) ) ).

% eq_add_iff1
tff(fact_1169_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3) )
        <=> ( C2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),E2)),D3) ) ) ) ).

% eq_add_iff2
tff(fact_1170_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [Xb: 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),Xb),Xb)),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),Xb),Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) ) ) ).

% square_diff_square_factored
tff(fact_1171_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Xb: 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),Xb),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),Xb),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),Xb),A2)),B2)) ) ) ).

% mult_diff_mult
tff(fact_1172_square__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [Xb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb) = one_one(A) )
        <=> ( ( Xb = one_one(A) )
            | ( Xb = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% square_eq_1_iff
tff(fact_1173_inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : ( aa(A,A,inverse_inverse(A),A2) = divide_divide(A,one_one(A),A2) ) ) ).

% inverse_eq_divide
tff(fact_1174_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),B2)),D3)
           => aa(A,$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),D3)) ) ) ) ).

% abs_mult_less
tff(fact_1175_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_1176_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),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_1177_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),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_1178_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),divide_divide(int,A2,B2)),zero_zero(int)) ) ) ).

% div_neg_pos_less0
tff(fact_1179_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_1180_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xaa: 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),Xaa))),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),Xaa))) ) ) ).

% mult_inverse_of_nat_commute
tff(fact_1181_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_1182_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xaa: 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),Xaa))),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),Xaa))) ) ) ).

% mult_inverse_of_int_commute
tff(fact_1183_pos__zmult__eq__1__iff__lemma,axiom,
    ! [Ma: int,Nb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),Ma),Nb) = one_one(int) )
     => ( ( Ma = one_one(int) )
        | ( Ma = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_1184_zmult__eq__1__iff,axiom,
    ! [Ma: int,Nb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),Ma),Nb) = one_one(int) )
    <=> ( ( ( Ma = one_one(int) )
          & ( Nb = one_one(int) ) )
        | ( ( Ma = aa(int,int,uminus_uminus(int),one_one(int)) )
          & ( Nb = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_1185_abs__zmult__eq__1,axiom,
    ! [Ma: int,Nb: int] :
      ( ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),Ma),Nb)) = one_one(int) )
     => ( aa(int,int,abs_abs(int),Ma) = one_one(int) ) ) ).

% abs_zmult_eq_1
tff(fact_1186_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q4: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,divide_divide(A,P3,Q4)))),Q4)) ) ) ).

% ceiling_divide_upper
tff(fact_1187_cpmi,axiom,
    ! [D7: int,P: fun(int,$o),P2: fun(int,$o),B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D7)
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Z4)
           => ( aa(int,$o,P,X)
            <=> aa(int,$o,P2,X) ) )
       => ( ! [X: int] :
              ( ! [Xa: int] :
                  ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D7))
                 => ! [Xb3: int] :
                      ( member(int,Xb3,B3)
                     => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa) ) ) )
             => ( aa(int,$o,P,X)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D7)) ) )
         => ( ! [X: int,K2: int] :
                ( aa(int,$o,P2,X)
              <=> aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D7))) )
           => ( ? [X_13: int] : aa(int,$o,P,X_13)
            <=> ( ? [X4: int] :
                    ( member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D7))
                    & aa(int,$o,P2,X4) )
                | ? [X4: int] :
                    ( member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D7))
                    & ? [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_1188_cppi,axiom,
    ! [D7: int,P: fun(int,$o),P2: fun(int,$o),A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D7)
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z4),X)
           => ( aa(int,$o,P,X)
            <=> aa(int,$o,P2,X) ) )
       => ( ! [X: int] :
              ( ! [Xa: int] :
                  ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D7))
                 => ! [Xb3: int] :
                      ( member(int,Xb3,A3)
                     => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa) ) ) )
             => ( aa(int,$o,P,X)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D7)) ) )
         => ( ! [X: int,K2: int] :
                ( aa(int,$o,P2,X)
              <=> aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D7))) )
           => ( ? [X_13: int] : aa(int,$o,P,X_13)
            <=> ( ? [X4: int] :
                    ( member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D7))
                    & aa(int,$o,P2,X4) )
                | ? [X4: int] :
                    ( member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D7))
                    & ? [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_1189_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q4: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q4)
         => aa(A,$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,divide_divide(A,P3,Q4)))),one_one(A))),Q4)),P3) ) ) ).

% ceiling_divide_lower
tff(fact_1190_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$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),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_pos
tff(fact_1191_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$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)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_nonpos_neg
tff(fact_1192_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$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)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_nonneg_pos
tff(fact_1193_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$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),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_neg
tff(fact_1194_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2))
        <=> ( ( aa(A,$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_1195_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( aa(A,$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),divide_divide(A,Xb,Z)),divide_divide(A,Y,W)) ) ) ) ) ) ).

% frac_less2
tff(fact_1196_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
           => ( aa(A,$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),divide_divide(A,Xb,Z)),divide_divide(A,Y,W)) ) ) ) ) ) ).

% frac_less
tff(fact_1197_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( aa(A,$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),divide_divide(A,Xb,Z)),divide_divide(A,Y,W)) ) ) ) ) ) ).

% frac_le
tff(fact_1198_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)),divide_divide(A,A2,B2)) ) ) ) ).

% div_positive
tff(fact_1199_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)
           => ( divide_divide(A,A2,B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1200_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B2,A2))
        <=> ( ( aa(A,$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_1201_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,A2)),one_one(A))
        <=> ( ( aa(A,$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_1202_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),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_1203_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),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_1204_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: 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_1205_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( divide_divide(A,aa(A,A,abs_abs(A),Xb),Y) = aa(A,A,abs_abs(A),divide_divide(A,Xb,Y)) ) ) ) ).

% abs_div_pos
tff(fact_1206_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_1207_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => ( aa(A,$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),D3)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1208_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => ( aa(A,$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),D3)) ) ) ) ) ) ).

% mult_le_less_imp_less
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__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_1211_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_1212_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_1213_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_1214_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => ( aa(A,$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),D3)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1215_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_1216_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_1217_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => ( aa(A,$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),D3)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1218_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_1219_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_1220_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_1221_aset_I2_J,axiom,
    ! [D7: int,A3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D7))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A3)
                 => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa) ) ) )
         => ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D7)) ) )
     => ( ! [X: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D7))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A3)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa) ) ) )
           => ( aa(int,$o,Q,X)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D7)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
           => ( ( aa(int,$o,P,X3)
                | aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D7))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D7)) ) ) ) ) ) ).

% aset(2)
tff(fact_1222_aset_I1_J,axiom,
    ! [D7: int,A3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D7))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A3)
                 => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa) ) ) )
         => ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D7)) ) )
     => ( ! [X: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D7))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A3)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa) ) ) )
           => ( aa(int,$o,Q,X)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D7)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
           => ( ( aa(int,$o,P,X3)
                & aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D7))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D7)) ) ) ) ) ) ).

% aset(1)
tff(fact_1223_bset_I2_J,axiom,
    ! [D7: int,B3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D7))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B3)
                 => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa) ) ) )
         => ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D7)) ) )
     => ( ! [X: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D7))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B3)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa) ) ) )
           => ( aa(int,$o,Q,X)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D7)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( ( aa(int,$o,P,X3)
                | aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D7))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D7)) ) ) ) ) ) ).

% bset(2)
tff(fact_1224_bset_I1_J,axiom,
    ! [D7: int,B3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D7))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B3)
                 => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa) ) ) )
         => ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D7)) ) )
     => ( ! [X: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D7))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B3)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa) ) ) )
           => ( aa(int,$o,Q,X)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D7)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( ( aa(int,$o,P,X3)
                & aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D7))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D7)) ) ) ) ) ) ).

% bset(1)
tff(fact_1225_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$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),Xb)),Xb) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_1226_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$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),Xb),Y)),Xb) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_1227_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_1228_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_1229_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Y: A] : aa(A,$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),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))) ) ).

% sum_squares_ge_zero
tff(fact_1230_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Y: A] : ~ aa(A,$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),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)) ) ).

% not_sum_squares_lt_zero
tff(fact_1231_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) = divide_divide(A,one_one(A),A2) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_1232_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3))
        <=> aa(A,$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)),E2)),C2)),D3) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1233_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3))
        <=> aa(A,$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)),E2)),D3)) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1234_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3))
        <=> aa(A,$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)),E2)),D3)) ) ) ).

% less_add_iff2
tff(fact_1235_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3))
        <=> aa(A,$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)),E2)),C2)),D3) ) ) ).

% less_add_iff1
tff(fact_1236_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ? [N: nat] : aa(A,$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)),Xb)) ) ) ).

% ex_less_of_nat_mult
tff(fact_1237_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),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),Xb),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),one_one(A))) ) ) ).

% square_diff_one_factored
tff(fact_1238_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_1239_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),Xb) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb)) ) ) ) ).

% abs_mult_pos
tff(fact_1240_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_1241_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_1242_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_1243_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_1244_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)),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_1245_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)),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_1246_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)),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_1247_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)),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_1248_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),divide_divide(int,A2,B2)),zero_zero(int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_1249_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),divide_divide(int,A2,B2)),zero_zero(int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_1250_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)),divide_divide(int,K,L)) ) ) ).

% div_positive_int
tff(fact_1251_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(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_1252_zdiv__mono2__neg,axiom,
    ! [A2: int,B6: 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)),B6)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,A2,B6)),divide_divide(int,A2,B2)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1253_zdiv__mono1__neg,axiom,
    ! [A2: int,A6: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),A6)
     => ( 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),divide_divide(int,A6,B2)),divide_divide(int,A2,B2)) ) ) ).

% zdiv_mono1_neg
tff(fact_1254_zdiv__eq__0__iff,axiom,
    ! [I: int,K: 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_1255_zdiv__mono2,axiom,
    ! [A2: int,B6: 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)),B6)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,A2,B2)),divide_divide(int,A2,B6)) ) ) ) ).

% zdiv_mono2
tff(fact_1256_zdiv__mono1,axiom,
    ! [A2: int,A6: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),A6)
     => ( 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),divide_divide(int,A2,B2)),divide_divide(int,A6,B2)) ) ) ).

% zdiv_mono1
tff(fact_1257_ex__gt__or__lt,axiom,
    ! [A: $tType] :
      ( condit5016429287641298734tinuum(A)
     => ! [A2: A] :
        ? [B4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B4)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),A2) ) ) ).

% ex_gt_or_lt
tff(fact_1258_int__div__less__self,axiom,
    ! [Xb: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Xb)
     => ( 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),divide_divide(int,Xb,K)),Xb) ) ) ).

% int_div_less_self
tff(fact_1259_pos__zmult__eq__1__iff,axiom,
    ! [Ma: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ma)
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),Ma),Nb) = one_one(int) )
      <=> ( ( Ma = one_one(int) )
          & ( Nb = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1260_plusinfinity,axiom,
    ! [D3: int,P2: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X: int,K2: int] :
            ( aa(int,$o,P2,X)
          <=> aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3))) )
       => ( ? [Z4: int] :
            ! [X: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z4),X)
             => ( aa(int,$o,P,X)
              <=> aa(int,$o,P2,X) ) )
         => ( ? [X_1: int] : aa(int,$o,P2,X_1)
           => ? [X_12: int] : aa(int,$o,P,X_12) ) ) ) ) ).

% plusinfinity
tff(fact_1261_minusinfinity,axiom,
    ! [D3: int,P1: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [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),D3))) )
       => ( ? [Z4: int] :
            ! [X: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Z4)
             => ( aa(int,$o,P,X)
              <=> aa(int,$o,P1,X) ) )
         => ( ? [X_1: int] : aa(int,$o,P1,X_1)
           => ? [X_12: int] : aa(int,$o,P,X_12) ) ) ) ) ).

% minusinfinity
tff(fact_1262_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,A2)),one_one(A))
        <=> ( ( aa(A,$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_1263_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A2))
        <=> ( ( aa(A,$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_1264_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_1265_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_1266_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_1267_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_1268_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_1269_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_1270_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_1271_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_1272_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( ! [Z2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),one_one(A))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Xb)),Y) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% field_le_mult_one_interval
tff(fact_1273_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [Xb: A,A2: A,Y: A,U: A,V2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2)
         => ( aa(A,$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,$o),ord_less_eq(A),zero_zero(A)),V2)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
                 => aa(A,$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),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1274_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_1275_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_1276_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_1277_verit__less__mono__div__int2,axiom,
    ! [A3: int,B3: int,Nb: 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),Nb))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,B3,Nb)),divide_divide(int,A3,Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_1278_div__eq__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( 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_1279_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_1280_q__pos__lemma,axiom,
    ! [B6: int,Q5: 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),B6),Q5)),R3))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R3),B6)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_1281_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q4: int,R2: int,B6: int,Q5: 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),Q4)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q5)),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),B6),Q5)),R3))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R3),B6)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q4),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1282_incr__mult__lemma,axiom,
    ! [D3: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X: int] :
            ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D3)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X3: int] :
              ( aa(int,$o,P,X3)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1283_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q4: int,R2: int,B6: int,Q5: 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),Q4)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q5)),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),B6),Q5)),R3)),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R3)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q4) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1284_unique__quotient__lemma,axiom,
    ! [B2: int,Q5: int,R3: int,Q4: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R3)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2))
     => ( 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),R2),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q4) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1285_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q5: int,R3: int,Q4: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R3)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q4),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1286_decr__mult__lemma,axiom,
    ! [D3: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X: int] :
            ( aa(int,$o,P,X)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D3)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X3: int] :
              ( aa(int,$o,P,X3)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1287_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
         => ~ ! [N: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)))),E2) ) ) ).

% nat_approx_posE
tff(fact_1288_aset_I7_J,axiom,
    ! [D7: int,A3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D7)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A3)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),X3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D7)) ) ) ) ).

% aset(7)
tff(fact_1289_aset_I5_J,axiom,
    ! [D7: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D7)
     => ( member(int,Ta,A3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X3),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D7)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_1290_aset_I4_J,axiom,
    ! [D7: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D7)
     => ( member(int,Ta,A3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
           => ( ( X3 != Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D7) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_1291_aset_I3_J,axiom,
    ! [D7: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D7)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int)),A3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
           => ( ( X3 = Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D7) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_1292_bset_I7_J,axiom,
    ! [D7: int,Ta: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D7)
     => ( member(int,Ta,B3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),X3)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D7)) ) ) ) ) ).

% bset(7)
tff(fact_1293_bset_I5_J,axiom,
    ! [D7: int,B3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D7)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B3)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X3),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D7)),Ta) ) ) ) ).

% bset(5)
tff(fact_1294_bset_I4_J,axiom,
    ! [D7: int,Ta: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D7)
     => ( member(int,Ta,B3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( ( X3 != Ta )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D7) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_1295_bset_I3_J,axiom,
    ! [D7: int,Ta: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D7)
     => ( member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Ta),one_one(int)),B3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B3)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( ( X3 = Ta )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D7) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_1296_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: nat,Ma: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
         => ( ( Nb != zero_zero(nat) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),Ma))),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),Nb))) ) ) ) ).

% inverse_of_nat_le
tff(fact_1297_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [Xb: A,A2: A,Y: A,U: A,V2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),A2)
         => ( aa(A,$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,$o),ord_less_eq(A),zero_zero(A)),V2)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
                 => aa(A,$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),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1298_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)
       => ( divide_divide(int,K,L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_1299_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)
       => ( divide_divide(int,K,L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_1300_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(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),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self4
tff(fact_1301_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(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),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self3
tff(fact_1302_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(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),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self2
tff(fact_1303_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(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),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self1
tff(fact_1304_div__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: 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_1305_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))
       => ( divide_divide(int,K,L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_1306_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: 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),divide_divide(A,A2,B2)) ) ) ).

% div_mult_mult1_if
tff(fact_1307_div__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : ( divide_divide(A,aa(A,A,uminus_uminus(A),A2),aa(A,A,uminus_uminus(A),B2)) = divide_divide(A,A2,B2) ) ) ).

% div_minus_minus
tff(fact_1308_div__by__Suc__0,axiom,
    ! [Ma: nat] : ( divide_divide(nat,Ma,aa(nat,nat,suc,zero_zero(nat))) = Ma ) ).

% div_by_Suc_0
tff(fact_1309_div__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
     => ( divide_divide(nat,Ma,Nb) = zero_zero(nat) ) ) ).

% div_less
tff(fact_1310_mult__is__0,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb) = zero_zero(nat) )
    <=> ( ( Ma = zero_zero(nat) )
        | ( Nb = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_1311_mult__0__right,axiom,
    ! [Ma: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),zero_zero(nat)) = zero_zero(nat) ) ).

% mult_0_right
tff(fact_1312_mult__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb) )
    <=> ( ( Ma = Nb )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_1313_mult__cancel2,axiom,
    ! [Ma: nat,K: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),K) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K) )
    <=> ( ( Ma = Nb )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_1314_nat__1__eq__mult__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb) )
    <=> ( ( Ma = one_one(nat) )
        & ( Nb = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_1315_nat__mult__eq__1__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb) = one_one(nat) )
    <=> ( ( Ma = one_one(nat) )
        & ( Nb = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_1316_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(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)) = divide_divide(A,A2,B2) ) ) ) ).

% div_mult_mult1
tff(fact_1317_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(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)) = divide_divide(A,A2,B2) ) ) ) ).

% div_mult_mult2
tff(fact_1318_one__eq__mult__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb) )
    <=> ( ( Ma = aa(nat,nat,suc,zero_zero(nat)) )
        & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_1319_mult__eq__1__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( Ma = aa(nat,nat,suc,zero_zero(nat)) )
        & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_1320_div__mult__self__is__m,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb),Nb) = Ma ) ) ).

% div_mult_self_is_m
tff(fact_1321_div__mult__self1__is__m,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Ma),Nb) = Ma ) ) ).

% div_mult_self1_is_m
tff(fact_1322_mult__less__cancel2,axiom,
    ! [Ma: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),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),Ma),Nb) ) ) ).

% mult_less_cancel2
tff(fact_1323_nat__0__less__mult__iff,axiom,
    ! [Ma: nat,Nb: 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),Ma),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_1324_mult__Suc__right,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb)) ) ).

% mult_Suc_right
tff(fact_1325_one__le__mult__iff,axiom,
    ! [Ma: nat,Nb: 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),Ma),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),Ma)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) ) ) ).

% one_le_mult_iff
tff(fact_1326_mult__le__cancel2,axiom,
    ! [Ma: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),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),Ma),Nb) ) ) ).

% mult_le_cancel2
tff(fact_1327_less__mult__imp__div__less,axiom,
    ! [Ma: nat,I: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,Ma,Nb)),I) ) ).

% less_mult_imp_div_less
tff(fact_1328_divide__real__def,axiom,
    ! [Xb: real,Y: real] : ( divide_divide(real,Xb,Y) = aa(real,real,aa(real,fun(real,real),times_times(real),Xb),aa(real,real,inverse_inverse(real),Y)) ) ).

% divide_real_def
tff(fact_1329_div__less__iff__less__mult,axiom,
    ! [Q4: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,Ma,Q4)),Nb)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q4)) ) ) ).

% div_less_iff_less_mult
tff(fact_1330_real__of__nat__div4,axiom,
    ! [Nb: nat,Xb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Nb,Xb))),divide_divide(real,aa(nat,real,semiring_1_of_nat(real),Nb),aa(nat,real,semiring_1_of_nat(real),Xb))) ).

% real_of_nat_div4
tff(fact_1331_div__nat__eqI,axiom,
    ! [Nb: nat,Q4: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q4)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Q4)))
       => ( divide_divide(nat,Ma,Nb) = Q4 ) ) ) ).

% div_nat_eqI
tff(fact_1332_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q4: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),divide_divide(nat,Nb,Q4))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Q4)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1333_split__div,axiom,
    ! [P: fun(nat,$o),Ma: nat,Nb: nat] :
      ( aa(nat,$o,P,divide_divide(nat,Ma,Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(nat,$o,P,zero_zero(nat)) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Nb)
             => ( ( Ma = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),I3)),J3) )
               => aa(nat,$o,P,I3) ) ) ) ) ) ).

% split_div
tff(fact_1334_dividend__less__div__times,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,Ma,Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_1335_dividend__less__times__div,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),divide_divide(nat,Ma,Nb)))) ) ).

% dividend_less_times_div
tff(fact_1336_Suc__mult__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Ma) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Nb) )
    <=> ( Ma = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_1337_mult__0,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),Nb) = zero_zero(nat) ) ).

% mult_0
tff(fact_1338_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_1339_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_1340_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_1341_le__square,axiom,
    ! [Ma: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Ma)) ).

% le_square
tff(fact_1342_le__cube,axiom,
    ! [Ma: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Ma))) ).

% le_cube
tff(fact_1343_add__mult__distrib,axiom,
    ! [Ma: nat,Nb: 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),Ma),Nb)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K)) ) ).

% add_mult_distrib
tff(fact_1344_add__mult__distrib2,axiom,
    ! [K: nat,Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) ) ).

% add_mult_distrib2
tff(fact_1345_diff__mult__distrib,axiom,
    ! [Ma: nat,Nb: 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),Ma),Nb)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K)) ) ).

% diff_mult_distrib
tff(fact_1346_diff__mult__distrib2,axiom,
    ! [K: nat,Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) ) ).

% diff_mult_distrib2
tff(fact_1347_nat__mult__1,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),Nb) = Nb ) ).

% nat_mult_1
tff(fact_1348_nat__mult__1__right,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),one_one(nat)) = Nb ) ).

% nat_mult_1_right
tff(fact_1349_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( divide_divide(nat,Ma,Nb) = zero_zero(nat) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
        | ( Nb = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_1350_Suc__div__le__mono,axiom,
    ! [Ma: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,Ma,Nb)),divide_divide(nat,aa(nat,nat,suc,Ma),Nb)) ).

% Suc_div_le_mono
tff(fact_1351_split__div_H,axiom,
    ! [P: fun(nat,$o),Ma: nat,Nb: nat] :
      ( aa(nat,$o,P,divide_divide(nat,Ma,Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
          & aa(nat,$o,P,zero_zero(nat)) )
        | ? [Q6: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q6)),Ma)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Q6)))
            & aa(nat,$o,P,Q6) ) ) ) ).

% split_div'
tff(fact_1352_real__of__nat__div2,axiom,
    ! [Nb: nat,Xb: 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),divide_divide(real,aa(nat,real,semiring_1_of_nat(real),Nb),aa(nat,real,semiring_1_of_nat(real),Xb))),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Nb,Xb)))) ).

% real_of_nat_div2
tff(fact_1353_ex__nat__less,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),Nb)
          & aa(nat,$o,P,M3) )
    <=> ? [X4: nat] :
          ( member(nat,X4,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P,X4) ) ) ).

% ex_nat_less
tff(fact_1354_all__nat__less,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),Nb)
         => aa(nat,$o,P,M3) )
    <=> ! [X4: nat] :
          ( member(nat,X4,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P,X4) ) ) ).

% all_nat_less
tff(fact_1355_real__of__nat__div3,axiom,
    ! [Nb: nat,Xb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),divide_divide(real,aa(nat,real,semiring_1_of_nat(real),Nb),aa(nat,real,semiring_1_of_nat(real),Xb))),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Nb,Xb)))),one_one(real)) ).

% real_of_nat_div3
tff(fact_1356_div__le__mono2,axiom,
    ! [Ma: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,K,Nb)),divide_divide(nat,K,Ma)) ) ) ).

% div_le_mono2
tff(fact_1357_div__greater__zero__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),divide_divide(nat,Ma,Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_1358_div__less__dividend,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,Ma,Nb)),Ma) ) ) ).

% div_less_dividend
tff(fact_1359_div__eq__dividend__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( ( divide_divide(nat,Ma,Nb) = Ma )
      <=> ( Nb = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_1360_Suc__mult__less__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: 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)),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_1361_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_1362_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_1363_Suc__mult__le__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: 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)),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_1364_mult__Suc,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Ma)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb)) ) ).

% mult_Suc
tff(fact_1365_mult__eq__self__implies__10,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( Ma = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb) )
     => ( ( Nb = one_one(nat) )
        | ( Ma = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_1366_real__minus__mult__self__le,axiom,
    ! [U: real,Xb: 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),Xb),Xb)) ).

% real_minus_mult_self_le
tff(fact_1367_zdiv__int,axiom,
    ! [A2: nat,B2: nat] : ( aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,A2,B2)) = 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_1368_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_1369_div__if,axiom,
    ! [Ma: nat,Nb: nat] :
      ( divide_divide(nat,Ma,Nb) = $ite(
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
          | ( Nb = zero_zero(nat) ) ),
          zero_zero(nat),
          aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb),Nb)) ) ) ).

% div_if
tff(fact_1370_nat__div__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,nat,nat2,divide_divide(int,Xb,Y)) = divide_divide(nat,aa(int,nat,nat2,Xb),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_1371_nat__div__distrib_H,axiom,
    ! [Y: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,nat,nat2,divide_divide(int,Xb,Y)) = divide_divide(nat,aa(int,nat,nat2,Xb),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_1372_one__less__mult,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Ma)
       => 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),Ma),Nb)) ) ) ).

% one_less_mult
tff(fact_1373_n__less__m__mult__n,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Ma)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_1374_n__less__n__mult__m,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Ma)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Ma)) ) ) ).

% n_less_n_mult_m
tff(fact_1375_reals__Archimedean3,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ! [Y4: real] :
        ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),Xb)) ) ).

% reals_Archimedean3
tff(fact_1376_real__of__int__div4,axiom,
    ! [Nb: int,Xb: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),divide_divide(int,Nb,Xb))),divide_divide(real,aa(int,real,ring_1_of_int(real),Nb),aa(int,real,ring_1_of_int(real),Xb))) ).

% real_of_int_div4
tff(fact_1377_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_1378_le__div__geq,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
       => ( divide_divide(nat,Ma,Nb) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb),Nb)) ) ) ) ).

% le_div_geq
tff(fact_1379_div__geq,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
       => ( divide_divide(nat,Ma,Nb) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb),Nb)) ) ) ) ).

% div_geq
tff(fact_1380_mult__eq__if,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb) = $ite(Ma = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),one_one(nat))),Nb))) ) ).

% mult_eq_if
tff(fact_1381_nat__mult__distrib,axiom,
    ! [Z: int,Z3: 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),Z3)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z3)) ) ) ).

% nat_mult_distrib
tff(fact_1382_div__abs__eq__div__nat,axiom,
    ! [K: int,L: 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),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_1383_real__archimedian__rdiv__eq__0,axiom,
    ! [Xb: real,C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C2)
       => ( ! [M: nat] :
              ( 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,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),Xb)),C2) )
         => ( Xb = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_1384_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z3: 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),Z3)) = 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),Z3))) ) ) ).

% nat_mult_distrib_neg
tff(fact_1385_real__of__int__div2,axiom,
    ! [Nb: int,Xb: 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),divide_divide(real,aa(int,real,ring_1_of_int(real),Nb),aa(int,real,ring_1_of_int(real),Xb))),aa(int,real,ring_1_of_int(real),divide_divide(int,Nb,Xb)))) ).

% real_of_int_div2
tff(fact_1386_real__of__int__div3,axiom,
    ! [Nb: int,Xb: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),divide_divide(real,aa(int,real,ring_1_of_int(real),Nb),aa(int,real,ring_1_of_int(real),Xb))),aa(int,real,ring_1_of_int(real),divide_divide(int,Nb,Xb)))),one_one(real)) ).

% real_of_int_div3
tff(fact_1387_div__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : ( divide_divide(A,A2,aa(A,A,uminus_uminus(A),B2)) = divide_divide(A,aa(A,A,uminus_uminus(A),A2),B2) ) ) ).

% div_minus_right
tff(fact_1388_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Ma: nat,Nb: nat] : ( aa(nat,A,semiring_1_of_nat(A),divide_divide(nat,Ma,Nb)) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Ma),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_1389_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,Ma: nat,Nb: nat] : ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,A,semiring_1_of_nat(A),Nb))) = divide_divide(A,divide_divide(A,A2,aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ).

% div_mult2_eq'
tff(fact_1390_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(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),divide_divide(A,A2,B2)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_1391_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(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),divide_divide(A,A2,B2)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_1392_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)
     => ( divide_divide(int,A2,aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = divide_divide(int,divide_divide(int,A2,B2),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_1393_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,Ma: nat,Nb: 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),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
    <=> ( 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),Ma),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1394_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
    <=> ( 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),Ma),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1395_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xb: 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),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1396_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Ma: nat,Nb: 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)),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> 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)),Ma)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_1397_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Ma: nat,Nb: 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)),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),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)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_1398_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) = $ite(K = zero_zero(nat),zero_zero(nat),divide_divide(nat,Ma,Nb)) ) ).

% nat_mult_div_cancel_disj
tff(fact_1399_bits__div__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : ( divide_divide(A,A2,one_one(A)) = A2 ) ) ).

% bits_div_by_1
tff(fact_1400_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : ( divide_divide(A,zero_zero(A),A2) = zero_zero(A) ) ) ).

% bits_div_0
tff(fact_1401_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : ( divide_divide(A,A2,zero_zero(A)) = zero_zero(A) ) ) ).

% bits_div_by_0
tff(fact_1402_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb) )
    <=> ( ( K = zero_zero(nat) )
        | ( Ma = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1403_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_1404_nat__mult__less__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: 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),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_1405_nat__mult__eq__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: 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),Ma) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb) )
      <=> ( Ma = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1406_nat__mult__div__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) = divide_divide(nat,Ma,Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_1407_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$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),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1408_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$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),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))
        <=> ( ( Xb != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1409_nat__mult__le__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: 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),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_1410_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Ma: nat,Nb: 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)),Ma) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb) )
      <=> ( 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)),Ma) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_1411_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Ma: nat,Nb: 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)),Ma) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb) )
      <=> ( Ma = 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)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1412_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Ma: nat,Nb: 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)),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> 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)),Ma)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_1413_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Ma: nat,Nb: 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)),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),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)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_1414_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,Ma: nat,Nb: 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)),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb)) = 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)),Ma)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_1415_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,Ma: nat,Nb: 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)),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),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)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_1416_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1417_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1418_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ) ).

% mult_less_iff1
tff(fact_1419_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_1420_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_1421_int__power__div__base,axiom,
    ! [Ma: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
       => ( divide_divide(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),K),Ma),K) = aa(nat,int,aa(int,fun(nat,int),power_power(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_1422_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),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_1423_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_1424_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_1425_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_1426_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_1427_power__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Nb: nat] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),Nb) = one_one(A) ) ) ).

% power_one
tff(fact_1428_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Xb: nat,B2: nat,W: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Xb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W) )
        <=> ( Xb = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W) ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_1429_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [B2: nat,W: nat,Xb: 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),Xb) )
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W) = Xb ) ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_1430_of__nat__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Ma: nat,Nb: nat] : ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Ma),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),Ma)),Nb) ) ) ).

% of_nat_power
tff(fact_1431_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_1432_gbinomial__1,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : ( aa(nat,A,gbinomial(A,A2),one_one(nat)) = A2 ) ) ).

% gbinomial_1
tff(fact_1433_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Ma: nat,Nb: nat] :
          ( aa(A,$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),Ma) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) )
          <=> ( Ma = Nb ) ) ) ) ).

% power_inject_exp
tff(fact_1434_power__0__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(nat,nat,suc,Nb)) = zero_zero(A) ) ) ).

% power_0_Suc
tff(fact_1435_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_1436_abs__power__minus,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: 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)),Nb)) = aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% abs_power_minus
tff(fact_1437_gbinomial__0_I2_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [K: nat] : ( aa(nat,A,gbinomial(A,zero_zero(A)),aa(nat,nat,suc,K)) = zero_zero(A) ) ) ).

% gbinomial_0(2)
tff(fact_1438_gbinomial__0_I1_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : ( aa(nat,A,gbinomial(A,A2),zero_zero(nat)) = one_one(A) ) ) ).

% gbinomial_0(1)
tff(fact_1439_gbinomial__Suc0,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : ( aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,zero_zero(nat))) = A2 ) ) ).

% gbinomial_Suc0
tff(fact_1440_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: int,B2: int,W: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Xb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W) )
        <=> ( Xb = aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_1441_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [B2: int,W: nat,Xb: 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),Xb) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) = Xb ) ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_1442_of__int__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int,Nb: nat] : ( aa(int,A,ring_1_of_int(A),aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),Z)),Nb) ) ) ).

% of_int_power
tff(fact_1443_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: 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))),Nb)),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))),Nb)),A2)) = A2 ) ) ).

% left_minus_one_mult_self
tff(fact_1444_minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: 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))),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Nb)) = one_one(A) ) ) ).

% minus_one_mult_self
tff(fact_1445_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xb: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y) ) ) ) ).

% power_strict_increasing_iff
tff(fact_1446_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,Nb: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% power_eq_0_iff
tff(fact_1447_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),Xb) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1448_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,B2: nat,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1449_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),Xb) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1450_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,B2: nat,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1451_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Ma: nat,Nb: nat] :
          ( aa(A,$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,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Ma)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1452_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xb: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Y) ) ) ) ).

% power_increasing_iff
tff(fact_1453_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$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,$o),ord_less(nat),zero_zero(nat)),Nb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ) ).

% power_mono_iff
tff(fact_1454_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$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)),Nb))
        <=> ( ( A2 != zero_zero(A) )
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_1455_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int,B2: int,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_1456_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,Xb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)),aa(int,A,ring_1_of_int(A),Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),Xb) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_1457_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,Xb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)),aa(int,A,ring_1_of_int(A),Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),Xb) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_1458_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int,B2: int,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_1459_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Ma: nat,Nb: nat] :
          ( aa(A,$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,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Ma)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma) ) ) ) ) ).

% power_decreasing_iff
tff(fact_1460_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),Xb)),Nb))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Xb)
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1461_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_1462_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_1463_Ints__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Nb: nat] :
          ( member(A,A2,ring_1_Ints(A))
         => member(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb),ring_1_Ints(A)) ) ) ).

% Ints_power
tff(fact_1464_of__nat__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Nb: nat,K: nat] : ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,gbinomial(nat,Nb),K)) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Nb)),K) ) ) ).

% of_nat_gbinomial
tff(fact_1465_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),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_1466_power__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,Nb: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) != zero_zero(A) ) ) ) ).

% power_not_zero
tff(fact_1467_Ints__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,zero_zero(A),ring_1_Ints(A)) ) ).

% Ints_0
tff(fact_1468_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_1469_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,one_one(A),ring_1_Ints(A)) ) ).

% Ints_1
tff(fact_1470_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_1471_power__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Nb: nat] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),A2)),Nb) = aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% power_inverse
tff(fact_1472_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_1473_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_1474_minus__in__Ints__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A] :
          ( member(A,aa(A,A,uminus_uminus(A),Xb),ring_1_Ints(A))
        <=> member(A,Xb,ring_1_Ints(A)) ) ) ).

% minus_in_Ints_iff
tff(fact_1475_Ints__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] : member(A,aa(nat,A,semiring_1_of_nat(A),Nb),ring_1_Ints(A)) ) ).

% Ints_of_nat
tff(fact_1476_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_1477_Ints__cases,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q4: A] :
          ( member(A,Q4,ring_1_Ints(A))
         => ~ ! [Z2: int] : ( Q4 != aa(int,A,ring_1_of_int(A),Z2) ) ) ) ).

% Ints_cases
tff(fact_1478_Ints__induct,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q4: A,P: fun(A,$o)] :
          ( member(A,Q4,ring_1_Ints(A))
         => ( ! [Z2: int] : aa(A,$o,P,aa(int,A,ring_1_of_int(A),Z2))
           => aa(A,$o,P,Q4) ) ) ) ).

% Ints_induct
tff(fact_1479_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_1480_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: 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),Nb))),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))),Nb)),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))),Nb)) ) ) ).

% gbinomial_index_swap
tff(fact_1481_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_1482_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_1483_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$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),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ).

% power_mono
tff(fact_1484_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$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),Nb)) ) ) ).

% zero_le_power
tff(fact_1485_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$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),Nb)) ) ) ).

% zero_less_power
tff(fact_1486_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$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),Nb)) ) ) ).

% one_le_power
tff(fact_1487_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xb: A,Y: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),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),Xb),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Nb)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_1488_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Nb: nat] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),divide_divide(A,one_one(A),A2)),Nb) = divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% power_one_over
tff(fact_1489_power__Suc,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A,Nb: nat] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% power_Suc
tff(fact_1490_power__Suc2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Nb: nat] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),A2) ) ) ).

% power_Suc2
tff(fact_1491_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_1492_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Nb)),K) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_1493_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Ma: nat,Nb: nat] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Ma)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% power_add
tff(fact_1494_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Ma: nat,Nb: nat] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Ma)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xb)),Nb)) = 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),Xb)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Ma)) ) ) ).

% power_mult_power_inverse_commute
tff(fact_1495_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Ma: nat] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Ma)),aa(A,A,inverse_inverse(A),Xb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Ma)) ) ) ).

% power_mult_inverse_distrib
tff(fact_1496_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_1497_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_1498_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_1499_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_1500_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_1501_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_1502_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
         => ( aa(A,$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_1503_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$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),Nb)),one_one(A)) ) ) ) ).

% power_le_one
tff(fact_1504_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,Nb)))
         => ( aa(A,$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_1505_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,Nb)) )
         => ( aa(A,$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_1506_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$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),Nb)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))) ) ) ).

% power_less_power_Suc
tff(fact_1507_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$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),Nb))) ) ) ).

% power_gt1_lemma
tff(fact_1508_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Nb) = $ite(Nb = zero_zero(nat),one_one(A),zero_zero(A)) ) ) ).

% power_0_left
tff(fact_1509_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$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,Nb))) ) ) ).

% power_gt1
tff(fact_1510_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Nb: nat] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb) = 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))),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% power_minus
tff(fact_1511_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,$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)),Nb)) ) ).

% zero_le_power_abs
tff(fact_1512_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Ma: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Ma)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ) ) ).

% power_less_imp_less_exp
tff(fact_1513_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N5: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N5)
         => ( aa(A,$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),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N5)) ) ) ) ).

% power_strict_increasing
tff(fact_1514_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N5: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N5)
         => ( aa(A,$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),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N5)) ) ) ) ).

% power_increasing
tff(fact_1515_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Nb) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_1516_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_1517_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_1518_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_1519_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Ma: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Ma)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),Ma)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Ma)),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),Ma),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_1520_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$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),Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ).

% power_Suc_less
tff(fact_1521_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$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,Nb))),A2) ) ) ) ).

% power_Suc_le_self
tff(fact_1522_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$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,Nb))),one_one(A)) ) ) ) ).

% power_Suc_less_one
tff(fact_1523_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N5: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N5)
         => ( aa(A,$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),N5)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1524_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N5: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N5)
         => ( aa(A,$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),N5)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ) ).

% power_decreasing
tff(fact_1525_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Ma: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Ma)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ) ) ).

% power_le_imp_le_exp
tff(fact_1526_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(A,$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),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb) )
              <=> ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1527_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb) )
         => ( aa(A,$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)),Nb)
               => ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1528_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ).

% self_le_power
tff(fact_1529_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ).

% one_less_power
tff(fact_1530_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,Nb: nat,Ma: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb)) = divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Ma),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ) ).

% power_diff
tff(fact_1531_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_1532_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( member(A,Xb,ring_1_Ints(A))
         => ( ( Xb != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),Xb)) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_1533_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( member(A,Xb,ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Xb)),one_one(A))
           => ( Xb = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_1534_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( member(A,Xb,ring_1_Ints(A))
         => ( member(A,Y,ring_1_Ints(A))
           => ( ( Xb = Y )
            <=> aa(A,$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),Xb),Y))),one_one(A)) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_1535_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)),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_1536_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),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_1537_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$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,$o),ord_less(nat),zero_zero(nat)),Nb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ) ).

% power_strict_mono
tff(fact_1538_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,Ma: nat,Nb: nat] :
          ( ( A2 != zero_zero(A) )
         => ( divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Ma),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb)),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),Nb),Ma)))) ) ) ) ).

% power_diff_power_eq
tff(fact_1539_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Ma: nat,Nb: nat] :
          ( ( Xb != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xb)),Ma)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_1540_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [P3: A,Ma: nat] :
          ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),Ma) = $ite(Ma = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),one_one(nat))))) ) ) ).

% power_eq_if
tff(fact_1541_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_1542_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( 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),Nb),one_one(nat)))),A2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) ) ) ) ).

% power_minus_mult
tff(fact_1543_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( 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),Nb),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),Nb),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_1544_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: A] :
          ( ( archimedean_frac(A,Xb) = A2 )
        <=> ( member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A2),ring_1_Ints(A))
            & aa(A,$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_1545_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_1546_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_1547_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( archimedean_frac(A,aa(A,A,uminus_uminus(A),Xb)) = $ite(member(A,Xb,ring_1_Ints(A)),zero_zero(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),archimedean_frac(A,Xb))) ) ) ).

% frac_neg
tff(fact_1548_Bernoulli__inequality,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => 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),Nb)),Xb))),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)),Xb)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_1549_arctan__add,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),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,Xb)),aa(real,real,arctan,Y)) = aa(real,real,arctan,divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),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),Xb),Y)))) ) ) ) ).

% arctan_add
tff(fact_1550_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : ( aa(set(int),nat,finite_card(int),set_or1337092689740270186AtMost(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)),one_one(int))) ) ).

% card_atLeastAtMost_int
tff(fact_1551_frac__frac,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : ( archimedean_frac(A,archimedean_frac(A,Xb)) = archimedean_frac(A,Xb) ) ) ).

% frac_frac
tff(fact_1552_pochhammer__1,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : ( comm_s3205402744901411588hammer(A,A2,one_one(nat)) = A2 ) ) ).

% pochhammer_1
tff(fact_1553_frac__in__Ints__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( member(A,archimedean_frac(A,Xb),ring_1_Ints(A))
        <=> member(A,Xb,ring_1_Ints(A)) ) ) ).

% frac_in_Ints_iff
tff(fact_1554_power__Suc__0,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) = aa(nat,nat,suc,zero_zero(nat)) ) ).

% power_Suc_0
tff(fact_1555_nat__power__eq__Suc__0__iff,axiom,
    ! [Xb: nat,Ma: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xb),Ma) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( Ma = zero_zero(nat) )
        | ( Xb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_1556_nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: 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),Xb),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Xb)
        | ( Nb = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_1557_pochhammer__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : ( comm_s3205402744901411588hammer(A,A2,zero_zero(nat)) = one_one(A) ) ) ).

% pochhammer_0
tff(fact_1558_pochhammer__Suc0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : ( comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,zero_zero(nat))) = A2 ) ) ).

% pochhammer_Suc0
tff(fact_1559_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_1560_frac__eq__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( ( archimedean_frac(A,Xb) = zero_zero(A) )
        <=> member(A,Xb,ring_1_Ints(A)) ) ) ).

% frac_eq_0_iff
tff(fact_1561_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),archimedean_frac(A,Xb))
        <=> ~ member(A,Xb,ring_1_Ints(A)) ) ) ).

% frac_gt_0_iff
tff(fact_1562_pochhammer__of__nat,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xb: nat,Nb: nat] : ( comm_s3205402744901411588hammer(A,aa(nat,A,semiring_1_of_nat(A),Xb),Nb) = aa(nat,A,semiring_1_of_nat(A),comm_s3205402744901411588hammer(nat,Xb,Nb)) ) ) ).

% pochhammer_of_nat
tff(fact_1563_pochhammer__of__int,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: int,Nb: nat] : ( comm_s3205402744901411588hammer(A,aa(int,A,ring_1_of_int(A),Xb),Nb) = aa(int,A,ring_1_of_int(A),comm_s3205402744901411588hammer(int,Xb,Nb)) ) ) ).

% pochhammer_of_int
tff(fact_1564_arctan__minus,axiom,
    ! [Xb: real] : ( aa(real,real,arctan,aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,arctan,Xb)) ) ).

% arctan_minus
tff(fact_1565_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xb,Nb)) ) ) ).

% pochhammer_pos
tff(fact_1566_nat__power__less__imp__less,axiom,
    ! [I: nat,Ma: nat,Nb: 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),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Nb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_1567_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Ma: nat,Nb: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Ma) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
           => ( comm_s3205402744901411588hammer(A,A2,Nb) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_1568_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat,Ma: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Nb) = zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
           => ( comm_s3205402744901411588hammer(A,A2,Ma) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_1569_real__arch__pow,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => ? [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),Xb),N)) ) ).

% real_arch_pow
tff(fact_1570_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,Xb)) ) ).

% frac_ge_0
tff(fact_1571_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),archimedean_frac(A,Xb)),one_one(A)) ) ).

% frac_lt_1
tff(fact_1572_frac__1__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),one_one(A))) = archimedean_frac(A,Xb) ) ) ).

% frac_1_eq
tff(fact_1573_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xb,Nb)) ) ) ).

% pochhammer_nonneg
tff(fact_1574_power__gt__expt,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),K)) ) ).

% power_gt_expt
tff(fact_1575_nat__one__le__power,axiom,
    ! [I: nat,Nb: 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),Nb)) ) ).

% nat_one_le_power
tff(fact_1576_pochhammer__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: nat] :
          ( comm_s3205402744901411588hammer(A,zero_zero(A),Nb) = $ite(Nb = zero_zero(nat),one_one(A),zero_zero(A)) ) ) ).

% pochhammer_0_left
tff(fact_1577_real__arch__pow__inv,axiom,
    ! [Y: real,Xb: 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),Xb),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),Xb),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_1578_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : ( comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = 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)),Nb)) ) ) ).

% pochhammer_rec
tff(fact_1579_pochhammer__rec_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,Nb: nat] : ( comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,Nb)) = 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),Nb))),comm_s3205402744901411588hammer(A,Z,Nb)) ) ) ).

% pochhammer_rec'
tff(fact_1580_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : ( comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A2,Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),Nb))) ) ) ).

% pochhammer_Suc
tff(fact_1581_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_1582_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [Nb: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K) = zero_zero(A) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_1583_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Nb) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),Nb)
              & ( A2 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_1584_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_1585_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,Nb: nat,Ma: nat] : ( comm_s3205402744901411588hammer(A,Z,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Ma)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,Nb)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),Nb)),Ma)) ) ) ).

% pochhammer_product'
tff(fact_1586_nat__power__eq,axiom,
    ! [Z: int,Nb: 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),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(int,nat,nat2,Z)),Nb) ) ) ).

% nat_power_eq
tff(fact_1587_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( ( archimedean_frac(A,Xb) = Xb )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),one_one(A)) ) ) ) ).

% frac_eq
tff(fact_1588_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),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,Xb)),archimedean_frac(A,Y))),one_one(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xb)),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,Xb)),archimedean_frac(A,Y))),one_one(A))) ) ) ).

% frac_add
tff(fact_1589_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Ma: nat,Nb: nat,Z: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( comm_s3205402744901411588hammer(A,Z,Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,Ma)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma))) ) ) ) ).

% pochhammer_product
tff(fact_1590_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R2: 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),R2),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R2),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R2)),one_one(A)),K)) ) ) ).

% pochhammer_absorb_comp
tff(fact_1591_linear__plus__1__le__power,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => 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),Nb)),Xb)),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),Xb),one_one(real))),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_1592_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_1593_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,A2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( 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),Nb) = A2 )
            & ! [Y4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y4)
                  & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),Nb) = A2 ) )
               => ( Y4 = X ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1594_realpow__pos__nth,axiom,
    ! [Nb: nat,A2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( 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),Nb) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_1595_realpow__pos__nth2,axiom,
    ! [A2: real,Nb: 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,Nb)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_1596_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_1597_arsinh__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arsinh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% arsinh_0
tff(fact_1598_powr__int,axiom,
    ! [Xb: real,I: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,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),Xb),aa(int,nat,nat2,I)),divide_divide(real,one_one(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),I))))) ) ) ).

% powr_int
tff(fact_1599_arsinh__minus__real,axiom,
    ! [Xb: real] : ( aa(real,real,arsinh(real),aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,arsinh(real),Xb)) ) ).

% arsinh_minus_real
tff(fact_1600_powr__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Z: A] : ( powr(A,zero_zero(A),Z) = zero_zero(A) ) ) ).

% powr_0
tff(fact_1601_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_1602_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_1603_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A] :
          ( powr(A,Xb,zero_zero(A)) = $ite(Xb = zero_zero(A),zero_zero(A),one_one(A)) ) ) ).

% powr_zero_eq_one
tff(fact_1604_powr__less__cancel__iff,axiom,
    ! [Xb: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xb,A2)),powr(real,Xb,B2))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2) ) ) ).

% powr_less_cancel_iff
tff(fact_1605_powr__eq__one__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( ( powr(real,A2,Xb) = one_one(real) )
      <=> ( Xb = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_1606_powr__one__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( ( powr(real,Xb,one_one(real)) = Xb )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb) ) ).

% powr_one_gt_zero_iff
tff(fact_1607_powr__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,one_one(real)) = Xb ) ) ).

% powr_one
tff(fact_1608_powr__le__cancel__iff,axiom,
    ! [Xb: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xb,A2)),powr(real,Xb,B2))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2) ) ) ).

% powr_le_cancel_iff
tff(fact_1609_artanh__minus__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,real,artanh(real),aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,artanh(real),Xb)) ) ) ).

% artanh_minus_real
tff(fact_1610_powr__less__mono,axiom,
    ! [A2: real,B2: real,Xb: 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)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xb,A2)),powr(real,Xb,B2)) ) ) ).

% powr_less_mono
tff(fact_1611_powr__less__cancel,axiom,
    ! [Xb: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xb,A2)),powr(real,Xb,B2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2) ) ) ).

% powr_less_cancel
tff(fact_1612_powr__mono,axiom,
    ! [A2: real,B2: real,Xb: 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)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xb,A2)),powr(real,Xb,B2)) ) ) ).

% powr_mono
tff(fact_1613_powr__inj,axiom,
    ! [A2: real,Xb: 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,Xb) = powr(real,A2,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% powr_inj
tff(fact_1614_gr__one__powr,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => ( 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,Xb,Y)) ) ) ).

% gr_one_powr
tff(fact_1615_ge__one__powr__ge__zero,axiom,
    ! [Xb: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => ( 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,Xb,A2)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_1616_powr__mono__both,axiom,
    ! [A2: real,B2: real,Xb: 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)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xb,A2)),powr(real,Y,B2)) ) ) ) ) ).

% powr_mono_both
tff(fact_1617_powr__le1,axiom,
    ! [A2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xb,A2)),one_one(real)) ) ) ) ).

% powr_le1
tff(fact_1618_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_1619_divide__powr__uminus,axiom,
    ! [A2: real,B2: real,C2: 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_1620_powr__add,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A,A2: A,B2: A] : ( powr(A,Xb,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,Xb,A2)),powr(A,Xb,B2)) ) ) ).

% powr_add
tff(fact_1621_powr__diff,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [W: A,Z1: A,Z22: A] : ( powr(A,W,aa(A,A,aa(A,fun(A,A),minus_minus(A),Z1),Z22)) = divide_divide(A,powr(A,W,Z1),powr(A,W,Z22)) ) ) ).

% powr_diff
tff(fact_1622_powr__minus,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A,A2: A] : ( powr(A,Xb,aa(A,A,uminus_uminus(A),A2)) = aa(A,A,inverse_inverse(A),powr(A,Xb,A2)) ) ) ).

% powr_minus
tff(fact_1623_powr__minus__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A,A2: A] : ( powr(A,Xb,aa(A,A,uminus_uminus(A),A2)) = divide_divide(A,one_one(A),powr(A,Xb,A2)) ) ) ).

% powr_minus_divide
tff(fact_1624_powr__neg__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(real,real,uminus_uminus(real),one_one(real))) = divide_divide(real,one_one(real),Xb) ) ) ).

% powr_neg_one
tff(fact_1625_powr__mult__base,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),Xb),powr(real,Xb,Y)) = powr(real,Xb,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Y)) ) ) ).

% powr_mult_base
tff(fact_1626_powr__real__of__int,axiom,
    ! [Xb: real,Nb: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(int,real,ring_1_of_int(real),Nb)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(int,nat,nat2,Nb)),aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Nb))))) ) ) ).

% powr_real_of_int
tff(fact_1627_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( ( archimedean_ceiling(real,aa(real,real,log2(B2),Xb)) = 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))),Xb)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),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_1628_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(nat,real,semiring_1_of_nat(real),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => 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)),divide_divide(real,Xb,aa(nat,real,semiring_1_of_nat(real),Nb)))),Nb)),exp(real,aa(real,real,uminus_uminus(real),Xb))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_1629_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xb: 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),Nb))),Xb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => 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)),divide_divide(real,Xb,aa(nat,real,semiring_1_of_nat(real),Nb)))),Nb)),exp(real,Xb)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_1630_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_1631_root__powr__inverse,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,real,root(Nb),Xb) = powr(real,Xb,divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_1632_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : ( aa(set(int),nat,finite_card(int),set_or5935395276787703475ssThan(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)))) ) ).

% card_greaterThanLessThan_int
tff(fact_1633_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : ( aa(nat,A,gbinomial(A,A2),K) = 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_1634_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : ( aa(nat,A,gbinomial(A,A2),K) = 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_1635_of__nat__fact,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : ( aa(nat,A,semiring_1_of_nat(A),semiring_char_0_fact(nat,Nb)) = semiring_char_0_fact(A,Nb) ) ) ).

% of_nat_fact
tff(fact_1636_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_1637_of__int__fact,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & ring_1(A) )
     => ! [Nb: nat] : ( aa(int,A,ring_1_of_int(A),semiring_char_0_fact(int,Nb)) = semiring_char_0_fact(A,Nb) ) ) ).

% of_int_fact
tff(fact_1638_exp__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( exp(A,zero_zero(A)) = one_one(A) ) ) ).

% exp_zero
tff(fact_1639_ln__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xb)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real)) ) ) ).

% ln_less_zero_iff
tff(fact_1640_ln__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb) ) ) ).

% ln_gt_zero_iff
tff(fact_1641_ln__eq__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( ( aa(real,real,ln_ln(real),Xb) = zero_zero(real) )
      <=> ( Xb = one_one(real) ) ) ) ).

% ln_eq_zero_iff
tff(fact_1642_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_1643_real__root__Suc__0,axiom,
    ! [Xb: real] : ( aa(real,real,root(aa(nat,nat,suc,zero_zero(nat))),Xb) = Xb ) ).

% real_root_Suc_0
tff(fact_1644_fact__1,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).

% fact_1
tff(fact_1645_root__0,axiom,
    ! [Xb: real] : ( aa(real,real,root(zero_zero(nat)),Xb) = zero_zero(real) ) ).

% root_0
tff(fact_1646_real__root__eq__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(real,real,root(Nb),Xb) = aa(real,real,root(Nb),Y) )
      <=> ( Xb = Y ) ) ) ).

% real_root_eq_iff
tff(fact_1647_log__one,axiom,
    ! [A2: real] : ( aa(real,real,log2(A2),one_one(real)) = zero_zero(real) ) ).

% log_one
tff(fact_1648_exp__eq__one__iff,axiom,
    ! [Xb: real] :
      ( ( exp(real,Xb) = one_one(real) )
    <=> ( Xb = zero_zero(real) ) ) ).

% exp_eq_one_iff
tff(fact_1649_ln__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real)) ) ) ).

% ln_le_zero_iff
tff(fact_1650_ln__ge__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb) ) ) ).

% ln_ge_zero_iff
tff(fact_1651_fact__Suc__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,aa(nat,nat,suc,zero_zero(nat))) = one_one(A) ) ) ).

% fact_Suc_0
tff(fact_1652_fact__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : ( semiring_char_0_fact(A,aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Nb))),semiring_char_0_fact(A,Nb)) ) ) ).

% fact_Suc
tff(fact_1653_real__root__eq__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(real,real,root(Nb),Xb) = zero_zero(real) )
      <=> ( Xb = zero_zero(real) ) ) ) ).

% real_root_eq_0_iff
tff(fact_1654_real__root__less__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ).

% real_root_less_iff
tff(fact_1655_real__root__le__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ).

% real_root_le_iff
tff(fact_1656_real__root__eq__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(real,real,root(Nb),Xb) = one_one(real) )
      <=> ( Xb = one_one(real) ) ) ) ).

% real_root_eq_1_iff
tff(fact_1657_real__root__one,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),one_one(real)) = one_one(real) ) ) ).

% real_root_one
tff(fact_1658_zero__less__log__cancel__iff,axiom,
    ! [A2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,log2(A2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_1659_log__less__zero__cancel__iff,axiom,
    ! [A2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log2(A2),Xb)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_1660_one__less__log__cancel__iff,axiom,
    ! [A2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,log2(A2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xb) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_1661_log__less__one__cancel__iff,axiom,
    ! [A2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log2(A2),Xb)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),A2) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_1662_log__less__cancel__iff,axiom,
    ! [A2: real,Xb: 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)),Xb)
       => ( 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,log2(A2),Xb)),aa(real,real,log2(A2),Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_1663_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,log2(A2),A2) = one_one(real) ) ) ) ).

% log_eq_one
tff(fact_1664_one__less__exp__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),exp(real,Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb) ) ).

% one_less_exp_iff
tff(fact_1665_exp__less__one__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xb)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ).

% exp_less_one_iff
tff(fact_1666_exp__le__one__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xb)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ).

% exp_le_one_iff
tff(fact_1667_one__le__exp__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),exp(real,Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb) ) ).

% one_le_exp_iff
tff(fact_1668_real__root__lt__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ) ).

% real_root_lt_0_iff
tff(fact_1669_real__root__gt__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_1670_real__root__le__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ) ).

% real_root_le_0_iff
tff(fact_1671_real__root__ge__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_1672_real__root__lt__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),one_one(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real)) ) ) ).

% real_root_lt_1_iff
tff(fact_1673_real__root__gt__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_1674_real__root__le__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),one_one(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real)) ) ) ).

% real_root_le_1_iff
tff(fact_1675_real__root__ge__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_1676_zero__le__log__cancel__iff,axiom,
    ! [A2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,log2(A2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_1677_log__le__zero__cancel__iff,axiom,
    ! [A2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log2(A2),Xb)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_1678_one__le__log__cancel__iff,axiom,
    ! [A2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,log2(A2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Xb) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_1679_log__le__one__cancel__iff,axiom,
    ! [A2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log2(A2),Xb)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),A2) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_1680_log__le__cancel__iff,axiom,
    ! [A2: real,Xb: 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)),Xb)
       => ( 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,log2(A2),Xb)),aa(real,real,log2(A2),Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_1681_powr__log__cancel,axiom,
    ! [A2: real,Xb: 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)),Xb)
         => ( powr(real,A2,aa(real,real,log2(A2),Xb)) = Xb ) ) ) ) ).

% powr_log_cancel
tff(fact_1682_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,log2(A2),powr(real,A2,Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_1683_real__root__pow__pos2,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos2
tff(fact_1684_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,log2(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_1685_log__ln,axiom,
    ! [Xb: real] : ( aa(real,real,ln_ln(real),Xb) = aa(real,real,log2(exp(real,one_one(real))),Xb) ) ).

% log_ln
tff(fact_1686_real__root__minus,axiom,
    ! [Nb: nat,Xb: real] : ( aa(real,real,root(Nb),aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,root(Nb),Xb)) ) ).

% real_root_minus
tff(fact_1687_exp__not__eq__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : ( exp(A,Xb) != zero_zero(A) ) ) ).

% exp_not_eq_zero
tff(fact_1688_fact__nonzero,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semiri3467727345109120633visors(A) )
     => ! [Nb: nat] : ( semiring_char_0_fact(A,Nb) != zero_zero(A) ) ) ).

% fact_nonzero
tff(fact_1689_real__root__inverse,axiom,
    ! [Nb: nat,Xb: real] : ( aa(real,real,root(Nb),aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,inverse_inverse(real),aa(real,real,root(Nb),Xb)) ) ).

% real_root_inverse
tff(fact_1690_ln__x__over__x__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),divide_divide(real,aa(real,real,ln_ln(real),Y),Y)),divide_divide(real,aa(real,real,ln_ln(real),Xb),Xb)) ) ) ).

% ln_x_over_x_mono
tff(fact_1691_fact__less__mono__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),semiring_char_0_fact(nat,Ma)),semiring_char_0_fact(nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_1692_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_ge_zero
tff(fact_1693_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,Nb)),zero_zero(A)) ) ).

% fact_not_neg
tff(fact_1694_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_gt_zero
tff(fact_1695_mult__exp__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,Xb)),exp(A,Y)) = exp(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) ) ) ).

% mult_exp_exp
tff(fact_1696_exp__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb) )
         => ( exp(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,Xb)),exp(A,Y)) ) ) ) ).

% exp_add_commuting
tff(fact_1697_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_ge_1
tff(fact_1698_exp__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : ( exp(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = divide_divide(A,exp(A,Xb),exp(A,Y)) ) ) ).

% exp_diff
tff(fact_1699_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A,A2: A] :
          ( powr(A,Xb,A2) = $ite(Xb = zero_zero(A),zero_zero(A),exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,ln_ln(A),Xb)))) ) ) ).

% powr_def
tff(fact_1700_exp__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( exp(A,aa(A,A,uminus_uminus(A),Xb)) = aa(A,A,inverse_inverse(A),exp(A,Xb)) ) ) ).

% exp_minus
tff(fact_1701_pochhammer__fact,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_semiring_1(A) )
     => ! [Nb: nat] : ( semiring_char_0_fact(A,Nb) = comm_s3205402744901411588hammer(A,one_one(A),Nb) ) ) ).

% pochhammer_fact
tff(fact_1702_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,Xb: 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)),Xb)
             => ( aa(real,real,log2(A2),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,ln_ln(real),B2),aa(real,real,ln_ln(real),A2))),aa(real,real,log2(B2),Xb)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_1703_ln__root,axiom,
    ! [Nb: nat,B2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( 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(Nb),B2)) = divide_divide(real,aa(real,real,ln_ln(real),B2),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% ln_root
tff(fact_1704_log__root,axiom,
    ! [Nb: nat,A2: real,B2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ( aa(real,real,log2(B2),aa(real,real,root(Nb),A2)) = divide_divide(real,aa(real,real,log2(B2),A2),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% log_root
tff(fact_1705_log__base__root,axiom,
    ! [Nb: nat,B2: real,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
       => ( aa(real,real,log2(aa(real,real,root(Nb),B2)),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log2(B2),Xb)) ) ) ) ).

% log_base_root
tff(fact_1706_ln__gt__zero__imp__gt__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_1707_ln__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xb)),zero_zero(real)) ) ) ).

% ln_less_zero
tff(fact_1708_ln__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb)) ) ).

% ln_gt_zero
tff(fact_1709_ln__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb)) ) ).

% ln_ge_zero
tff(fact_1710_real__root__less__mono,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y)) ) ) ).

% real_root_less_mono
tff(fact_1711_real__root__le__mono,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y)) ) ) ).

% real_root_le_mono
tff(fact_1712_fact__ge__Suc__0__nat,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,Nb)) ).

% fact_ge_Suc_0_nat
tff(fact_1713_real__root__power,axiom,
    ! [Nb: nat,Xb: real,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),K)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),K) ) ) ).

% real_root_power
tff(fact_1714_exp__gt__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),exp(real,Xb)) ) ).

% exp_gt_one
tff(fact_1715_real__root__abs,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),aa(real,real,abs_abs(real),Xb)) = aa(real,real,abs_abs(real),aa(real,real,root(Nb),Xb)) ) ) ).

% real_root_abs
tff(fact_1716_exp__ge__add__one__self,axiom,
    ! [Xb: 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)),Xb)),exp(real,Xb)) ).

% exp_ge_add_one_self
tff(fact_1717_exp__minus__inverse,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb))) = one_one(A) ) ) ).

% exp_minus_inverse
tff(fact_1718_exp__of__nat2__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Nb: nat] : ( exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(nat,A,semiring_1_of_nat(A),Nb))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,Xb)),Nb) ) ) ).

% exp_of_nat2_mult
tff(fact_1719_exp__of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Nb: nat,Xb: A] : ( exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),Xb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,Xb)),Nb) ) ) ).

% exp_of_nat_mult
tff(fact_1720_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ma: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,Ma)),semiring_char_0_fact(A,Nb)) ) ) ) ).

% fact_less_mono
tff(fact_1721_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,Nb)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),Nb))) ) ).

% fact_le_power
tff(fact_1722_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or5935395276787703475ssThan(A,C2,D3))
        <=> ( aa(A,$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),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_1723_ln__ge__zero__imp__ge__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_1724_real__root__gt__zero,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,root(Nb),Xb)) ) ) ).

% real_root_gt_zero
tff(fact_1725_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N5: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N5)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(N5),Xb)),aa(real,real,root(Nb),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_1726_ln__add__one__self__le__self,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => 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)),Xb))),Xb) ) ).

% ln_add_one_self_le_self
tff(fact_1727_ln__eq__minus__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( ( aa(real,real,ln_ln(real),Xb) = aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),one_one(real)) )
       => ( Xb = one_one(real) ) ) ) ).

% ln_eq_minus_one
tff(fact_1728_log__base__change,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log2(B2),Xb) = divide_divide(real,aa(real,real,log2(A2),Xb),aa(real,real,log2(A2),B2)) ) ) ) ).

% log_base_change
tff(fact_1729_ln__inverse,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,ln_ln(real),aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,ln_ln(real),Xb)) ) ) ).

% ln_inverse
tff(fact_1730_root__abs__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,abs_abs(real),aa(real,real,root(Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),Nb))) = aa(real,real,abs_abs(real),Y) ) ) ).

% root_abs_power
tff(fact_1731_exp__ge__add__one__self__aux,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => 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)),Xb)),exp(real,Xb)) ) ).

% exp_ge_add_one_self_aux
tff(fact_1732_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y)
     => ? [X: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),minus_minus(real),Y),one_one(real)))
          & ( exp(real,X) = Y ) ) ) ).

% lemma_exp_total
tff(fact_1733_powr__less__iff,axiom,
    ! [B2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,Y)),Xb)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,log2(B2),Xb)) ) ) ) ).

% powr_less_iff
tff(fact_1734_less__powr__iff,axiom,
    ! [B2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),powr(real,B2,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log2(B2),Xb)),Y) ) ) ) ).

% less_powr_iff
tff(fact_1735_log__less__iff,axiom,
    ! [B2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log2(B2),Xb)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),powr(real,B2,Y)) ) ) ) ).

% log_less_iff
tff(fact_1736_less__log__iff,axiom,
    ! [B2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,log2(B2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,Y)),Xb) ) ) ) ).

% less_log_iff
tff(fact_1737_log__of__power__eq,axiom,
    ! [Ma: nat,B2: real,Nb: nat] :
      ( ( aa(nat,real,semiring_1_of_nat(real),Ma) = aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Nb) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( aa(nat,real,semiring_1_of_nat(real),Nb) = aa(real,real,log2(B2),aa(nat,real,semiring_1_of_nat(real),Ma)) ) ) ) ).

% log_of_power_eq
tff(fact_1738_less__log__of__power,axiom,
    ! [B2: real,Nb: nat,Ma: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Nb)),Ma)
     => ( 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),Nb)),aa(real,real,log2(B2),Ma)) ) ) ).

% less_log_of_power
tff(fact_1739_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,semiring_char_0_fact(nat,Nb),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),R2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_1740_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D3))
        <=> ( aa(A,$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),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_1741_real__root__pos__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Nb),Xb)) ) ) ).

% real_root_pos_pos
tff(fact_1742_real__root__strict__increasing,axiom,
    ! [Nb: nat,N5: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N5)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(N5),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_1743_real__root__decreasing,axiom,
    ! [Nb: nat,N5: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N5)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(N5),Xb)),aa(real,real,root(Nb),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_1744_real__root__pow__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos
tff(fact_1745_real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
       => ( aa(real,real,root(Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb)) = Xb ) ) ) ).

% real_root_power_cancel
tff(fact_1746_real__root__pos__unique,axiom,
    ! [Nb: nat,Y: real,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( 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),Nb) = Xb )
         => ( aa(real,real,root(Nb),Xb) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_1747_ln__le__minus__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),one_one(real))) ) ).

% ln_le_minus_one
tff(fact_1748_ln__add__one__self__le__self2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => 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)),Xb))),Xb) ) ).

% ln_add_one_self_le_self2
tff(fact_1749_log__mult,axiom,
    ! [A2: real,Xb: 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)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
           => ( aa(real,real,log2(A2),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log2(A2),Xb)),aa(real,real,log2(A2),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_1750_le__log__iff,axiom,
    ! [B2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,log2(B2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Y)),Xb) ) ) ) ).

% le_log_iff
tff(fact_1751_log__le__iff,axiom,
    ! [B2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log2(B2),Xb)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),powr(real,B2,Y)) ) ) ) ).

% log_le_iff
tff(fact_1752_le__powr__iff,axiom,
    ! [B2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),powr(real,B2,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log2(B2),Xb)),Y) ) ) ) ).

% le_powr_iff
tff(fact_1753_powr__le__iff,axiom,
    ! [B2: real,Xb: 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)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Y)),Xb)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,log2(B2),Xb)) ) ) ) ).

% powr_le_iff
tff(fact_1754_log__divide,axiom,
    ! [A2: real,Xb: 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)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
           => ( aa(real,real,log2(A2),divide_divide(real,Xb,Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log2(A2),Xb)),aa(real,real,log2(A2),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_1755_le__log__of__power,axiom,
    ! [B2: real,Nb: nat,Ma: 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),Nb)),Ma)
     => ( 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),Nb)),aa(real,real,log2(B2),Ma)) ) ) ).

% le_log_of_power
tff(fact_1756_log__inverse,axiom,
    ! [A2: real,Xb: 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)),Xb)
         => ( aa(real,real,log2(A2),aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,log2(A2),Xb)) ) ) ) ) ).

% log_inverse
tff(fact_1757_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Nb: nat,Xb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,divide_divide(A,Xb,aa(nat,A,semiring_1_of_nat(A),Nb)))),Nb) = exp(A,Xb) ) ) ) ).

% exp_divide_power_eq
tff(fact_1758_real__root__increasing,axiom,
    ! [Nb: nat,N5: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N5)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(N5),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_1759_ln__one__minus__pos__upper__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),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)),Xb))),aa(real,real,uminus_uminus(real),Xb)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_1760_ln__powr__bound,axiom,
    ! [Xb: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => ( 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),Xb)),divide_divide(real,powr(real,Xb,A2),A2)) ) ) ).

% ln_powr_bound
tff(fact_1761_ln__powr__bound2,axiom,
    ! [Xb: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => ( 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),Xb),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A2,A2)),Xb)) ) ) ).

% ln_powr_bound2
tff(fact_1762_log__of__power__less,axiom,
    ! [Ma: nat,B2: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Ma)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Nb))
     => ( 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)),Ma)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log2(B2),aa(nat,real,semiring_1_of_nat(real),Ma))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% log_of_power_less
tff(fact_1763_add__log__eq__powr,axiom,
    ! [B2: real,Xb: 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)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Y),aa(real,real,log2(B2),Xb)) = aa(real,real,log2(B2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B2,Y)),Xb)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_1764_log__add__eq__powr,axiom,
    ! [B2: real,Xb: 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)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log2(B2),Xb)),Y) = aa(real,real,log2(B2),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),powr(real,B2,Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_1765_minus__log__eq__powr,axiom,
    ! [B2: real,Xb: 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)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),Y),aa(real,real,log2(B2),Xb)) = aa(real,real,log2(B2),divide_divide(real,powr(real,B2,Y),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_1766_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Ma: nat] :
          ( semiring_char_0_fact(A,Ma) = $ite(Ma = 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),Ma)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),one_one(nat))))) ) ) ).

% fact_num_eq_if
tff(fact_1767_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( semiring_char_0_fact(A,Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_1768_pochhammer__same,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_ring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [Nb: nat] : ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),Nb) = 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))),Nb)),semiring_char_0_fact(A,Nb)) ) ) ).

% pochhammer_same
tff(fact_1769_log__of__power__le,axiom,
    ! [Ma: nat,B2: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Ma)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Nb))
     => ( 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)),Ma)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log2(B2),aa(nat,real,semiring_1_of_nat(real),Ma))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% log_of_power_le
tff(fact_1770_log__minus__eq__powr,axiom,
    ! [B2: real,Xb: 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)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log2(B2),Xb)),Y) = aa(real,real,log2(B2),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),powr(real,B2,aa(real,real,uminus_uminus(real),Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_1771_fact__diff__Suc,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Ma))
     => ( semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Ma)),Nb)) = 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,Ma)),Nb)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_1772_floor__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( ( aa(real,int,archim6421214686448440834_floor(real),aa(real,real,log2(B2),Xb)) = 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))),Xb)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),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_1773_tanh__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( aa(A,A,tanh(A),Xb) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb))),aa(A,A,aa(A,fun(A,A),plus_plus(A),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb)))) ) ) ).

% tanh_altdef
tff(fact_1774_split__root,axiom,
    ! [P: fun(real,$o),Nb: nat,Xb: real] :
      ( aa(real,$o,P,aa(real,real,root(Nb),Xb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(real,$o,P,zero_zero(real)) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ! [Y5: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y5)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y5)),Nb)) = Xb )
             => aa(real,$o,P,Y5) ) ) ) ) ).

% split_root
tff(fact_1775_power_Opower__eq__if,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),P3: A,Ma: nat] :
      ( power2(A,One,Times,P3,Ma) = $ite(Ma = zero_zero(nat),One,aa(A,A,aa(A,fun(A,A),Times,P3),power2(A,One,Times,P3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),one_one(nat))))) ) ).

% power.power_eq_if
tff(fact_1776_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q4: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),P3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),aa(A,int,archim6421214686448440834_floor(A),divide_divide(A,P3,Q4)))),one_one(A))),Q4)) ) ) ).

% floor_divide_upper
tff(fact_1777_split__pos__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),P,divide_divide(int,Nb,K)),modulo_modulo(int,Nb,K))
      <=> ! [I3: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
              & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I3),J3) ) ) ) ).

% split_pos_lemma
tff(fact_1778_split__neg__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),P,divide_divide(int,Nb,K)),modulo_modulo(int,Nb,K))
      <=> ! [I3: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
              & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I3),J3) ) ) ) ).

% split_neg_lemma
tff(fact_1779_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : ( aa(A,A,sgn_sgn(A),aa(A,A,sgn_sgn(A),A2)) = aa(A,A,sgn_sgn(A),A2) ) ) ).

% sgn_sgn
tff(fact_1780_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : ( modulo_modulo(A,zero_zero(A),A2) = zero_zero(A) ) ) ).

% mod_0
tff(fact_1781_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : ( modulo_modulo(A,A2,zero_zero(A)) = A2 ) ) ).

% mod_by_0
tff(fact_1782_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : ( modulo_modulo(A,A2,A2) = zero_zero(A) ) ) ).

% mod_self
tff(fact_1783_bits__mod__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : ( modulo_modulo(A,zero_zero(A),A2) = zero_zero(A) ) ) ).

% bits_mod_0
tff(fact_1784_mod__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),B2) = modulo_modulo(A,A2,B2) ) ) ).

% mod_add_self2
tff(fact_1785_mod__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2),B2) = modulo_modulo(A,A2,B2) ) ) ).

% mod_add_self1
tff(fact_1786_minus__mod__self2,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),B2) = modulo_modulo(A,A2,B2) ) ) ).

% minus_mod_self2
tff(fact_1787_mod__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : ( modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,B2)) ) ) ).

% mod_minus_minus
tff(fact_1788_sgn__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_0
tff(fact_1789_sgn__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),one_one(A)) = one_one(A) ) ) ).

% sgn_1
tff(fact_1790_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : ( aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),A2)) ) ) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_1791_inverse__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] : ( aa(A,A,inverse_inverse(A),aa(A,A,sgn_sgn(A),A2)) = aa(A,A,sgn_sgn(A),A2) ) ) ).

% inverse_sgn
tff(fact_1792_sgn__inverse,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A] : ( aa(A,A,sgn_sgn(A),aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),aa(A,A,sgn_sgn(A),A2)) ) ) ).

% sgn_inverse
tff(fact_1793_of__int__floor__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( ( aa(int,A,ring_1_of_int(A),aa(A,int,archim6421214686448440834_floor(A),Xb)) = Xb )
        <=> ? [N4: int] : ( Xb = aa(int,A,ring_1_of_int(A),N4) ) ) ) ).

% of_int_floor_cancel
tff(fact_1794_floor__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int] : ( aa(A,int,archim6421214686448440834_floor(A),aa(int,A,ring_1_of_int(A),Z)) = Z ) ) ).

% floor_of_int
tff(fact_1795_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_1796_tanh__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( aa(A,A,tanh(A),aa(A,A,uminus_uminus(A),Xb)) = aa(A,A,uminus_uminus(A),aa(A,A,tanh(A),Xb)) ) ) ).

% tanh_minus
tff(fact_1797_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: 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_1798_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: 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_1799_mod__by__1,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : ( modulo_modulo(A,A2,one_one(A)) = zero_zero(A) ) ) ).

% mod_by_1
tff(fact_1800_bits__mod__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : ( modulo_modulo(A,A2,one_one(A)) = zero_zero(A) ) ) ).

% bits_mod_by_1
tff(fact_1801_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : ( divide_divide(A,modulo_modulo(A,A2,B2),B2) = zero_zero(A) ) ) ).

% mod_div_trivial
tff(fact_1802_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] : ( divide_divide(A,modulo_modulo(A,A2,B2),B2) = zero_zero(A) ) ) ).

% bits_mod_div_trivial
tff(fact_1803_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: 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) = modulo_modulo(A,A2,B2) ) ) ).

% mod_mult_self4
tff(fact_1804_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: 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) = modulo_modulo(A,A2,B2) ) ) ).

% mod_mult_self3
tff(fact_1805_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C2: 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) = modulo_modulo(A,A2,B2) ) ) ).

% mod_mult_self2
tff(fact_1806_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: 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) = modulo_modulo(A,A2,B2) ) ) ).

% mod_mult_self1
tff(fact_1807_minus__mod__self1,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [B2: A,A2: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) ) ) ).

% minus_mod_self1
tff(fact_1808_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,sgn_sgn(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% sgn_less
tff(fact_1809_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,sgn_sgn(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% sgn_greater
tff(fact_1810_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( aa(A,int,archim6421214686448440834_floor(A),zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_1811_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( aa(A,int,archim6421214686448440834_floor(A),one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_1812_floor__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: nat] : ( aa(A,int,archim6421214686448440834_floor(A),aa(nat,A,semiring_1_of_nat(A),Nb)) = aa(nat,int,semiring_1_of_nat(int),Nb) ) ) ).

% floor_of_nat
tff(fact_1813_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A] : ( modulo_modulo(A,A2,aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% mod_minus1_right
tff(fact_1814_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,A,sgn_sgn(A),A2) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_1815_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_1816_floor__uminus__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int] : ( aa(A,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_1817_floor__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] : ( aa(A,int,archim6421214686448440834_floor(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),Z) ) ) ).

% floor_diff_of_int
tff(fact_1818_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)
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_1819_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)
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_1820_floor__add2,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( ( member(A,Xb,ring_1_Ints(A))
            | member(A,Y,ring_1_Ints(A)) )
         => ( aa(A,int,archim6421214686448440834_floor(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),aa(A,int,archim6421214686448440834_floor(A),Y)) ) ) ) ).

% floor_add2
tff(fact_1821_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_1822_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(A,int,archim6421214686448440834_floor(A),Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb) ) ) ).

% zero_le_floor
tff(fact_1823_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A)) ) ) ).

% floor_less_zero
tff(fact_1824_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(A,int,archim6421214686448440834_floor(A),Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ).

% zero_less_floor
tff(fact_1825_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),one_one(A)) ) ) ).

% floor_le_zero
tff(fact_1826_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),aa(A,int,archim6421214686448440834_floor(A),Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ).

% one_le_floor
tff(fact_1827_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),one_one(A)) ) ) ).

% floor_less_one
tff(fact_1828_floor__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : ( aa(A,int,archim6421214686448440834_floor(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),one_one(int)) ) ) ).

% floor_diff_one
tff(fact_1829_of__nat__mod,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Ma: nat,Nb: nat] : ( aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,Ma,Nb)) = modulo_modulo(A,aa(nat,A,semiring_1_of_nat(A),Ma),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ).

% of_nat_mod
tff(fact_1830_tanh__real__bounds,axiom,
    ! [Xb: real] : member(real,aa(real,real,tanh(real),Xb),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real))) ).

% tanh_real_bounds
tff(fact_1831_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_1832_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_1833_mod__add__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) ) ) ).

% mod_add_eq
tff(fact_1834_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A6: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A6,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B6,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),B6),C2) ) ) ) ) ).

% mod_add_cong
tff(fact_1835_mod__add__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) ) ) ).

% mod_add_left_eq
tff(fact_1836_mod__add__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C2: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) ) ) ).

% mod_add_right_eq
tff(fact_1837_mod__diff__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C2) ) ) ).

% mod_diff_eq
tff(fact_1838_mod__diff__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,A6: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A6,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B6,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A6),B6),C2) ) ) ) ) ).

% mod_diff_cong
tff(fact_1839_mod__diff__left__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C2) ) ) ).

% mod_diff_left_eq
tff(fact_1840_mod__diff__right__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,C2: A] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C2) ) ) ).

% mod_diff_right_eq
tff(fact_1841_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: A] : ( aa(A,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),aa(A,A,sgn_sgn(A),A2)),aa(A,A,sgn_sgn(A),B2)) ) ) ).

% sgn_mult
tff(fact_1842_mod__minus__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : ( modulo_modulo(A,aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,B2)),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) ) ) ).

% mod_minus_eq
tff(fact_1843_mod__minus__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,A6: A] :
          ( ( modulo_modulo(A,A2,B2) = modulo_modulo(A,A6,B2) )
         => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A6),B2) ) ) ) ).

% mod_minus_cong
tff(fact_1844_mod__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : ( modulo_modulo(A,A2,aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2)) ) ) ).

% mod_minus_right
tff(fact_1845_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A2) )
         => ( aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,sgn_sgn(A),A2) ) ) ) ).

% same_sgn_sgn_add
tff(fact_1846_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),aa(A,int,archim6421214686448440834_floor(A),Y)) ) ) ).

% floor_mono
tff(fact_1847_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),aa(A,int,archim6421214686448440834_floor(A),Xb))),Xb) ) ).

% of_int_floor_le
tff(fact_1848_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),aa(A,int,archim6421214686448440834_floor(A),Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ).

% floor_less_cancel
tff(fact_1849_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),modulo_modulo(A,A2,B2)),A2) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1850_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),modulo_modulo(A,A2,B2)),B2) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1851_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = A2 )
        <=> ( divide_divide(A,A2,B2) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1852_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,B2,C2) )
         => ~ ! [D5: 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),D5)) ) ) ) ).

% mod_eqE
tff(fact_1853_div__add1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C2: 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),divide_divide(A,A2,C2)),divide_divide(A,B2,C2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2)) ) ) ).

% div_add1_eq
tff(fact_1854_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) != aa(A,A,sgn_sgn(A),A2) )
         => ( ( aa(A,A,sgn_sgn(A),A2) != zero_zero(A) )
           => ( ( aa(A,A,sgn_sgn(A),B2) != zero_zero(A) )
             => ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_1855_sgn__minus__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% sgn_minus_1
tff(fact_1856_mult__sgn__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),Xb)),aa(A,A,abs_abs(A),Xb)) = Xb ) ) ).

% mult_sgn_abs
tff(fact_1857_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,abs_abs(A),A2)) = A2 ) ) ).

% sgn_mult_abs
tff(fact_1858_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)),aa(A,A,sgn_sgn(A),A2)) = A2 ) ) ).

% abs_mult_sgn
tff(fact_1859_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K: A] : ( aa(A,A,abs_abs(A),K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,sgn_sgn(A),K)) ) ) ).

% linordered_idom_class.abs_sgn
tff(fact_1860_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A2) )
         => ( aa(A,A,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_1861_floor__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),archimedean_ceiling(A,Xb)) ) ).

% floor_le_ceiling
tff(fact_1862_zmod__le__nonneg__dividend,axiom,
    ! [Ma: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ma)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),modulo_modulo(int,Ma,K)),Ma) ) ).

% zmod_le_nonneg_dividend
tff(fact_1863_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),modulo_modulo(int,K,L)) ) ).

% neg_mod_bound
tff(fact_1864_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),modulo_modulo(int,K,L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_1865_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo(int,K,aa(int,int,uminus_uminus(int),L)) != zero_zero(int) )
     => ( modulo_modulo(int,K,L) != zero_zero(int) ) ) ).

% zmod_zminus2_not_zero
tff(fact_1866_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K),L) != zero_zero(int) )
     => ( modulo_modulo(int,K,L) != zero_zero(int) ) ) ).

% zmod_zminus1_not_zero
tff(fact_1867_zmod__eq__0__iff,axiom,
    ! [Ma: int,D3: int] :
      ( ( modulo_modulo(int,Ma,D3) = zero_zero(int) )
    <=> ? [Q6: int] : ( Ma = aa(int,int,aa(int,fun(int,int),times_times(int),D3),Q6) ) ) ).

% zmod_eq_0_iff
tff(fact_1868_zmod__eq__0D,axiom,
    ! [Ma: int,D3: int] :
      ( ( modulo_modulo(int,Ma,D3) = zero_zero(int) )
     => ? [Q2: int] : ( Ma = aa(int,int,aa(int,fun(int,int),times_times(int),D3),Q2) ) ) ).

% zmod_eq_0D
tff(fact_1869_power_Opower_Opower__Suc,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A2: A,Nb: nat] : ( power2(A,One,Times,A2,aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),Times,A2),power2(A,One,Times,A2,Nb)) ) ).

% power.power.power_Suc
tff(fact_1870_power_Opower_Opower__0,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A2: A] : ( power2(A,One,Times,A2,zero_zero(nat)) = One ) ).

% power.power.power_0
tff(fact_1871_tanh__real__lt__1,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),Xb)),one_one(real)) ).

% tanh_real_lt_1
tff(fact_1872_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),aa(A,int,archim6421214686448440834_floor(A),Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),Xb) ) ) ).

% le_floor_iff
tff(fact_1873_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% floor_less_iff
tff(fact_1874_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)
           => ( modulo_modulo(A,A2,B2) = A2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1875_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)),modulo_modulo(A,A2,B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1876_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),aa(A,int,archim6421214686448440834_floor(A),Y))),aa(A,int,archim6421214686448440834_floor(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))) ) ).

% le_floor_add
tff(fact_1877_int__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] : ( aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),aa(A,int,archim6421214686448440834_floor(A),Xb)) = aa(A,int,archim6421214686448440834_floor(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),Xb)) ) ) ).

% int_add_floor
tff(fact_1878_floor__add__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] : ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),Z) = aa(A,int,archim6421214686448440834_floor(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(int,A,ring_1_of_int(A),Z))) ) ) ).

% floor_add_int
tff(fact_1879_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = one_one(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% sgn_1_pos
tff(fact_1880_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C2: 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),divide_divide(A,B2,C2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),modulo_modulo(A,B2,C2)),C2)) ) ) ).

% div_mult1_eq
tff(fact_1881_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),divide_divide(A,A2,B2))),modulo_modulo(A,A2,B2)) = A2 ) ) ).

% mult_div_mod_eq
tff(fact_1882_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),modulo_modulo(A,A2,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A2,B2))) = A2 ) ) ).

% mod_mult_div_eq
tff(fact_1883_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),modulo_modulo(A,A2,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),B2)) = A2 ) ) ).

% mod_div_mult_eq
tff(fact_1884_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),divide_divide(A,A2,B2)),B2)),modulo_modulo(A,A2,B2)) = A2 ) ) ).

% div_mult_mod_eq
tff(fact_1885_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),divide_divide(A,A2,B2)),B2)),modulo_modulo(A,A2,B2)) ) ) ).

% mod_div_decomp
tff(fact_1886_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),divide_divide(A,A2,B2)),B2)),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_1887_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),divide_divide(A,A2,B2))),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_1888_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),divide_divide(A,A2,B2))) = modulo_modulo(A,A2,B2) ) ) ).

% minus_mult_div_eq_mod
tff(fact_1889_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),modulo_modulo(A,A2,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A2,B2)) ) ) ).

% minus_mod_eq_mult_div
tff(fact_1890_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),modulo_modulo(A,A2,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),B2) ) ) ).

% minus_mod_eq_div_mult
tff(fact_1891_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),divide_divide(A,A2,B2)),B2)) = modulo_modulo(A,A2,B2) ) ) ).

% minus_div_mult_eq_mod
tff(fact_1892_floor__divide__of__int__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [K: int,L: int] : ( aa(A,int,archim6421214686448440834_floor(A),divide_divide(A,aa(int,A,ring_1_of_int(A),K),aa(int,A,ring_1_of_int(A),L))) = divide_divide(int,K,L) ) ) ).

% floor_divide_of_int_eq
tff(fact_1893_ceiling__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : ( archimedean_ceiling(A,Xb) = aa(int,int,uminus_uminus(int),aa(A,int,archim6421214686448440834_floor(A),aa(A,A,uminus_uminus(A),Xb))) ) ) ).

% ceiling_def
tff(fact_1894_floor__minus,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : ( aa(A,int,archim6421214686448440834_floor(A),aa(A,A,uminus_uminus(A),Xb)) = aa(int,int,uminus_uminus(int),archimedean_ceiling(A,Xb)) ) ) ).

% floor_minus
tff(fact_1895_ceiling__minus,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : ( archimedean_ceiling(A,aa(A,A,uminus_uminus(A),Xb)) = aa(int,int,uminus_uminus(int),aa(A,int,archim6421214686448440834_floor(A),Xb)) ) ) ).

% ceiling_minus
tff(fact_1896_sgn__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,sgn_sgn(real),aa(real,real,root(Nb),Xb)) = aa(real,real,sgn_sgn(real),Xb) ) ) ).

% sgn_root
tff(fact_1897_floor__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Nb: nat] :
          ( ( Xb = aa(int,A,ring_1_of_int(A),aa(A,int,archim6421214686448440834_floor(A),Xb)) )
         => ( aa(A,int,archim6421214686448440834_floor(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),Nb) ) ) ) ).

% floor_power
tff(fact_1898_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ) ).

% abs_sgn_eq
tff(fact_1899_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [Ma: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( modulo_modulo(A,semiring_char_0_fact(A,Nb),semiring_char_0_fact(A,Ma)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_1900_frac__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : ( archimedean_frac(A,Xb) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(int,A,ring_1_of_int(A),aa(A,int,archim6421214686448440834_floor(A),Xb))) ) ) ).

% frac_def
tff(fact_1901_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),modulo_modulo(int,K,L)),zero_zero(int)) ) ).

% neg_mod_sign
tff(fact_1902_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)),modulo_modulo(int,K,L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_1903_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),modulo_modulo(int,A2,B2)),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),modulo_modulo(int,A2,B2)) ) ) ).

% neg_mod_conj
tff(fact_1904_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)),modulo_modulo(int,A2,B2))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),modulo_modulo(int,A2,B2)),B2) ) ) ).

% pos_mod_conj
tff(fact_1905_zmod__trivial__iff,axiom,
    ! [I: int,K: 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_1906_zdiv__mono__strict,axiom,
    ! [A3: int,B3: int,Nb: 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)),Nb)
       => ( ( modulo_modulo(int,A3,Nb) = zero_zero(int) )
         => ( ( modulo_modulo(int,B3,Nb) = zero_zero(int) )
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,A3,Nb)),divide_divide(int,B3,Nb)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_1907_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      ( modulo_modulo(int,aa(int,int,uminus_uminus(int),A2),B2) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),modulo_modulo(int,A2,B2))) ) ).

% zmod_zminus1_eq_if
tff(fact_1908_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      ( modulo_modulo(int,A2,aa(int,int,uminus_uminus(int),B2)) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),modulo_modulo(int,A2,B2)),B2)) ) ).

% zmod_zminus2_eq_if
tff(fact_1909_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),modulo_modulo(int,K,L))),aa(int,int,abs_abs(int),L)) ) ).

% abs_mod_less
tff(fact_1910_div__mod__decomp__int,axiom,
    ! [A3: int,Nb: int] : ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),divide_divide(int,A3,Nb)),Nb)),modulo_modulo(int,A3,Nb)) ) ).

% div_mod_decomp_int
tff(fact_1911_tanh__real__gt__neg1,axiom,
    ! [Xb: 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),Xb)) ).

% tanh_real_gt_neg1
tff(fact_1912_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(A,int,archim6421214686448440834_floor(A),R2)))),R2) ) ) ).

% of_nat_floor
tff(fact_1913_one__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),one_one(int)) = aa(A,int,archim6421214686448440834_floor(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),one_one(A))) ) ) ).

% one_add_floor
tff(fact_1914_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,aa(A,int,archim6421214686448440834_floor(A),A2))),aa(int,nat,nat2,aa(A,int,archim6421214686448440834_floor(A),B2)))),aa(int,nat,nat2,aa(A,int,archim6421214686448440834_floor(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))) ) ).

% le_mult_nat_floor
tff(fact_1915_floor__divide__of__nat__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Ma: nat,Nb: nat] : ( aa(A,int,archim6421214686448440834_floor(A),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Ma),aa(nat,A,semiring_1_of_nat(A),Nb))) = aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Ma,Nb)) ) ) ).

% floor_divide_of_nat_eq
tff(fact_1916_nat__floor__neg,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
     => ( aa(int,nat,nat2,aa(real,int,archim6421214686448440834_floor(real),Xb)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_1917_sgn__real__def,axiom,
    ! [A2: real] :
      ( aa(real,real,sgn_sgn(real),A2) = $ite(
          A2 = zero_zero(real),
          zero_zero(real),
          $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ) ) ).

% sgn_real_def
tff(fact_1918_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% sgn_1_neg
tff(fact_1919_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(A,A,sgn_sgn(A),Xb) = $ite(
              Xb = zero_zero(A),
              zero_zero(A),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ) ).

% sgn_if
tff(fact_1920_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,Ma: nat,Nb: nat] : ( modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,A,semiring_1_of_nat(A),Nb))) = 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),Ma)),modulo_modulo(A,divide_divide(A,A2,aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,A,semiring_1_of_nat(A),Nb)))),modulo_modulo(A,A2,aa(nat,A,semiring_1_of_nat(A),Ma))) ) ) ).

% mod_mult2_eq'
tff(fact_1921_floor__eq3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb)))
       => ( aa(int,nat,nat2,aa(real,int,archim6421214686448440834_floor(real),Xb)) = Nb ) ) ) ).

% floor_eq3
tff(fact_1922_le__nat__floor,axiom,
    ! [Xb: nat,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Xb)),A2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),aa(int,nat,nat2,aa(real,int,archim6421214686448440834_floor(real),A2))) ) ).

% le_nat_floor
tff(fact_1923_ceiling__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( archimedean_ceiling(A,Xb) = $ite(Xb = aa(int,A,ring_1_of_int(A),aa(A,int,archim6421214686448440834_floor(A),Xb)),aa(A,int,archim6421214686448440834_floor(A),Xb),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),one_one(int))) ) ) ).

% ceiling_altdef
tff(fact_1924_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,Xb)),aa(A,int,archim6421214686448440834_floor(A),Xb))),one_one(int)) ) ).

% ceiling_diff_floor_le_1
tff(fact_1925_floor__eq,axiom,
    ! [Nb: int,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(int,real,ring_1_of_int(real),Nb)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Nb)),one_one(real)))
       => ( aa(real,int,archim6421214686448440834_floor(real),Xb) = Nb ) ) ) ).

% floor_eq
tff(fact_1926_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),aa(real,int,archim6421214686448440834_floor(real),R2))),one_one(real))) ).

% real_of_int_floor_add_one_gt
tff(fact_1927_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),aa(real,int,archim6421214686448440834_floor(real),R2))),one_one(real))) ).

% real_of_int_floor_add_one_ge
tff(fact_1928_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),one_one(real))),aa(int,real,ring_1_of_int(real),aa(real,int,archim6421214686448440834_floor(real),R2))) ).

% real_of_int_floor_gt_diff_one
tff(fact_1929_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),one_one(real))),aa(int,real,ring_1_of_int(real),aa(real,int,archim6421214686448440834_floor(real),R2))) ).

% real_of_int_floor_ge_diff_one
tff(fact_1930_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))
       => ( modulo_modulo(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_1931_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)
       => ( modulo_modulo(int,K,L) = modulo_modulo(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L),L) ) ) ) ).

% mod_pos_geq
tff(fact_1932_real__of__int__div__aux,axiom,
    ! [Xb: int,D3: int] : ( divide_divide(real,aa(int,real,ring_1_of_int(real),Xb),aa(int,real,ring_1_of_int(real),D3)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),divide_divide(int,Xb,D3))),divide_divide(real,aa(int,real,ring_1_of_int(real),modulo_modulo(int,Xb,D3)),aa(int,real,ring_1_of_int(real),D3))) ) ).

% real_of_int_div_aux
tff(fact_1933_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))
           => ( aa(A,int,archim6421214686448440834_floor(A),Xb) = Z ) ) ) ) ).

% floor_unique
tff(fact_1934_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: int] :
          ( ( aa(A,int,archim6421214686448440834_floor(A),Xb) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),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_1935_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,aa(A,int,archim6421214686448440834_floor(A),Ta))
        <=> ! [I3: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I3)),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I3)),one_one(A))) )
             => aa(int,$o,P,I3) ) ) ) ).

% floor_split
tff(fact_1936_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),aa(A,int,archim6421214686448440834_floor(A),A2)),aa(A,int,archim6421214686448440834_floor(A),B2))),aa(A,int,archim6421214686448440834_floor(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ) ).

% le_mult_floor
tff(fact_1937_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(A,int,archim6421214686448440834_floor(A),Xb))
        <=> aa(A,$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))),Xb) ) ) ).

% less_floor_iff
tff(fact_1938_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),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_1939_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),aa(A,int,archim6421214686448440834_floor(A),Xb))),Xb)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),one_one(int)))) ) ) ).

% floor_correct
tff(fact_1940_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)
         => ( 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),modulo_modulo(A,divide_divide(A,A2,B2),C2))),modulo_modulo(A,A2,B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_1941_floor__eq4,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Nb)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb)))
       => ( aa(int,nat,nat2,aa(real,int,archim6421214686448440834_floor(real),Xb)) = Nb ) ) ) ).

% floor_eq4
tff(fact_1942_sgn__power__injE,axiom,
    ! [A2: real,Nb: nat,Xb: real,B2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),A2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),A2)),Nb)) = Xb )
     => ( ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),B2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),B2)),Nb)) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_1943_floor__eq2,axiom,
    ! [Nb: int,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Nb)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Nb)),one_one(real)))
       => ( aa(real,int,archim6421214686448440834_floor(real),Xb) = Nb ) ) ) ).

% floor_eq2
tff(fact_1944_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)
     => ( aa(real,int,archim6421214686448440834_floor(real),divide_divide(real,A2,aa(int,real,ring_1_of_int(real),B2))) = divide_divide(int,aa(real,int,archim6421214686448440834_floor(real),A2),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_1945_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q4: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
         => ( modulo_modulo(int,A2,B2) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_1946_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q4: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R2)
         => ( modulo_modulo(int,A2,B2) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_1947_split__zmod,axiom,
    ! [P: fun(int,$o),Nb: int,K: int] :
      ( aa(int,$o,P,modulo_modulo(int,Nb,K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,Nb) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => aa(int,$o,P,J3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => aa(int,$o,P,J3) ) ) ) ) ).

% split_zmod
tff(fact_1948_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)
     => ( 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))),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_1949_zmod__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( 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_1950_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)
     => ( 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),modulo_modulo(int,divide_divide(int,A2,B2),C2))),modulo_modulo(int,A2,B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_1951_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( divide_divide(int,A2,aa(int,int,uminus_uminus(int),B2)) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),aa(int,int,uminus_uminus(int),divide_divide(int,A2,B2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),divide_divide(int,A2,B2))),one_one(int))) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_1952_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( divide_divide(int,aa(int,int,uminus_uminus(int),A2),B2) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),aa(int,int,uminus_uminus(int),divide_divide(int,A2,B2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),divide_divide(int,A2,B2))),one_one(int))) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_1953_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q4: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q4)
         => aa(A,$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),aa(A,int,archim6421214686448440834_floor(A),divide_divide(A,P3,Q4)))),Q4)),P3) ) ) ).

% floor_divide_lower
tff(fact_1954_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),aa(A,int,archim6421214686448440834_floor(A),A2)),aa(A,int,archim6421214686448440834_floor(A),B2)))),aa(int,B,ring_1_of_int(B),aa(A,int,archim6421214686448440834_floor(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_1955_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,int,archim6421214686448440834_floor(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),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,Xb)),archimedean_frac(A,Y))),one_one(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),aa(A,int,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),aa(A,int,archim6421214686448440834_floor(A),Xb)),aa(A,int,archim6421214686448440834_floor(A),Y))),one_one(int))) ) ) ).

% floor_add
tff(fact_1956_root__sgn__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y)),Nb))) = Y ) ) ).

% root_sgn_power
tff(fact_1957_sgn__power__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),aa(real,real,root(Nb),Xb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),aa(real,real,root(Nb),Xb))),Nb)) = Xb ) ) ).

% sgn_power_root
tff(fact_1958_verit__le__mono__div__int,axiom,
    ! [A3: int,B3: int,Nb: 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)),Nb)
       => aa(int,$o,
            aa(int,fun(int,$o),ord_less_eq(int),
              aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,A3,Nb)),
                $ite(modulo_modulo(int,B3,Nb) = zero_zero(int),one_one(int),zero_zero(int)))),
            divide_divide(int,B3,Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_1959_sgn__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( aa(A,A,sgn_sgn(A),one_one(A)) = one_one(A) ) ) ).

% sgn_one
tff(fact_1960_sgn__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_zero
tff(fact_1961_Real__Vector__Spaces_Osgn__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] : ( aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),Xb)) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),Xb)) ) ) ).

% Real_Vector_Spaces.sgn_minus
tff(fact_1962_sgn__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( ( aa(A,A,sgn_sgn(A),Xb) = zero_zero(A) )
        <=> ( Xb = zero_zero(A) ) ) ) ).

% sgn_zero_iff
tff(fact_1963_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cosh(A,Xb) != zero_zero(A) )
         => ( ( cosh(A,Y) != zero_zero(A) )
           => ( aa(A,A,tanh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tanh(A),Xb)),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),Xb)),aa(A,A,tanh(A),Y)))) ) ) ) ) ).

% tanh_add
tff(fact_1964_verit__le__mono__div,axiom,
    ! [A3: nat,B3: nat,Nb: 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)),Nb)
       => aa(nat,$o,
            aa(nat,fun(nat,$o),ord_less_eq(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,A3,Nb)),
                $ite(modulo_modulo(nat,B3,Nb) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            divide_divide(nat,B3,Nb)) ) ) ).

% verit_le_mono_div
tff(fact_1965_Cauchy__iff2,axiom,
    ! [X6: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X6)
    <=> ! [J3: nat] :
        ? [M6: nat] :
        ! [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M6),M3)
         => ! [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M6),N4)
             => 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,X6,M3)),aa(nat,real,X6,N4)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_1966_mod__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
     => ( modulo_modulo(nat,Ma,Nb) = Ma ) ) ).

% mod_less
tff(fact_1967_cosh__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : ( cosh(A,aa(A,A,uminus_uminus(A),Xb)) = cosh(A,Xb) ) ) ).

% cosh_minus
tff(fact_1968_cosh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cosh(A,zero_zero(A)) = one_one(A) ) ) ).

% cosh_0
tff(fact_1969_mod__by__Suc__0,axiom,
    ! [Ma: nat] : ( modulo_modulo(nat,Ma,aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ) ).

% mod_by_Suc_0
tff(fact_1970_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K: nat,Ma: 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),Nb),K)),Ma)),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Ma),Nb) ) ).

% Suc_mod_mult_self4
tff(fact_1971_Suc__mod__mult__self3,axiom,
    ! [K: nat,Nb: nat,Ma: 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),Nb)),Ma)),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Ma),Nb) ) ).

% Suc_mod_mult_self3
tff(fact_1972_Suc__mod__mult__self2,axiom,
    ! [Ma: nat,Nb: nat,K: nat] : ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Ma),Nb) ) ).

% Suc_mod_mult_self2
tff(fact_1973_Suc__mod__mult__self1,axiom,
    ! [Ma: nat,K: nat,Nb: nat] : ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Ma),Nb) ) ).

% Suc_mod_mult_self1
tff(fact_1974_mod__Suc__Suc__eq,axiom,
    ! [Ma: nat,Nb: nat] : ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,modulo_modulo(nat,Ma,Nb))),Nb) = modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,Ma)),Nb) ) ).

% mod_Suc_Suc_eq
tff(fact_1975_mod__Suc__eq,axiom,
    ! [Ma: nat,Nb: nat] : ( modulo_modulo(nat,aa(nat,nat,suc,modulo_modulo(nat,Ma,Nb)),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Ma),Nb) ) ).

% mod_Suc_eq
tff(fact_1976_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L2: int] : ( K != aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L2)),aa(nat,int,semiring_1_of_nat(int),N)) ) ).

% int_sgnE
tff(fact_1977_mod__Suc,axiom,
    ! [Ma: nat,Nb: nat] :
      ( modulo_modulo(nat,aa(nat,nat,suc,Ma),Nb) = $ite(aa(nat,nat,suc,modulo_modulo(nat,Ma,Nb)) = Nb,zero_zero(nat),aa(nat,nat,suc,modulo_modulo(nat,Ma,Nb))) ) ).

% mod_Suc
tff(fact_1978_mod__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat,P3: nat,Ma: nat] :
      ( aa(nat,$o,P,Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),P3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),P3)
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),P3)
               => ( aa(nat,$o,P,N)
                 => aa(nat,$o,P,modulo_modulo(nat,aa(nat,nat,suc,N),P3)) ) )
           => aa(nat,$o,P,Ma) ) ) ) ) ).

% mod_induct
tff(fact_1979_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),Ma: nat,Nb: nat] :
      ( ! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,M),zero_zero(nat))
     => ( ! [M: nat,N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),P,N),modulo_modulo(nat,M,N))
             => aa(nat,$o,aa(nat,fun(nat,$o),P,M),N) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),P,Ma),Nb) ) ) ).

% gcd_nat_induct
tff(fact_1980_mod__less__divisor,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),modulo_modulo(nat,Ma,Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_1981_mod__Suc__le__divisor,axiom,
    ! [Ma: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,Ma,aa(nat,nat,suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_1982_mod__eq__0D,axiom,
    ! [Ma: nat,D3: nat] :
      ( ( modulo_modulo(nat,Ma,D3) = zero_zero(nat) )
     => ? [Q2: nat] : ( Ma = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D3),Q2) ) ) ).

% mod_eq_0D
tff(fact_1983_mod__geq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
     => ( modulo_modulo(nat,Ma,Nb) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb),Nb) ) ) ).

% mod_geq
tff(fact_1984_mod__if,axiom,
    ! [Ma: nat,Nb: nat] :
      ( modulo_modulo(nat,Ma,Nb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb),Ma,modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb),Nb)) ) ).

% mod_if
tff(fact_1985_le__mod__geq,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
     => ( modulo_modulo(nat,Ma,Nb) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb),Nb) ) ) ).

% le_mod_geq
tff(fact_1986_cosh__real__ge__1,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),cosh(real,Xb)) ).

% cosh_real_ge_1
tff(fact_1987_nat__mod__eq__iff,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( modulo_modulo(nat,Xb,Nb) = modulo_modulo(nat,Y,Nb) )
    <=> ? [Q1: nat,Q22: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q22)) ) ) ).

% nat_mod_eq_iff
tff(fact_1988_zmod__int,axiom,
    ! [A2: nat,B2: nat] : ( aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,A2,B2)) = 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_1989_mod__le__divisor,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,Ma,Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_1990_div__less__mono,axiom,
    ! [A3: nat,B3: nat,Nb: 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)),Nb)
       => ( ( modulo_modulo(nat,A3,Nb) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B3,Nb) = zero_zero(nat) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,A3,Nb)),divide_divide(nat,B3,Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_1991_mod__eq__nat1E,axiom,
    ! [Ma: nat,Q4: nat,Nb: nat] :
      ( ( modulo_modulo(nat,Ma,Q4) = modulo_modulo(nat,Nb,Q4) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
       => ~ ! [S2: nat] : ( Ma != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q4),S2)) ) ) ) ).

% mod_eq_nat1E
tff(fact_1992_mod__eq__nat2E,axiom,
    ! [Ma: nat,Q4: nat,Nb: nat] :
      ( ( modulo_modulo(nat,Ma,Q4) = modulo_modulo(nat,Nb,Q4) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
       => ~ ! [S2: nat] : ( Nb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q4),S2)) ) ) ) ).

% mod_eq_nat2E
tff(fact_1993_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( modulo_modulo(nat,Xb,Nb) = modulo_modulo(nat,Y,Nb) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xb)
       => ? [Q2: nat] : ( Xb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)) ) ) ) ).

% nat_mod_eq_lemma
tff(fact_1994_mod__mult2__eq,axiom,
    ! [Ma: nat,Nb: nat,Q4: nat] : ( modulo_modulo(nat,Ma,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),modulo_modulo(nat,divide_divide(nat,Ma,Nb),Q4))),modulo_modulo(nat,Ma,Nb)) ) ).

% mod_mult2_eq
tff(fact_1995_div__mod__decomp,axiom,
    ! [A3: nat,Nb: nat] : ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,A3,Nb)),Nb)),modulo_modulo(nat,A3,Nb)) ) ).

% div_mod_decomp
tff(fact_1996_modulo__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : ( modulo_modulo(nat,Ma,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,Ma,Nb)),Nb)) ) ).

% modulo_nat_def
tff(fact_1997_zsgn__def,axiom,
    ! [I: int] :
      ( aa(int,int,sgn_sgn(int),I) = $ite(
          I = zero_zero(int),
          zero_zero(int),
          $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),I),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ) ) ).

% zsgn_def
tff(fact_1998_div__sgn__abs__cancel,axiom,
    ! [V2: int,K: int,L: int] :
      ( ( V2 != zero_zero(int) )
     => ( divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V2)),aa(int,int,abs_abs(int),K)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V2)),aa(int,int,abs_abs(int),L))) = divide_divide(int,aa(int,int,abs_abs(int),K),aa(int,int,abs_abs(int),L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_1999_field__char__0__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Ma: nat,Nb: nat] : ( aa(nat,A,semiring_1_of_nat(A),divide_divide(nat,Ma,Nb)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,Ma,Nb))),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ).

% field_char_0_class.of_nat_div
tff(fact_2000_split__mod,axiom,
    ! [P: fun(nat,$o),Ma: nat,Nb: nat] :
      ( aa(nat,$o,P,modulo_modulo(nat,Ma,Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(nat,$o,P,Ma) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Nb)
             => ( ( Ma = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),I3)),J3) )
               => aa(nat,$o,P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_2001_real__of__nat__div__aux,axiom,
    ! [Xb: nat,D3: nat] : ( divide_divide(real,aa(nat,real,semiring_1_of_nat(real),Xb),aa(nat,real,semiring_1_of_nat(real),D3)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Xb,D3))),divide_divide(real,aa(nat,real,semiring_1_of_nat(real),modulo_modulo(nat,Xb,D3)),aa(nat,real,semiring_1_of_nat(real),D3))) ) ).

% real_of_nat_div_aux
tff(fact_2002_nat__mod__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => ( aa(int,nat,nat2,modulo_modulo(int,Xb,Y)) = modulo_modulo(nat,aa(int,nat,nat2,Xb),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_2003_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: 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),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_2004_Suc__times__mod__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Ma)
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb)),Ma) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_2005_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))),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_2006_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),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_2007_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A,W: A,Ma: nat] :
          ( aa(real,$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),Ma)),aa(nat,A,aa(A,fun(nat,A),power_power(A),W),Ma)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Ma)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),W)))) ) ) ) ).

% norm_power_diff
tff(fact_2008_count__notin,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(A,nat,count_list(A,Xs),Xb) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_2009_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [J3: nat] :
            ? [M6: nat] :
            ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M6),M3)
             => ! [N4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M6),N4)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M3),aa(nat,A,X6,N4))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_2010_powr__real__of__int_H,axiom,
    ! [Xb: real,Nb: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( ( ( Xb != zero_zero(real) )
          | aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb) )
       => ( powr(real,Xb,aa(int,real,ring_1_of_int(real),Nb)) = power_int(real,Xb,Nb) ) ) ) ).

% powr_real_of_int'
tff(fact_2011_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: A,Ma: int] : aa(A,$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),Ma)))) ) ).

% round_diff_minimal
tff(fact_2012_numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Ma: num,Nb: num] :
          ( ( aa(num,A,numeral_numeral(A),Ma) = aa(num,A,numeral_numeral(A),Nb) )
        <=> ( Ma = Nb ) ) ) ).

% numeral_eq_iff
tff(fact_2013_int__eq__iff__numeral,axiom,
    ! [Ma: nat,V2: num] :
      ( ( aa(nat,int,semiring_1_of_nat(int),Ma) = aa(num,int,numeral_numeral(int),V2) )
    <=> ( Ma = aa(num,nat,numeral_numeral(nat),V2) ) ) ).

% int_eq_iff_numeral
tff(fact_2014_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_2015_power__int__mult__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Ma: num,Nb: num] : ( power_int(A,power_int(A,Xb,aa(num,int,numeral_numeral(int),Ma)),aa(num,int,numeral_numeral(int),Nb)) = power_int(A,Xb,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),times_times(num),Ma),Nb))) ) ) ).

% power_int_mult_numeral
tff(fact_2016_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Ma: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Ma)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Ma),Nb) ) ) ).

% numeral_le_iff
tff(fact_2017_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Ma: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Ma)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Ma),Nb) ) ) ).

% numeral_less_iff
tff(fact_2018_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Ma: num,Nb: num] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),Ma)),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),Ma),Nb)) ) ) ).

% numeral_times_numeral
tff(fact_2019_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [V2: num,W: num,Z: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W))),Z) ) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_2020_add__numeral__left,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [V2: num,W: num,Z: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),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),V2),W))),Z) ) ) ).

% add_numeral_left
tff(fact_2021_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Ma: num,Nb: num] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Ma)),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),Nb)) ) ) ).

% numeral_plus_numeral
tff(fact_2022_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_2023_neg__numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Ma: num,Nb: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) )
        <=> ( Ma = Nb ) ) ) ).

% neg_numeral_eq_iff
tff(fact_2024_of__nat__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: num] : ( aa(nat,A,semiring_1_of_nat(A),aa(num,nat,numeral_numeral(nat),Nb)) = aa(num,A,numeral_numeral(A),Nb) ) ) ).

% of_nat_numeral
tff(fact_2025_abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : ( aa(A,A,abs_abs(A),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),Nb) ) ) ).

% abs_numeral
tff(fact_2026_of__int__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int,Nb: num] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = aa(num,A,numeral_numeral(A),Nb) )
        <=> ( Z = aa(num,int,numeral_numeral(int),Nb) ) ) ) ).

% of_int_eq_numeral_iff
tff(fact_2027_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_2028_norm__minus__cancel,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] : ( real_V7770717601297561774m_norm(A,aa(A,A,uminus_uminus(A),Xb)) = real_V7770717601297561774m_norm(A,Xb) ) ) ).

% norm_minus_cancel
tff(fact_2029_floor__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num] : ( aa(A,int,archim6421214686448440834_floor(A),aa(num,A,numeral_numeral(A),V2)) = aa(num,int,numeral_numeral(int),V2) ) ) ).

% floor_numeral
tff(fact_2030_power__int__add__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Ma: num,Nb: num] : ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,aa(num,int,numeral_numeral(int),Ma))),power_int(A,Xb,aa(num,int,numeral_numeral(int),Nb))) = power_int(A,Xb,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),Nb))) ) ) ).

% power_int_add_numeral
tff(fact_2031_power__int__add__numeral2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Ma: num,Nb: num,B2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,aa(num,int,numeral_numeral(int),Ma))),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,aa(num,int,numeral_numeral(int),Nb))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),Nb)))),B2) ) ) ).

% power_int_add_numeral2
tff(fact_2032_power__int__1__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Nb: int] : ( power_int(A,one_one(A),Nb) = one_one(A) ) ) ).

% power_int_1_left
tff(fact_2033_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_2034_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_2035_power__int__numeral,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [Xb: A,Nb: num] : ( power_int(A,Xb,aa(num,int,numeral_numeral(int),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),Nb)) ) ) ).

% power_int_numeral
tff(fact_2036_ceiling__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num] : ( archimedean_ceiling(A,aa(num,A,numeral_numeral(A),V2)) = aa(num,int,numeral_numeral(int),V2) ) ) ).

% ceiling_numeral
tff(fact_2037_power__int__1__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & monoid_mult(A) )
     => ! [Y: A] : ( power_int(A,Y,one_one(int)) = Y ) ) ).

% power_int_1_right
tff(fact_2038_power__int__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,Nb: int] : ( aa(A,A,sgn_sgn(A),power_int(A,A2,Nb)) = power_int(A,aa(A,A,sgn_sgn(A),A2),Nb) ) ) ).

% power_int_sgn
tff(fact_2039_round__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: num] : ( archimedean_round(A,aa(num,A,numeral_numeral(A),Nb)) = aa(num,int,numeral_numeral(int),Nb) ) ) ).

% round_numeral
tff(fact_2040_round__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: int] : ( archimedean_round(A,aa(int,A,ring_1_of_int(A),Nb)) = Nb ) ) ).

% round_of_int
tff(fact_2041_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Nb),Ma) ) ) ).

% neg_numeral_le_iff
tff(fact_2042_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A2: A,B2: A,V2: 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),V2)) = 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),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% distrib_right_numeral
tff(fact_2043_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V2: num,B2: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),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),V2)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),C2)) ) ) ).

% distrib_left_numeral
tff(fact_2044_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Nb),Ma) ) ) ).

% neg_numeral_less_iff
tff(fact_2045_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V2: num,B2: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),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),V2)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),C2)) ) ) ).

% right_diff_distrib_numeral
tff(fact_2046_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A2: A,B2: A,V2: 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),V2)) = 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),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% left_diff_distrib_numeral
tff(fact_2047_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Ma: num,Nb: num] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),Ma),Nb)) ) ) ).

% mult_neg_numeral_simps(1)
tff(fact_2048_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Ma: num,Nb: num] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),aa(num,A,numeral_numeral(A),Nb)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),Ma),Nb))) ) ) ).

% mult_neg_numeral_simps(2)
tff(fact_2049_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Ma: num,Nb: num] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),Ma)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),Ma),Nb))) ) ) ).

% mult_neg_numeral_simps(3)
tff(fact_2050_semiring__norm_I172_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W: num,Y: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W))),Y) ) ) ).

% semiring_norm(172)
tff(fact_2051_semiring__norm_I171_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W: num,Y: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W)))),Y) ) ) ).

% semiring_norm(171)
tff(fact_2052_semiring__norm_I170_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W: num,Y: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W)))),Y) ) ) ).

% semiring_norm(170)
tff(fact_2053_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Ma: num,Nb: num] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Ma)),aa(num,A,numeral_numeral(A),Nb))) ) ) ).

% add_neg_numeral_simps(3)
tff(fact_2054_semiring__norm_I168_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V2: 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),V2))),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),V2),W)))),Y) ) ) ).

% semiring_norm(168)
tff(fact_2055_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Ma: num,Nb: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),aa(num,A,numeral_numeral(A),Nb)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),Nb))) ) ) ).

% diff_numeral_simps(3)
tff(fact_2056_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Ma: num,Nb: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),Ma)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),Nb)) ) ) ).

% diff_numeral_simps(2)
tff(fact_2057_abs__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : ( aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(num,A,numeral_numeral(A),Nb) ) ) ).

% abs_neg_numeral
tff(fact_2058_norm__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( real_V7770717601297561774m_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).

% norm_zero
tff(fact_2059_norm__eq__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( ( real_V7770717601297561774m_norm(A,Xb) = zero_zero(real) )
        <=> ( Xb = zero_zero(A) ) ) ) ).

% norm_eq_zero
tff(fact_2060_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_2061_norm__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( real_V7770717601297561774m_norm(A,one_one(A)) = one_one(real) ) ) ).

% norm_one
tff(fact_2062_power__int__mult__distrib__numeral2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,W: num,Ma: int] : ( power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(num,A,numeral_numeral(A),W)),Ma) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,Ma)),power_int(A,aa(num,A,numeral_numeral(A),W),Ma)) ) ) ).

% power_int_mult_distrib_numeral2
tff(fact_2063_power__int__mult__distrib__numeral1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [W: num,Y: A,Ma: int] : ( power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Y),Ma) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(num,A,numeral_numeral(A),W),Ma)),power_int(A,Y,Ma)) ) ) ).

% power_int_mult_distrib_numeral1
tff(fact_2064_power__int__0__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Ma: int] :
          ( ( Ma != zero_zero(int) )
         => ( power_int(A,zero_zero(A),Ma) = zero_zero(A) ) ) ) ).

% power_int_0_left
tff(fact_2065_power__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Nb: int] :
          ( ( power_int(A,Xb,Nb) = zero_zero(A) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Nb != zero_zero(int) ) ) ) ) ).

% power_int_eq_0_iff
tff(fact_2066_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Nb: 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),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),W)),Nb) ) ).

% numeral_less_real_of_nat_iff
tff(fact_2067_real__of__nat__less__numeral__iff,axiom,
    ! [Nb: nat,W: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(num,real,numeral_numeral(real),W))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(num,nat,numeral_numeral(nat),W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_2068_power__int__0__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [Xb: A] : ( power_int(A,Xb,zero_zero(int)) = one_one(A) ) ) ).

% power_int_0_right
tff(fact_2069_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,Ma: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),Nb)),aa(nat,real,semiring_1_of_nat(real),Ma))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Nb)),Ma) ) ).

% numeral_le_real_of_nat_iff
tff(fact_2070_dist__0__norm,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] : ( real_V557655796197034286t_dist(A,zero_zero(A),Xb) = real_V7770717601297561774m_norm(A,Xb) ) ) ).

% dist_0_norm
tff(fact_2071_abs__power__int__minus,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,Nb: int] : ( aa(A,A,abs_abs(A),power_int(A,aa(A,A,uminus_uminus(A),A2),Nb)) = aa(A,A,abs_abs(A),power_int(A,A2,Nb)) ) ) ).

% abs_power_int_minus
tff(fact_2072_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_2073_norm__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Nb: nat] : ( real_V7770717601297561774m_norm(A,aa(nat,A,semiring_1_of_nat(A),Nb)) = aa(nat,real,semiring_1_of_nat(real),Nb) ) ) ).

% norm_of_nat
tff(fact_2074_dist__diff_I2_J,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : ( real_V557655796197034286t_dist(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),A2) = real_V7770717601297561774m_norm(A,B2) ) ) ).

% dist_diff(2)
tff(fact_2075_dist__diff_I1_J,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : ( real_V557655796197034286t_dist(A,A2,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = real_V7770717601297561774m_norm(A,B2) ) ) ).

% dist_diff(1)
tff(fact_2076_power__int__of__nat,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [Xb: A,Nb: nat] : ( power_int(A,Xb,aa(nat,int,semiring_1_of_nat(int),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb) ) ) ).

% power_int_of_nat
tff(fact_2077_diff__nat__numeral,axiom,
    ! [V2: num,V3: num] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),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),V2)),aa(num,int,numeral_numeral(int),V3))) ) ).

% diff_nat_numeral
tff(fact_2078_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb) = aa(int,nat,nat2,Y) )
    <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_2079_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa(int,nat,nat2,Y) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb) )
    <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_2080_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_2081_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_2082_round__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: nat] : ( archimedean_round(A,aa(nat,A,semiring_1_of_nat(A),Nb)) = aa(nat,int,semiring_1_of_nat(int),Nb) ) ) ).

% round_of_nat
tff(fact_2083_floor__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : ( aa(real,int,archim6421214686448440834_floor(real),divide_divide(real,aa(num,real,numeral_numeral(real),A2),aa(num,real,numeral_numeral(real),B2))) = divide_divide(int,aa(num,int,numeral_numeral(int),A2),aa(num,int,numeral_numeral(int),B2)) ) ).

% floor_divide_eq_div_numeral
tff(fact_2084_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,$o),ord_less_eq(A),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W))),A2)
        <=> aa(A,$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_2085_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,$o),ord_less_eq(A),A2),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$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_2086_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W: num] :
          ( ( A2 = 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_2087_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A2: 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_2088_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,$o),ord_less(A),A2),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$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_2089_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,$o),ord_less(A),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W))),A2)
        <=> aa(A,$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_2090_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)) = divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),W)) ) ) ).

% inverse_eq_divide_numeral
tff(fact_2091_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V7770717601297561774m_norm(A,Xb))
        <=> ( Xb != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_2092_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Xb)),zero_zero(real))
        <=> ( Xb = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_2093_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Nb)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Nb)),Z) ) ) ).

% of_int_numeral_le_iff
tff(fact_2094_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),aa(num,int,numeral_numeral(int),Nb)) ) ) ).

% of_int_le_numeral_iff
tff(fact_2095_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(num,int,numeral_numeral(int),Nb)) ) ) ).

% of_int_less_numeral_iff
tff(fact_2096_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Nb)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),Nb)),Z) ) ) ).

% of_int_numeral_less_iff
tff(fact_2097_power__int__minus__one__mult__self_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Ma: int,B2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),Ma)),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),Ma)),B2)) = B2 ) ) ).

% power_int_minus_one_mult_self'
tff(fact_2098_power__int__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Ma: int] : ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),Ma)),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),Ma)) = one_one(A) ) ) ).

% power_int_minus_one_mult_self
tff(fact_2099_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Y: nat,Xb: num,Nb: 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),Xb)),Nb) )
        <=> ( Y = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb) ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_2100_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Xb: num,Nb: nat,Y: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb) = 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),Xb)),Nb) = Y ) ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_2101_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),aa(A,int,archim6421214686448440834_floor(A),Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),V2)),Xb) ) ) ).

% numeral_le_floor
tff(fact_2102_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(num,A,numeral_numeral(A),V2)) ) ) ).

% floor_less_numeral
tff(fact_2103_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(num,A,numeral_numeral(A),V2)) ) ) ).

% ceiling_le_numeral
tff(fact_2104_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),V2)),Xb) ) ) ).

% numeral_less_ceiling
tff(fact_2105_floor__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num] : ( aa(A,int,archim6421214686448440834_floor(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)) ) ) ).

% floor_neg_numeral
tff(fact_2106_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] : ( archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V2)) ) ) ).

% ceiling_add_numeral
tff(fact_2107_floor__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] : ( aa(A,int,archim6421214686448440834_floor(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),aa(num,int,numeral_numeral(int),V2)) ) ) ).

% floor_diff_numeral
tff(fact_2108_ceiling__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num] : ( archimedean_ceiling(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)) ) ) ).

% ceiling_neg_numeral
tff(fact_2109_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: num,Nb: nat,Y: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb) = 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),Xb)),Nb) = Y ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_2110_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,Xb: num,Nb: 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),Xb)),Nb) )
        <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_2111_ceiling__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] : ( archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V2)) ) ) ).

% ceiling_diff_numeral
tff(fact_2112_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) != one_one(nat) )
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),K)),Nb)),aa(num,nat,numeral_numeral(nat),K)) = one_one(nat) ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_2113_floor__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: num,Nb: nat] : ( aa(A,int,archim6421214686448440834_floor(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) ) ) ).

% floor_numeral_power
tff(fact_2114_ceiling__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: num,Nb: nat] : ( archimedean_ceiling(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) ) ) ).

% ceiling_numeral_power
tff(fact_2115_power__int__minus1__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & monoid_mult(A) )
     => ! [Y: A] : ( power_int(A,Y,aa(int,int,uminus_uminus(int),one_one(int))) = aa(A,A,inverse_inverse(A),Y) ) ) ).

% power_int_minus1_right
tff(fact_2116_round__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: num] : ( archimedean_round(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb)) ) ) ).

% round_neg_numeral
tff(fact_2117_norm__of__int,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Z: int] : ( real_V7770717601297561774m_norm(A,aa(int,A,ring_1_of_int(A),Z)) = aa(real,real,abs_abs(real),aa(int,real,ring_1_of_int(real),Z)) ) ) ).

% norm_of_int
tff(fact_2118_ceiling__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : ( archimedean_ceiling(real,divide_divide(real,aa(num,real,numeral_numeral(real),A2),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,uminus_uminus(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_2119_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,$o),ord_less_eq(A),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A2)
        <=> aa(A,$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_2120_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,$o),ord_less_eq(A),A2),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$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_2121_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A2: 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_2122_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W: num] :
          ( ( A2 = 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_2123_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,$o),ord_less(A),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A2)
        <=> aa(A,$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_2124_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,$o),ord_less(A),A2),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$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_2125_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_2126_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_2127_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))) = 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_2128_power__int__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,Nb: int] :
          ( aa(A,$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,$o),ord_less(int),zero_zero(int)),Nb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A2,Nb)),power_int(A,B2,Nb))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ) ).

% power_int_mono_iff
tff(fact_2129_nat__numeral__diff__1,axiom,
    ! [V2: num] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V2)),one_one(int))) ) ).

% nat_numeral_diff_1
tff(fact_2130_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: 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),Xb)),Nb)),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),Xb)),Nb)),A2) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_2131_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,Nb: 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),Xb)),Nb))
    <=> 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),Xb)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_2132_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,Nb: 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),Xb)),Nb))
    <=> 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),Xb)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_2133_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: 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),Xb)),Nb)),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),Xb)),Nb)),A2) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_2134_floor__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : ( aa(real,int,archim6421214686448440834_floor(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),B2))) = divide_divide(int,one_one(int),aa(num,int,numeral_numeral(int),B2)) ) ).

% floor_one_divide_eq_div_numeral
tff(fact_2135_floor__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : ( aa(real,int,archim6421214686448440834_floor(real),aa(real,real,uminus_uminus(real),divide_divide(real,aa(num,real,numeral_numeral(real),A2),aa(num,real,numeral_numeral(real),B2)))) = 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_2136_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : ( archimedean_ceiling(real,aa(real,real,uminus_uminus(real),divide_divide(real,aa(num,real,numeral_numeral(real),A2),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,uminus_uminus(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_2137_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,I: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),Nb)) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_2138_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Nb: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$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)),Nb)),Xb) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_2139_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Nb: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$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)),Nb)),Xb) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_2140_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,I: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),Nb)) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_2141_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V2)),aa(A,int,archim6421214686448440834_floor(A),Xb))
        <=> aa(A,$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),V2)),one_one(A))),Xb) ) ) ).

% numeral_less_floor
tff(fact_2142_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A))) ) ) ).

% floor_le_numeral
tff(fact_2143_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A))) ) ) ).

% ceiling_less_numeral
tff(fact_2144_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,Xb))
        <=> aa(A,$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),V2)),one_one(A))),Xb) ) ) ).

% numeral_le_ceiling
tff(fact_2145_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),aa(A,int,archim6421214686448440834_floor(A),Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),Xb) ) ) ).

% neg_numeral_le_floor
tff(fact_2146_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% floor_less_neg_numeral
tff(fact_2147_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_2148_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb))
        <=> aa(int,$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),Xb)),Nb)) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_2149_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$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),Xb)),Nb)),A2) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_2150_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),Xb) ) ) ).

% neg_numeral_less_ceiling
tff(fact_2151_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb))
        <=> aa(int,$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),Xb)),Nb)) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_2152_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$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),Xb)),Nb)),A2) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_2153_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: num,Nb: 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),Xb))),Nb) = 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),Xb))),Nb) = Y ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_2154_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,Xb: num,Nb: 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),Xb))),Nb) )
        <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),Nb) ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_2155_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : ( aa(real,int,archim6421214686448440834_floor(real),aa(real,real,uminus_uminus(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),B2)))) = 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_2156_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),aa(A,int,archim6421214686448440834_floor(A),Xb))
        <=> aa(A,$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),V2))),one_one(A))),Xb) ) ) ).

% neg_numeral_less_floor
tff(fact_2157_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))) ) ) ).

% floor_le_neg_numeral
tff(fact_2158_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_2159_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,Xb))
        <=> aa(A,$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),V2))),one_one(A))),Xb) ) ) ).

% neg_numeral_le_ceiling
tff(fact_2160_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$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),Xb))),Nb)),A2) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_2161_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb))
        <=> aa(int,$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),Xb))),Nb)) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_2162_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb))
        <=> aa(int,$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),Xb))),Nb)) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_2163_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$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),Xb))),Nb)),A2) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_2164_dist__norm,axiom,
    ! [A: $tType] :
      ( real_V6936659425649961206t_norm(A)
     => ! [Xb: A,Y: A] : ( real_V557655796197034286t_dist(A,Xb,Y) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) ) ) ).

% dist_norm
tff(fact_2165_norm__conv__dist,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] : ( real_V7770717601297561774m_norm(A,Xb) = real_V557655796197034286t_dist(A,Xb,zero_zero(A)) ) ) ).

% norm_conv_dist
tff(fact_2166_int__ops_I3_J,axiom,
    ! [Nb: num] : ( aa(nat,int,semiring_1_of_nat(int),aa(num,nat,numeral_numeral(nat),Nb)) = aa(num,int,numeral_numeral(int),Nb) ) ).

% int_ops(3)
tff(fact_2167_nat__numeral__as__int,axiom,
    ! [X3: num] : ( aa(num,nat,numeral_numeral(nat),X3) = aa(int,nat,nat2,aa(num,int,numeral_numeral(int),X3)) ) ).

% nat_numeral_as_int
tff(fact_2168_power__int__commutes,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Nb: int] : ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,Nb)),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),power_int(A,Xb,Nb)) ) ) ).

% power_int_commutes
tff(fact_2169_power__int__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Y: A,Ma: int] : ( power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y),Ma) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,Ma)),power_int(A,Y,Ma)) ) ) ).

% power_int_mult_distrib
tff(fact_2170_power__int__divide__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Y: A,Ma: int] : ( power_int(A,divide_divide(A,Xb,Y),Ma) = divide_divide(A,power_int(A,Xb,Ma),power_int(A,Y,Ma)) ) ) ).

% power_int_divide_distrib
tff(fact_2171_power__int__abs,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,Nb: int] : ( aa(A,A,abs_abs(A),power_int(A,A2,Nb)) = power_int(A,aa(A,A,abs_abs(A),A2),Nb) ) ) ).

% power_int_abs
tff(fact_2172_power__int__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Nb: int] : ( power_int(A,aa(A,A,inverse_inverse(A),Xb),Nb) = aa(A,A,inverse_inverse(A),power_int(A,Xb,Nb)) ) ) ).

% power_int_inverse
tff(fact_2173_power__int__mult,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Ma: int,Nb: int] : ( power_int(A,Xb,aa(int,int,aa(int,fun(int,int),times_times(int),Ma),Nb)) = power_int(A,power_int(A,Xb,Ma),Nb) ) ) ).

% power_int_mult
tff(fact_2174_norm__minus__commute,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : ( real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) ) ) ).

% norm_minus_commute
tff(fact_2175_zero__neq__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: num] : ( zero_zero(A) != aa(num,A,numeral_numeral(A),Nb) ) ) ).

% zero_neq_numeral
tff(fact_2176_neg__numeral__neq__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Ma: num,Nb: num] : ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma)) != aa(num,A,numeral_numeral(A),Nb) ) ) ).

% neg_numeral_neq_numeral
tff(fact_2177_numeral__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Ma: num,Nb: num] : ( aa(num,A,numeral_numeral(A),Ma) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) ) ) ).

% numeral_neq_neg_numeral
tff(fact_2178_Ints__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: num] : member(A,aa(num,A,numeral_numeral(A),Nb),ring_1_Ints(A)) ) ).

% Ints_numeral
tff(fact_2179_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_2180_zero__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),power_int(A,Xb,Nb)) ) ) ).

% zero_le_power_int
tff(fact_2181_zero__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),power_int(A,Xb,Nb)) ) ) ).

% zero_less_power_int
tff(fact_2182_power__int__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Nb: int] : ( power_int(A,divide_divide(A,one_one(A),Xb),Nb) = divide_divide(A,one_one(A),power_int(A,Xb,Nb)) ) ) ).

% power_int_one_over
tff(fact_2183_power__int__not__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Nb: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( Nb = zero_zero(int) ) )
         => ( power_int(A,Xb,Nb) != zero_zero(A) ) ) ) ).

% power_int_not_zero
tff(fact_2184_power__int__minus,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Nb: int] : ( power_int(A,Xb,aa(int,int,uminus_uminus(int),Nb)) = aa(A,A,inverse_inverse(A),power_int(A,Xb,Nb)) ) ) ).

% power_int_minus
tff(fact_2185_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% zero_le_numeral
tff(fact_2186_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Nb)),zero_zero(A)) ) ).

% not_numeral_le_zero
tff(fact_2187_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% zero_less_numeral
tff(fact_2188_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Nb)),zero_zero(A)) ) ).

% not_numeral_less_zero
tff(fact_2189_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% one_le_numeral
tff(fact_2190_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Nb)),one_one(A)) ) ).

% not_numeral_less_one
tff(fact_2191_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num,Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),aa(num,A,numeral_numeral(A),Nb)) ) ).

% neg_numeral_le_numeral
tff(fact_2192_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num,Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Ma)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_numeral_le_neg_numeral
tff(fact_2193_zero__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] : ( zero_zero(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) ) ) ).

% zero_neq_neg_numeral
tff(fact_2194_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num,Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),aa(num,A,numeral_numeral(A),Nb)) ) ).

% neg_numeral_less_numeral
tff(fact_2195_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num,Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Ma)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_numeral_less_neg_numeral
tff(fact_2196_one__plus__numeral__commute,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Xb: num] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),Xb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Xb)),one_one(A)) ) ) ).

% one_plus_numeral_commute
tff(fact_2197_numeral__times__minus__swap,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [W: num,Xb: 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),Xb)) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ) ).

% numeral_times_minus_swap
tff(fact_2198_numeral__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] : ( aa(num,A,numeral_numeral(A),Nb) != aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% numeral_neq_neg_one
tff(fact_2199_one__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] : ( one_one(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) ) ) ).

% one_neq_neg_numeral
tff(fact_2200_norm__inverse,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: A] : ( real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),A2)) = aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,A2)) ) ) ).

% norm_inverse
tff(fact_2201_power__int__0__left__If,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Ma: int] :
          ( power_int(A,zero_zero(A),Ma) = $ite(Ma = zero_zero(int),one_one(A),zero_zero(A)) ) ) ).

% power_int_0_left_If
tff(fact_2202_power__int__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N5: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),N5)
         => ( aa(A,$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),power_int(A,A2,Nb)),power_int(A,A2,N5)) ) ) ) ).

% power_int_increasing
tff(fact_2203_power__int__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N5: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),N5)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A2,Nb)),power_int(A,A2,N5)) ) ) ) ).

% power_int_strict_increasing
tff(fact_2204_power__int__minus__one__minus,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Nb: int] : ( power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,uminus_uminus(int),Nb)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),Nb) ) ) ).

% power_int_minus_one_minus
tff(fact_2205_power__int__diff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Ma: int,Nb: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( Ma != Nb ) )
         => ( power_int(A,Xb,aa(int,int,aa(int,fun(int,int),minus_minus(int),Ma),Nb)) = divide_divide(A,power_int(A,Xb,Ma),power_int(A,Xb,Nb)) ) ) ) ).

% power_int_diff
tff(fact_2206_power__int__minus__one__diff__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: int,B2: int] : ( power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,aa(int,fun(int,int),minus_minus(int),A2),B2)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),A2)) ) ) ).

% power_int_minus_one_diff_commute
tff(fact_2207_norm__uminus__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A] : ( real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),Xb)),Y)) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) ) ) ).

% norm_uminus_minus
tff(fact_2208_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,divide_divide(A,A2,B2)) = divide_divide(real,real_V7770717601297561774m_norm(A,A2),real_V7770717601297561774m_norm(A,B2)) ) ) ) ).

% nonzero_norm_divide
tff(fact_2209_power__eq__imp__eq__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,Nb: nat,Z: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => ( real_V7770717601297561774m_norm(A,W) = real_V7770717601297561774m_norm(A,Z) ) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_2210_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),E2) ) ) ).

% norm_triangle_lt
tff(fact_2211_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,R2: real,Y: A,S: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),S)) ) ) ) ).

% norm_add_less
tff(fact_2212_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_2213_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),E2) ) ) ).

% norm_triangle_le
tff(fact_2214_norm__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A] : aa(real,$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),Xb),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_triangle_ineq
tff(fact_2215_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,R2: real,B2: A,S: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,A2)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),S)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),S)) ) ) ) ).

% norm_triangle_mono
tff(fact_2216_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y))),E1)
         => ( aa(real,$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),Xb),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_less
tff(fact_2217_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,Xb)),archimedean_round(A,Y)) ) ) ).

% round_mono
tff(fact_2218_norm__triangle__sub,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Xb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Y)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)))) ) ).

% norm_triangle_sub
tff(fact_2219_norm__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))) ) ).

% norm_triangle_ineq4
tff(fact_2220_norm__diff__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y))),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z))),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_le
tff(fact_2221_norm__triangle__le__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y))),E2) ) ) ).

% norm_triangle_le_diff
tff(fact_2222_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))),zero_zero(A)) ) ).

% neg_numeral_le_zero
tff(fact_2223_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : ~ aa(A,$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),Nb))) ) ).

% not_zero_le_neg_numeral
tff(fact_2224_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))),zero_zero(A)) ) ).

% neg_numeral_less_zero
tff(fact_2225_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_zero_less_neg_numeral
tff(fact_2226_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_2227_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) = 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_2228_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( 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_2229_norm__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))) ) ).

% norm_triangle_ineq2
tff(fact_2230_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),one_one(A)) ) ).

% neg_numeral_le_one
tff(fact_2231_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num] : aa(A,$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),Ma)) ) ).

% neg_one_le_numeral
tff(fact_2232_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% neg_numeral_le_neg_one
tff(fact_2233_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Ma)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_le_neg_one
tff(fact_2234_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num] : ~ aa(A,$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),Ma))) ) ).

% not_one_le_neg_numeral
tff(fact_2235_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),one_one(A)) ) ).

% neg_numeral_less_one
tff(fact_2236_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),Ma)) ) ).

% neg_one_less_numeral
tff(fact_2237_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Ma)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_less_neg_one
tff(fact_2238_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))) ) ).

% not_one_less_neg_numeral
tff(fact_2239_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num] : ~ aa(A,$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),Ma))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_2240_nonzero__norm__inverse,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),A2)) = aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,A2)) ) ) ) ).

% nonzero_norm_inverse
tff(fact_2241_floor__le__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),archimedean_round(A,Xb)) ) ).

% floor_le_round
tff(fact_2242_ceiling__ge__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,Xb)),archimedean_ceiling(A,Xb)) ) ).

% ceiling_ge_round
tff(fact_2243_powr__neg__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),Nb))) = divide_divide(real,one_one(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),Nb))) ) ) ).

% powr_neg_numeral
tff(fact_2244_power__int__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N5: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),N5)
         => ( aa(A,$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),power_int(A,A2,N5)),power_int(A,A2,Nb)) ) ) ) ) ).

% power_int_strict_decreasing
tff(fact_2245_power__int__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,Xb,Nb)),power_int(A,Y,Nb)) ) ) ) ) ).

% power_int_mono
tff(fact_2246_power__int__strict__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),zero_zero(int))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,B2,Nb)),power_int(A,A2,Nb)) ) ) ) ) ).

% power_int_strict_antimono
tff(fact_2247_one__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),power_int(A,Xb,Nb)) ) ) ) ).

% one_le_power_int
tff(fact_2248_one__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),power_int(A,A2,Nb)) ) ) ) ).

% one_less_power_int
tff(fact_2249_power__int__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Ma: int,Nb: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( aa(int,int,aa(int,fun(int,int),plus_plus(int),Ma),Nb) != zero_zero(int) ) )
         => ( power_int(A,Xb,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ma),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,Ma)),power_int(A,Xb,Nb)) ) ) ) ).

% power_int_add
tff(fact_2250_power__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,Nb: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W),Nb) = one_one(A) )
         => ( ( real_V7770717601297561774m_norm(A,W) = one_one(real) )
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% power_eq_1_iff
tff(fact_2251_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: A,D3: A] : aa(real,$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),D3)))),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),D3)))) ) ).

% norm_diff_triangle_ineq
tff(fact_2252_norm__sgn,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( real_V7770717601297561774m_norm(A,aa(A,A,sgn_sgn(A),Xb)) = $ite(Xb = zero_zero(A),zero_zero(real),one_one(real)) ) ) ).

% norm_sgn
tff(fact_2253_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),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_2254_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)),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_2255_dist__of__int,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Ma: int,Nb: int] : ( real_V557655796197034286t_dist(A,aa(int,A,ring_1_of_int(A),Ma),aa(int,A,ring_1_of_int(A),Nb)) = 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),Ma),Nb))) ) ) ).

% dist_of_int
tff(fact_2256_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( 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_2257_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)) = 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_2258_norm__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))) ) ).

% norm_triangle_ineq3
tff(fact_2259_power__int__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),zero_zero(int))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,B2,Nb)),power_int(A,A2,Nb)) ) ) ) ) ).

% power_int_antimono
tff(fact_2260_power__int__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A2,Nb)),power_int(A,B2,Nb)) ) ) ) ) ).

% power_int_strict_mono
tff(fact_2261_power__int__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N5: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),N5)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
             => ( ( ( A2 != zero_zero(A) )
                  | ( N5 != zero_zero(int) )
                  | ( Nb = zero_zero(int) ) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A2,N5)),power_int(A,A2,Nb)) ) ) ) ) ) ).

% power_int_decreasing
tff(fact_2262_power__int__le__one,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,Xb,Nb)),one_one(A)) ) ) ) ) ).

% power_int_le_one
tff(fact_2263_power__int__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Ma: int,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,Xb,Ma)),power_int(A,Xb,Nb))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ma),Nb) ) ) ) ) ).

% power_int_le_imp_le_exp
tff(fact_2264_power__int__le__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Ma: int,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,Xb,Ma)),power_int(A,Xb,Nb))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ma),Nb) ) ) ) ) ).

% power_int_le_imp_less_exp
tff(fact_2265_power__int__minus__mult,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Nb: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( Nb != zero_zero(int) ) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,aa(int,int,aa(int,fun(int,int),minus_minus(int),Nb),one_one(int)))),Xb) = power_int(A,Xb,Nb) ) ) ) ).

% power_int_minus_mult
tff(fact_2266_power__int__add__1_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Ma: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( Ma != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,Xb,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ma),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),power_int(A,Xb,Ma)) ) ) ) ).

% power_int_add_1'
tff(fact_2267_power__int__add__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Ma: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( Ma != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,Xb,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ma),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,Ma)),Xb) ) ) ) ).

% power_int_add_1
tff(fact_2268_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,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M6: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M6),M3)
                 => ! [N4: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N4)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,F2,M3),aa(nat,A,F2,N4))),E3) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_2269_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [M7: nat] :
                ! [M: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),M)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M),aa(nat,A,X6,N))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% CauchyI'
tff(fact_2270_dist__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Ma: nat,Nb: nat] : ( real_V557655796197034286t_dist(A,aa(nat,A,semiring_1_of_nat(A),Ma),aa(nat,A,semiring_1_of_nat(A),Nb)) = 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),Ma)),aa(nat,int,semiring_1_of_nat(int),Nb)))) ) ) ).

% dist_of_nat
tff(fact_2271_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(A,B)] :
          ( ? [K5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K5)
              & ! [N4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N4))),K5) )
        <=> ? [N6: nat] :
            ! [N4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N4))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6))) ) ) ).

% lemma_NBseq_def
tff(fact_2272_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(A,B)] :
          ( ? [K5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K5)
              & ! [N4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N4))),K5) )
        <=> ? [N6: nat] :
            ! [N4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N4))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6))) ) ) ).

% lemma_NBseq_def2
tff(fact_2273_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)),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_2274_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),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_2275_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),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_2276_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))),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_2277_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R2),real_V7770717601297561774m_norm(A,Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),Xb))),aa(real,real,inverse_inverse(real),R2)) ) ) ) ).

% norm_inverse_le_norm
tff(fact_2278_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M6: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M6),M3)
                 => ! [N4: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M6),N4)
                     => aa(real,$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,X6,M3)),aa(nat,A,X6,N4)))),E3) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_2279_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [M7: nat] :
                ! [M: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),M)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),N)
                     => aa(real,$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,X6,M)),aa(nat,A,X6,N)))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% CauchyI
tff(fact_2280_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),E2: real] :
          ( topolo3814608138187158403Cauchy(A,X6)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
           => ? [M8: nat] :
              ! [M4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M4)
               => ! [N3: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N3)
                   => aa(real,$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,X6,M4)),aa(nat,A,X6,N3)))),E2) ) ) ) ) ) ).

% CauchyD
tff(fact_2281_power__int__def,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [Xb: A,Nb: int] :
          ( power_int(A,Xb,Nb) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(int,nat,nat2,Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xb)),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Nb)))) ) ) ).

% power_int_def
tff(fact_2282_power__int__numeral__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ma: num,Nb: num] : ( power_int(A,aa(num,A,numeral_numeral(A),Ma),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),pow(Ma,Nb))) ) ) ).

% power_int_numeral_neg_numeral
tff(fact_2283_enat__ord__number_I2_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),Ma)),aa(num,extended_enat,numeral_numeral(extended_enat),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),Ma)),aa(num,nat,numeral_numeral(nat),Nb)) ) ).

% enat_ord_number(2)
tff(fact_2284_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K6: real,Nb: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(real,$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),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)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb)),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),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),Nb)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),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),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),real_V7770717601297561774m_norm(A,H))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_2285_complex__mod__minus__le__complex__mod,axiom,
    ! [Xb: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,Xb))),real_V7770717601297561774m_norm(complex,Xb)) ).

% complex_mod_minus_le_complex_mod
tff(fact_2286_norm__of__real__add1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: real] : ( real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,Xb)),one_one(A))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),one_one(real))) ) ) ).

% norm_of_real_add1
tff(fact_2287_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Nb: 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,log2(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),Nb)),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),Nb)),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),Nb),one_one(nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_2288_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Ma: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),Ma)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(Ma)) ) ) ).

% diff_numeral_special(6)
tff(fact_2289_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( bit0(X2) = bit0(Y2) )
    <=> ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_2290_pow__sum,axiom,
    ! [A2: nat,B2: 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_2291_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xb: nat,Nb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Tree),Xb)
     => ( vEBT_invar_vebt(Tree,Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ) ) ).

% member_bound
tff(fact_2292_bit__concat__def,axiom,
    ! [H: nat,L: nat,D3: nat] : ( vEBT_VEBT_bit_concat(H,L,D3) = 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))),D3))),L) ) ).

% bit_concat_def
tff(fact_2293_numeral__eq__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: num] :
          ( ( aa(num,A,numeral_numeral(A),Nb) = one_one(A) )
        <=> ( Nb = one2 ) ) ) ).

% numeral_eq_one_iff
tff(fact_2294_one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),Nb) )
        <=> ( one2 = Nb ) ) ) ).

% one_eq_numeral_iff
tff(fact_2295_of__real__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real] :
          ( ( real_Vector_of_real(A,Xb) = zero_zero(A) )
        <=> ( Xb = zero_zero(real) ) ) ) ).

% of_real_eq_0_iff
tff(fact_2296_of__real__0,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ( real_Vector_of_real(A,zero_zero(real)) = zero_zero(A) ) ) ).

% of_real_0
tff(fact_2297_of__real__1,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ( real_Vector_of_real(A,one_one(real)) = one_one(A) ) ) ).

% of_real_1
tff(fact_2298_of__real__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real] :
          ( ( real_Vector_of_real(A,Xb) = one_one(A) )
        <=> ( Xb = one_one(real) ) ) ) ).

% of_real_eq_1_iff
tff(fact_2299_of__real__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Y: real] : ( real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,Xb)),real_Vector_of_real(A,Y)) ) ) ).

% of_real_add
tff(fact_2300_of__real__minus,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real] : ( real_Vector_of_real(A,aa(real,real,uminus_uminus(real),Xb)) = aa(A,A,uminus_uminus(A),real_Vector_of_real(A,Xb)) ) ) ).

% of_real_minus
tff(fact_2301_minus__of__real__eq__of__real__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Y: real] :
          ( ( aa(A,A,uminus_uminus(A),real_Vector_of_real(A,Xb)) = real_Vector_of_real(A,Y) )
        <=> ( aa(real,real,uminus_uminus(real),Xb) = Y ) ) ) ).

% minus_of_real_eq_of_real_iff
tff(fact_2302_of__real__eq__minus__of__real__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Y: real] :
          ( ( real_Vector_of_real(A,Xb) = aa(A,A,uminus_uminus(A),real_Vector_of_real(A,Y)) )
        <=> ( Xb = aa(real,real,uminus_uminus(real),Y) ) ) ) ).

% of_real_eq_minus_of_real_iff
tff(fact_2303_of__real__diff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Y: real] : ( real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),real_Vector_of_real(A,Xb)),real_Vector_of_real(A,Y)) ) ) ).

% of_real_diff
tff(fact_2304_of__real__of__nat__eq,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Nb: nat] : ( real_Vector_of_real(A,aa(nat,real,semiring_1_of_nat(real),Nb)) = aa(nat,A,semiring_1_of_nat(A),Nb) ) ) ).

% of_real_of_nat_eq
tff(fact_2305_num__double,axiom,
    ! [Nb: num] : ( aa(num,num,aa(num,fun(num,num),times_times(num),bit0(one2)),Nb) = bit0(Nb) ) ).

% num_double
tff(fact_2306_of__real__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [Xb: real] : ( real_Vector_of_real(A,aa(real,real,inverse_inverse(real),Xb)) = aa(A,A,inverse_inverse(A),real_Vector_of_real(A,Xb)) ) ) ).

% of_real_inverse
tff(fact_2307_of__real__of__int__eq,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Z: int] : ( real_Vector_of_real(A,aa(int,real,ring_1_of_int(real),Z)) = aa(int,A,ring_1_of_int(A),Z) ) ) ).

% of_real_of_int_eq
tff(fact_2308_numeral__eq__neg__one__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( Nb = one2 ) ) ) ).

% numeral_eq_neg_one_iff
tff(fact_2309_neg__one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] :
          ( ( aa(A,A,uminus_uminus(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) )
        <=> ( Nb = one2 ) ) ) ).

% neg_one_eq_numeral_iff
tff(fact_2310_Suc__numeral,axiom,
    ! [Nb: num] : ( aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),Nb)) = aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),Nb),one2)) ) ).

% Suc_numeral
tff(fact_2311_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num] :
          ( ~ aa(A,$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),Ma)))
        <=> ( Ma != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_2312_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),aa(A,A,uminus_uminus(A),one_one(A)))
        <=> ( Ma != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_2313_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_2314_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_2315_one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% one_mod_two_eq_one
tff(fact_2316_bits__one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% bits_one_mod_two_eq_one
tff(fact_2317_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_2318_add__2__eq__Suc,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) = aa(nat,nat,suc,aa(nat,nat,suc,Nb)) ) ).

% add_2_eq_Suc
tff(fact_2319_add__2__eq__Suc_H,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,Nb)) ) ).

% add_2_eq_Suc'
tff(fact_2320_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% Suc_1
tff(fact_2321_div2__Suc__Suc,axiom,
    ! [Ma: nat] : ( divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Ma)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,divide_divide(nat,Ma,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% div2_Suc_Suc
tff(fact_2322_add__self__div__2,axiom,
    ! [Ma: nat] : ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Ma),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Ma ) ).

% add_self_div_2
tff(fact_2323_mod2__Suc__Suc,axiom,
    ! [Ma: nat] : ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,Ma)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = modulo_modulo(nat,Ma,aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% mod2_Suc_Suc
tff(fact_2324_one__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Nb)) ) ) ).

% one_plus_numeral
tff(fact_2325_numeral__plus__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Nb)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Nb),one2)) ) ) ).

% numeral_plus_one
tff(fact_2326_of__real__neg__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : ( 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_2327_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Nb)),one_one(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Nb),one2) ) ) ).

% numeral_le_one_iff
tff(fact_2328_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),Nb) ) ) ).

% one_less_numeral_iff
tff(fact_2329_add__neg__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: 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),Nb))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Nb))) ) ) ).

% add_neg_numeral_special(5)
tff(fact_2330_add__neg__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Ma: num] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Ma))) ) ) ).

% add_neg_numeral_special(6)
tff(fact_2331_diff__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: 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),Nb)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Nb))) ) ) ).

% diff_numeral_special(5)
tff(fact_2332_inrange,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),vEBT_VEBT_set_vebt(Ta)),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))),Nb)),one_one(nat)))) ) ).

% inrange
tff(fact_2333_bits__1__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% bits_1_div_2
tff(fact_2334_one__div__two__eq__zero,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(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_2335_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$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),Xb),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))) )
            <=> ( Xb = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2336_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_2337_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_2338_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_2339_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_2340_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_2341_minus__1__div__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(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_2342_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: 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),Xb),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) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2343_not__mod__2__eq__0__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(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_2344_not__mod__2__eq__1__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(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_2345_bits__minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% bits_minus_1_mod_2_eq
tff(fact_2346_minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(A)
     => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% minus_1_mod_2_eq
tff(fact_2347_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Nb: 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))),Nb)) = 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))),Nb)) ) ) ).

% Power.ring_1_class.power_minus_even
tff(fact_2348_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Nb: nat] :
      ( ( modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))) != aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_2349_diff__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: 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),Nb))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Nb)) ) ) ).

% diff_numeral_special(3)
tff(fact_2350_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Ma: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),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),Ma),one2))) ) ) ).

% diff_numeral_special(4)
tff(fact_2351_add__self__mod__2,axiom,
    ! [Ma: nat] : ( modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Ma),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ) ).

% add_self_mod_2
tff(fact_2352_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(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_2353_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(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_2354_power__minus1__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: 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))),Nb)) = one_one(A) ) ) ).

% power_minus1_even
tff(fact_2355_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),aa(A,int,archim6421214686448440834_floor(A),Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb) ) ) ).

% one_less_floor
tff(fact_2356_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(A,int,archim6421214686448440834_floor(A),Xb)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% floor_le_one
tff(fact_2357_mod2__gr__0,axiom,
    ! [Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,Ma,aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> ( modulo_modulo(nat,Ma,aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_2358_norm__of__real__addn,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: real,B2: num] : ( real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,Xb)),aa(num,A,numeral_numeral(A),B2))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(num,real,numeral_numeral(real),B2))) ) ) ).

% norm_of_real_addn
tff(fact_2359_square__powr__half,axiom,
    ! [Xb: real] : ( powr(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(real,real,abs_abs(real),Xb) ) ).

% square_powr_half
tff(fact_2360__C3_Ohyps_C_I2_J,axiom,
    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) ).

% "3.hyps"(2)
tff(fact_2361_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] : ( one2 != bit0(X2) ) ).

% verit_eq_simplify(10)
tff(fact_2362_num__induct,axiom,
    ! [P: fun(num,$o),Xb: num] :
      ( aa(num,$o,P,one2)
     => ( ! [X: num] :
            ( aa(num,$o,P,X)
           => aa(num,$o,P,inc(X)) )
       => aa(num,$o,P,Xb) ) ) ).

% num_induct
tff(fact_2363_pow_Osimps_I1_J,axiom,
    ! [Xb: num] : ( pow(Xb,one2) = Xb ) ).

% pow.simps(1)
tff(fact_2364_inc_Osimps_I1_J,axiom,
    inc(one2) = bit0(one2) ).

% inc.simps(1)
tff(fact_2365_add__One,axiom,
    ! [Xb: num] : ( aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),one2) = inc(Xb) ) ).

% add_One
tff(fact_2366_add__inc,axiom,
    ! [Xb: num,Y: num] : ( aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),inc(Y)) = inc(aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),Y)) ) ).

% add_inc
tff(fact_2367_add__One__commute,axiom,
    ! [Nb: num] : ( aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Nb) = aa(num,num,aa(num,fun(num,num),plus_plus(num),Nb),one2) ) ).

% add_One_commute
tff(fact_2368_le__num__One__iff,axiom,
    ! [Xb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Xb),one2)
    <=> ( Xb = one2 ) ) ).

% le_num_One_iff
tff(fact_2369_mult__inc,axiom,
    ! [Xb: num,Y: num] : ( aa(num,num,aa(num,fun(num,num),times_times(num),Xb),inc(Y)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),times_times(num),Xb),Y)),Xb) ) ).

% mult_inc
tff(fact_2370_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_2371_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_2372_power2__commute,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: 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),Xb),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),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ).

% power2_commute
tff(fact_2373_power2__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),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))) )
        <=> ( ( Xb = Y )
            | ( Xb = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% power2_eq_iff
tff(fact_2374_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_2375_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_2376_double__not__eq__Suc__double,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma) != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ) ).

% double_not_eq_Suc_double
tff(fact_2377_Suc__double__not__eq__double,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma)) != aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ) ).

% Suc_double_not_eq_double
tff(fact_2378_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_2379_less__exp,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% less_exp
tff(fact_2380_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_2381_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$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),Xb),Y) ) ) ) ).

% power2_le_imp_le
tff(fact_2382_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
             => ( Xb = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2383_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_2384_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_2385_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_2386_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_2387_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_2388_field__sum__of__halves,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xb,aa(num,A,numeral_numeral(A),bit0(one2)))),divide_divide(A,Xb,aa(num,A,numeral_numeral(A),bit0(one2)))) = Xb ) ) ).

% field_sum_of_halves
tff(fact_2389_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_2390_less__2__cases,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))
     => ( ( Nb = zero_zero(nat) )
        | ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_2391_less__2__cases__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))
    <=> ( ( Nb = zero_zero(nat) )
        | ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_2392_abs__square__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
        <=> ( aa(A,A,abs_abs(A),Xb) = one_one(A) ) ) ) ).

% abs_square_eq_1
tff(fact_2393_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,$o)),Xb: 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),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_sqrt_wlog
tff(fact_2394_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_2395_nat__induct2,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$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,Nb) ) ) ) ).

% nat_induct2
tff(fact_2396_two__realpow__ge__one,axiom,
    ! [Nb: 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))),Nb)) ).

% two_realpow_ge_one
tff(fact_2397_realpow__square__minus__le,axiom,
    ! [U: real,Xb: 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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% realpow_square_minus_le
tff(fact_2398_diff__le__diff__pow,axiom,
    ! [K: nat,Ma: nat,Nb: 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),Ma),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Nb))) ) ).

% diff_le_diff_pow
tff(fact_2399_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_2400_not__exp__less__eq__0__int,axiom,
    ! [Nb: 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))),Nb)),zero_zero(int)) ).

% not_exp_less_eq_0_int
tff(fact_2401_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$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),Xb),Y) ) ) ) ).

% power2_less_imp_less
tff(fact_2402_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% half_gt_zero_iff
tff(fact_2403_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)),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% half_gt_zero
tff(fact_2404_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] : aa(A,$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),Xb),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_2405_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$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),Xb),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))
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2406_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] : ~ aa(A,$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),Xb),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_2407_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$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),Xb),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)))))
        <=> ( ( Xb != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2408_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% field_less_half_sum
tff(fact_2409_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xb: 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),Xb),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),Xb),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))),Xb)),Y)) ) ) ).

% power2_sum
tff(fact_2410_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ) ) ).

% square_le_1
tff(fact_2411_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),Y) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2412_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) ) ).

% of_nat_less_two_power
tff(fact_2413_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ma: nat,Nb: 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),Ma),Nb)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2414_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ma: nat,Nb: 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),Ma),Nb)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ma) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2415_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,$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))),Nb))) ) ).

% zero_le_even_power'
tff(fact_2416_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat,Ma: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) != 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),Nb),Ma)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2417_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),one_one(A)) ) ) ).

% abs_square_le_1
tff(fact_2418_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Xb)),one_one(A)) ) ) ).

% abs_square_less_1
tff(fact_2419_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Ma: nat,Nb: nat] : ( divide_divide(A,divide_divide(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ma)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = 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),Ma),Nb))) ) ) ).

% div_exp_eq
tff(fact_2420_minus__power__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,Nb: 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)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb)) = 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))),Nb)) ) ) ).

% minus_power_mult_self
tff(fact_2421_power__odd__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Nb: 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))),Nb))) = 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),Nb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% power_odd_eq
tff(fact_2422_nat__bit__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$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,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_2423_square__norm__one,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
         => ( real_V7770717601297561774m_norm(A,Xb) = one_one(real) ) ) ) ).

% square_norm_one
tff(fact_2424_div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% div_2_gt_zero
tff(fact_2425_Suc__n__div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),divide_divide(nat,aa(nat,nat,suc,Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2426_numeral__Bit0,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] : ( aa(num,A,numeral_numeral(A),bit0(Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Nb)),aa(num,A,numeral_numeral(A),Nb)) ) ) ).

% numeral_Bit0
tff(fact_2427_exp__half__le2,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(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_2428_power__minus__Bit0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A,K: num] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xb)),aa(num,nat,numeral_numeral(nat),bit0(K))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(K))) ) ) ).

% power_minus_Bit0
tff(fact_2429_minus__1__div__exp__eq__int,axiom,
    ! [Nb: nat] : ( 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))),Nb)) = aa(int,int,uminus_uminus(int),one_one(int)) ) ).

% minus_1_div_exp_eq_int
tff(fact_2430_exp__plus__inverse__exp,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),exp(real,Xb)),aa(real,real,inverse_inverse(real),exp(real,Xb)))) ).

% exp_plus_inverse_exp
tff(fact_2431_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_2432_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_2433_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_2434_divide__numeral__1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] : ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),one2)) = A2 ) ) ).

% divide_numeral_1
tff(fact_2435_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_2436_Suc__nat__number__of__add,axiom,
    ! [V2: num,Nb: nat] : ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),Nb)) = 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),V2),one2))),Nb) ) ).

% Suc_nat_number_of_add
tff(fact_2437_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_2438_triangle__def,axiom,
    ! [Nb: nat] : ( nat_triangle(Nb) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% triangle_def
tff(fact_2439_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] : aa(A,$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))),Xb)),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),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_2440_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),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)
           => ( modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)) = modulo_modulo(A,A2,B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2441_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: 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),Xb),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),Xb),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))),Xb)),Y)) ) ) ).

% power2_diff
tff(fact_2442_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: 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),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_2443_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2444_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$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))),Nb)))),zero_zero(A)) ) ) ).

% odd_power_less_zero
tff(fact_2445_power__minus1__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: 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))),Nb))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% power_minus1_odd
tff(fact_2446_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat,Ma: nat] : ( modulo_modulo(A,divide_divide(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ma)) = divide_divide(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),Nb),Ma))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) ) ) ).

% div_exp_mod_exp_eq
tff(fact_2447_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_2448_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_2449_plus__inverse__ge__2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => 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),Xb),aa(real,real,inverse_inverse(real),Xb))) ) ).

% plus_inverse_ge_2
tff(fact_2450_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,Z))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% exp_bound_half
tff(fact_2451_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_2452_less__log2__of__power,axiom,
    ! [Nb: nat,Ma: 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))),Nb)),Ma)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log2(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Ma))) ) ).

% less_log2_of_power
tff(fact_2453_arsinh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A] : ( aa(A,A,arsinh(A),Xb) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),powr(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)),real_Vector_of_real(A,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))))) ) ) ).

% arsinh_def
tff(fact_2454_nonzero__of__real__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [Xb: real] :
          ( ( Xb != zero_zero(real) )
         => ( real_Vector_of_real(A,aa(real,real,inverse_inverse(real),Xb)) = aa(A,A,inverse_inverse(A),real_Vector_of_real(A,Xb)) ) ) ) ).

% nonzero_of_real_inverse
tff(fact_2455_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),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))),divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))) = divide_divide(A,A2,B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2456_arcosh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A] : ( aa(A,A,arcosh(A),Xb) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),powr(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)),real_Vector_of_real(A,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))))) ) ) ).

% arcosh_def
tff(fact_2457_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ma: nat,Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( 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))),Ma)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(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),Nb),Ma)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ma)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_2458_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cosh(A,Xb) = zero_zero(A) )
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_2459_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num,Q4: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(Nb)),aa(num,A,numeral_numeral(A),bit0(Q4))) = zero_zero(A) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q4)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2460_cosh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : ( cosh(A,Z) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),exp(A,Z)),exp(A,aa(A,A,uminus_uminus(A),Z))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% cosh_field_def
tff(fact_2461_log2__of__power__less,axiom,
    ! [Ma: nat,Nb: 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))),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log2(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Ma))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ).

% log2_of_power_less
tff(fact_2462_exp__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xb)),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)),Xb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% exp_bound
tff(fact_2463_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))
     => ( 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)) = 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_2464_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)
     => ( 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)) = divide_divide(int,B2,A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_2465_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : ( archimedean_round(A,Xb) = aa(A,int,archim6421214686448440834_floor(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ) ).

% round_def
tff(fact_2466_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)
     => ( 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))),modulo_modulo(int,B2,A2))) ) ) ).

% pos_zmod_mult_2
tff(fact_2467_real__le__x__sinh,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),exp(real,Xb)),aa(real,real,inverse_inverse(real),exp(real,Xb))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% real_le_x_sinh
tff(fact_2468_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_2469_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_2470_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_2471_real__le__abs__sinh,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,abs_abs(real),divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),exp(real,Xb)),aa(real,real,inverse_inverse(real),exp(real,Xb))),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% real_le_abs_sinh
tff(fact_2472_cong__exp__iff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),one2)) = zero_zero(A) ) ) ).

% cong_exp_iff_simps(1)
tff(fact_2473_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,Xb: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),U),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => ( aa(A,$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),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% arith_geo_mean
tff(fact_2474_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Ma: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ma)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => ( ( modulo_modulo(A,Xb,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ma)) = modulo_modulo(A,Xb,Ma) )
              | ( modulo_modulo(A,Xb,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ma)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,Xb,Ma)),Ma) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_2475_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),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),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) = modulo_modulo(A,A2,B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_2476_norm__less__p1,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xb: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,real_V7770717601297561774m_norm(A,Xb))),one_one(A)))) ) ).

% norm_less_p1
tff(fact_2477_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xb))),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% of_int_round_le
tff(fact_2478_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),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,Xb))) ) ).

% of_int_round_ge
tff(fact_2479_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),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,Xb))) ) ).

% of_int_round_gt
tff(fact_2480_log2__of__power__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log2(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Ma))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ).

% log2_of_power_le
tff(fact_2481_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),real_V7770717601297561774m_norm(A,Z)))) ) ) ).

% exp_bound_lemma
tff(fact_2482_real__exp__bound__lemma,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xb)),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))),Xb))) ) ) ).

% real_exp_bound_lemma
tff(fact_2483_exp__lower__Taylor__quadratic,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => 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)),Xb)),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,real,numeral_numeral(real),bit0(one2))))),exp(real,Xb)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_2484_ln__one__plus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),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),Xb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),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)),Xb))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_2485_artanh__def,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A] : ( aa(A,A,artanh(A),Xb) = divide_divide(A,aa(A,A,ln_ln(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),Xb),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% artanh_def
tff(fact_2486_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))
     => ( 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))),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_2487_numeral__inc,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Xb: num] : ( aa(num,A,numeral_numeral(A),inc(Xb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Xb)),one_one(A)) ) ) ).

% numeral_inc
tff(fact_2488_cosh__ln__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( cosh(real,aa(real,real,ln_ln(real),Xb)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,inverse_inverse(real),Xb)),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% cosh_ln_real
tff(fact_2489_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(real,int,archim6421214686448440834_floor(real),aa(real,real,log2(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Nb))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(real,int,archim6421214686448440834_floor(real),aa(real,real,log2(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_2490_fact__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Nb: 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))),Nb)) = 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))),Nb))),comm_s3205402744901411588hammer(A,divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb))),semiring_char_0_fact(A,Nb)) ) ) ).

% fact_double
tff(fact_2491_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$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,Xb))),Xb))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% of_int_round_abs_le
tff(fact_2492_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Nb: int] :
          ( aa(A,$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),Xb),aa(int,A,ring_1_of_int(A),Nb)))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))
         => ( archimedean_round(A,Xb) = Nb ) ) ) ).

% round_unique'
tff(fact_2493_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),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)),Xb))),Xb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_2494_arctan__double,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),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,Xb)) = aa(real,real,arctan,divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),Xb),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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% arctan_double
tff(fact_2495_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( archimedean_round(A,Xb) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,Xb)),archimedean_ceiling(A,Xb),aa(A,int,archim6421214686448440834_floor(A),Xb)) ) ) ).

% round_altdef
tff(fact_2496_tanh__real__altdef,axiom,
    ! [Xb: real] : ( aa(real,real,tanh(real),Xb) = divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(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)))),Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(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)))),Xb)))) ) ).

% tanh_real_altdef
tff(fact_2497_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),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))),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)) = divide_divide(A,A2,B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_2498_pochhammer__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,Nb: 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))),Nb)) = 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))),Nb)))),comm_s3205402744901411588hammer(A,Z,Nb))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),Nb)) ) ) ).

% pochhammer_double
tff(fact_2499_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),Y))
         => ( aa(A,$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),Xb),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))))
           => ( archimedean_round(A,Xb) = Y ) ) ) ) ).

% round_unique
tff(fact_2500_norm__of__real__diff,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [B2: real,A2: real] : aa(real,$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),real_Vector_of_real(A,B2)),real_Vector_of_real(A,A2)))),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2))) ) ).

% norm_of_real_diff
tff(fact_2501_ln__one__minus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),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),Xb)),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),Xb),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)),Xb))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_2502_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),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)),Xb))),Xb))),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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_2503_tanh__ln__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,tanh(real),aa(real,real,ln_ln(real),Xb)) = 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),Xb),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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))) ) ) ).

% tanh_ln_real
tff(fact_2504_floor__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: 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),Nb)),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),Nb),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
         => ( aa(real,int,archim6421214686448440834_floor(real),aa(real,real,log2(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),Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_2505_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Nb: 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)
       => ( ( aa(real,int,archim6421214686448440834_floor(real),aa(real,real,log2(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),Nb) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),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),Nb),one_one(nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_2506_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( archimedean_ceiling(real,aa(real,real,log2(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Nb))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(real,aa(real,real,log2(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),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))))),one_one(int)) ) ) ).

% ceiling_log2_div2
tff(fact_2507_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),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)),Xb))),Xb))),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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_2508_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: 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),Nb)),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),Nb),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,log2(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),Nb)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_2509_set__n__deg__not__0,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X,Nb) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_2510_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: 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))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ) ).

% set_bit_0
tff(fact_2511_low__inv,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
     => ( 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))),Nb))),Xb),Nb) = Xb ) ) ).

% low_inv
tff(fact_2512_high__inv,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
     => ( 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))),Nb))),Xb),Nb) = Y ) ) ).

% high_inv
tff(fact_2513_high__bound__aux,axiom,
    ! [Ma: nat,Nb: nat,Mb: 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),Nb),Mb)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Ma,Nb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb)) ) ).

% high_bound_aux
tff(fact_2514_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: 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))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% unset_bit_0
tff(fact_2515_bit__split__inv,axiom,
    ! [Xb: nat,D3: nat] : ( vEBT_VEBT_bit_concat(vEBT_VEBT_high(Xb,D3),vEBT_VEBT_low(Xb,D3),D3) = Xb ) ).

% bit_split_inv
tff(fact_2516_high__def,axiom,
    ! [Xb: nat,Nb: nat] : ( vEBT_VEBT_high(Xb,Nb) = divide_divide(nat,Xb,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ) ).

% high_def
tff(fact_2517_low__def,axiom,
    ! [Xb: nat,Nb: nat] : ( vEBT_VEBT_low(Xb,Nb) = modulo_modulo(nat,Xb,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ) ).

% low_def
tff(fact_2518_unset__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se2638667681897837118et_bit(int,Nb,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_2519_set__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se5668285175392031749et_bit(int,Nb,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_2520_unset__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se2638667681897837118et_bit(int,Nb,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_2521_set__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se5668285175392031749et_bit(int,Nb,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_2522_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,nat,size_size(A),Xb) != aa(A,nat,size_size(A),Y) )
         => ( Xb != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_2523_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_2524_unset__bit__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : ( bit_se2638667681897837118et_bit(nat,Ma,Nb) = aa(int,nat,nat2,bit_se2638667681897837118et_bit(int,Ma,aa(nat,int,semiring_1_of_nat(int),Nb))) ) ).

% unset_bit_nat_def
tff(fact_2525_length__pos__if__in__set,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_pos_if_in_set
tff(fact_2526_card__length,axiom,
    ! [A: $tType,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% card_length
tff(fact_2527_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xb: nat,Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),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),Nb),Ma)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,Nb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_2528_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xb: nat,Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),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),Nb),Ma)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_low(Xb,Nb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_2529_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,Ma: nat,Dega: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X,Nb) )
     => ( vEBT_invar_vebt(Summarya,Ma)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma) )
         => ( ( Ma = Nb )
           => ( ( Dega = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Ma) )
             => ( ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X_12)
               => ( ! [X: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_12) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_2530_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,Ma: nat,Dega: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X,Nb) )
     => ( vEBT_invar_vebt(Summarya,Ma)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma) )
         => ( ( Ma = aa(nat,nat,suc,Nb) )
           => ( ( Dega = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Ma) )
             => ( ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X_12)
               => ( ! [X: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_12) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_2531_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : ( bit_se2638667681897837118et_bit(A,aa(nat,nat,suc,Nb),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(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))),bit_se2638667681897837118et_bit(A,Nb,divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ) ).

% unset_bit_Suc
tff(fact_2532_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : ( bit_se5668285175392031749et_bit(A,aa(nat,nat,suc,Nb),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(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))),bit_se5668285175392031749et_bit(A,Nb,divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ) ).

% set_bit_Suc
tff(fact_2533_both__member__options__ding,axiom,
    ! [Info2: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info2,Dega,TreeLista,Summarya),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Dega))
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(Info2,Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_2534_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : ( bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,Nb),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(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))),bit_se8732182000553998342ip_bit(A,Nb,divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ) ).

% flip_bit_Suc
tff(fact_2535_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] :
          ( bit_ri4674362597316999326ke_bit(A,Nb,A2) = $ite(Nb = zero_zero(nat),aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(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))),bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))))) ) ) ).

% signed_take_bit_rec
tff(fact_2536_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_2537_log__base__10__eq1,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,log2(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,ln_ln(real),exp(real,one_one(real))),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))))),aa(real,real,ln_ln(real),Xb)) ) ) ).

% log_base_10_eq1
tff(fact_2538_inthall,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Nb: nat] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% inthall
tff(fact_2539_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_2540_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : ( aa(set(nat),nat,finite_card(nat),set_or5935395276787703475ssThan(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),aa(nat,nat,suc,L)) ) ).

% card_greaterThanLessThan
tff(fact_2541_flip__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se8732182000553998342ip_bit(int,Nb,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_2542_flip__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se8732182000553998342ip_bit(int,Nb,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_2543_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : ( bit_ri4674362597316999326ke_bit(A,Nb,zero_zero(A)) = zero_zero(A) ) ) ).

% signed_take_bit_of_0
tff(fact_2544_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_2545_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : ( aa(set(nat),nat,finite_card(nat),set_or1337092689740270186AtMost(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,U)),L) ) ).

% card_atLeastAtMost
tff(fact_2546_signed__take__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : ( bit_ri4674362597316999326ke_bit(A,Nb,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_2547_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : ( bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,Nb),one_one(A)) = one_one(A) ) ) ).

% signed_take_bit_Suc_1
tff(fact_2548_signed__take__bit__numeral__of__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : ( bit_ri4674362597316999326ke_bit(A,aa(num,nat,numeral_numeral(nat),K),one_one(A)) = one_one(A) ) ) ).

% signed_take_bit_numeral_of_1
tff(fact_2549_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_2550_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_2551_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_2552_signed__take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : ( bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb),aa(num,int,numeral_numeral(int),bit0(K))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,Nb,aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ).

% signed_take_bit_Suc_bit0
tff(fact_2553_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_2554_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_2555_div__Suc__eq__div__add3,axiom,
    ! [Ma: nat,Nb: nat] : ( divide_divide(nat,Ma,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Nb)))) = divide_divide(nat,Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Nb)) ) ).

% div_Suc_eq_div_add3
tff(fact_2556_Suc__div__eq__add3__div__numeral,axiom,
    ! [Ma: nat,V2: num] : ( divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Ma))),aa(num,nat,numeral_numeral(nat),V2)) = 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))),Ma),aa(num,nat,numeral_numeral(nat),V2)) ) ).

% Suc_div_eq_add3_div_numeral
tff(fact_2557_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : ( bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,Nb,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_2558_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [Ma: nat,V2: num] : ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Ma))),aa(num,nat,numeral_numeral(nat),V2)) = 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))),Ma),aa(num,nat,numeral_numeral(nat),V2)) ) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_2559_mod__Suc__eq__mod__add3,axiom,
    ! [Ma: nat,Nb: nat] : ( modulo_modulo(nat,Ma,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Nb)))) = modulo_modulo(nat,Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Nb)) ) ).

% mod_Suc_eq_mod_add3
tff(fact_2560_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_2561_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : ( bit_ri4674362597316999326ke_bit(A,zero_zero(nat),A2) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% signed_take_bit_0
tff(fact_2562_signed__take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : ( bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb),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),bit_ri4674362597316999326ke_bit(int,Nb,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_2563_zmod__numeral__Bit1,axiom,
    ! [V2: num,W: num] : ( modulo_modulo(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,V2)),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))),modulo_modulo(int,aa(num,int,numeral_numeral(int),V2),aa(num,int,numeral_numeral(int),W)))),one_one(int)) ) ).

% zmod_numeral_Bit1
tff(fact_2564_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : ( bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb),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),bit_ri4674362597316999326ke_bit(int,Nb,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_2565_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] : ( bit0(X2) != aa(num,num,bit1,X32) ) ).

% verit_eq_simplify(14)
tff(fact_2566_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : ( one2 != aa(num,num,bit1,X32) ) ).

% verit_eq_simplify(12)
tff(fact_2567_signed__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : ( bit_ri4674362597316999326ke_bit(int,Nb,aa(int,int,uminus_uminus(int),bit_ri4674362597316999326ke_bit(int,Nb,K))) = bit_ri4674362597316999326ke_bit(int,Nb,aa(int,int,uminus_uminus(int),K)) ) ).

% signed_take_bit_minus
tff(fact_2568_signed__take__bit__add,axiom,
    ! [Nb: nat,K: int,L: int] : ( bit_ri4674362597316999326ke_bit(int,Nb,aa(int,int,aa(int,fun(int,int),plus_plus(int),bit_ri4674362597316999326ke_bit(int,Nb,K)),bit_ri4674362597316999326ke_bit(int,Nb,L))) = bit_ri4674362597316999326ke_bit(int,Nb,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% signed_take_bit_add
tff(fact_2569_signed__take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : ( bit_ri4674362597316999326ke_bit(int,Nb,aa(int,int,aa(int,fun(int,int),minus_minus(int),bit_ri4674362597316999326ke_bit(int,Nb,K)),bit_ri4674362597316999326ke_bit(int,Nb,L))) = bit_ri4674362597316999326ke_bit(int,Nb,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)) ) ).

% signed_take_bit_diff
tff(fact_2570_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_2571_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
     => ( ! [I2: nat] :
            ( aa(nat,$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) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_2572_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: fun(nat,fun(A,$o))] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K)
         => ? [X_13: A] : aa(A,$o,aa(nat,fun(A,$o),P,I3),X_13) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K)
             => aa(A,$o,aa(nat,fun(A,$o),P,I3),aa(nat,A,nth(A,Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_2573_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs = Ys2 )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
        & ! [I3: nat] :
            ( aa(nat,$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) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2574_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one2 )
     => ( ! [X23: num] : ( Y != bit0(X23) )
       => ~ ! [X33: num] : ( Y != aa(num,num,bit1,X33) ) ) ) ).

% num.exhaust
tff(fact_2575_dbl__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xb: A] : ( neg_numeral_dbl(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb) ) ) ).

% dbl_def
tff(fact_2576_inc_Osimps_I2_J,axiom,
    ! [Xb: num] : ( inc(bit0(Xb)) = aa(num,num,bit1,Xb) ) ).

% inc.simps(2)
tff(fact_2577_inc_Osimps_I3_J,axiom,
    ! [Xb: num] : ( inc(aa(num,num,bit1,Xb)) = bit0(inc(Xb)) ) ).

% inc.simps(3)
tff(fact_2578_nth__mem,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => member(A,aa(nat,A,nth(A,Xs),Nb),aa(list(A),set(A),set2(A),Xs)) ) ).

% nth_mem
tff(fact_2579_list__ball__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),P: fun(A,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Xs))
           => aa(A,$o,P,X) )
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_2580_in__set__conv__nth,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
          & ( aa(nat,A,nth(A,Xs),I3) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_2581_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Xb: A] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => aa(A,$o,P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_2582_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X4) )
    <=> ! [I3: nat] :
          ( aa(nat,$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)) ) ) ).

% all_set_conv_all_nth
tff(fact_2583_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] : ( aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)) = 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),Nb)),aa(num,A,numeral_numeral(A),Nb))),one_one(A)) ) ) ).

% numeral_Bit1
tff(fact_2584_eval__nat__numeral_I3_J,axiom,
    ! [Nb: num] : ( aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Nb)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bit0(Nb))) ) ).

% eval_nat_numeral(3)
tff(fact_2585_power__minus__Bit1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A,K: num] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xb)),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K)))) ) ) ).

% power_minus_Bit1
tff(fact_2586_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num,Q4: num] : ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),bit0(Q4))) != zero_zero(A) ) ) ).

% cong_exp_iff_simps(3)
tff(fact_2587_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_2588_Suc3__eq__add__3,axiom,
    ! [Nb: nat] : ( aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Nb))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Nb) ) ).

% Suc3_eq_add_3
tff(fact_2589_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_2590_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] : ( aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Leaf((X21),(X22))) = zero_zero(nat) ) ).

% VEBT.size(4)
tff(fact_2591_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Ma: num,Q4: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Ma)),aa(num,A,numeral_numeral(A),bit0(Q4))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q4))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Ma),aa(num,A,numeral_numeral(A),Q4)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_2592_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q4: num,Nb: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q4))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),bit0(Q4))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q4)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_2593_Suc__div__eq__add3__div,axiom,
    ! [Ma: nat,Nb: nat] : ( divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Ma))),Nb) = 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))),Ma),Nb) ) ).

% Suc_div_eq_add3_div
tff(fact_2594_Suc__mod__eq__add3__mod,axiom,
    ! [Ma: nat,Nb: nat] : ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Ma))),Nb) = 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))),Ma),Nb) ) ).

% Suc_mod_eq_add3_mod
tff(fact_2595_signed__take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_ri4674362597316999326ke_bit(int,Nb,K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ).

% signed_take_bit_int_less_exp
tff(fact_2596_exp__le,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,one_one(real))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))) ).

% exp_le
tff(fact_2597_mod__exhaust__less__4,axiom,
    ! [Ma: nat] :
      ( ( modulo_modulo(nat,Ma,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = zero_zero(nat) )
      | ( modulo_modulo(nat,Ma,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = one_one(nat) )
      | ( modulo_modulo(nat,Ma,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
      | ( modulo_modulo(nat,Ma,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_2598_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),bit_ri4674362597316999326ke_bit(int,Nb,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))),Nb)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_2599_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_ri4674362597316999326ke_bit(int,Nb,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))),Nb)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_2600_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Nb: 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))),Nb))),bit_ri4674362597316999326ke_bit(int,Nb,K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_2601_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),bit_ri4674362597316999326ke_bit(int,Nb,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))),Nb))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_2602_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),bit_ri4674362597316999326ke_bit(int,Nb,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))),Nb))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_2603_signed__take__bit__int__less__eq,axiom,
    ! [Nb: 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))),Nb)),K)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),bit_ri4674362597316999326ke_bit(int,Nb,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,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_2604_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( bit_ri4674362597316999326ke_bit(int,Nb,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))),Nb))),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))),Nb)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_2605_signed__take__bit__int__eq__self,axiom,
    ! [Nb: 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))),Nb))),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))),Nb))
       => ( bit_ri4674362597316999326ke_bit(int,Nb,K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2606_num_Osize_I5_J,axiom,
    ! [X2: num] : ( aa(num,nat,size_size(num),bit0(X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X2)),aa(nat,nat,suc,zero_zero(nat))) ) ).

% num.size(5)
tff(fact_2607_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: 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))),Nb)))
     => 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,Nb)))),bit_ri4674362597316999326ke_bit(int,Nb,K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2608_log__base__10__eq2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,log2(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,log2(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),exp(real,one_one(real)))),aa(real,real,ln_ln(real),Xb)) ) ) ).

% log_base_10_eq2
tff(fact_2609_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : ( bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,Nb),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(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))),bit_ri4674362597316999326ke_bit(A,Nb,divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ) ).

% signed_take_bit_Suc
tff(fact_2610_in__children__def,axiom,
    ! [Nb: nat,TreeLista: list(vEBT_VEBT),Xb: nat] :
      ( vEBT_V5917875025757280293ildren(Nb,TreeLista,Xb)
    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,Nb))),vEBT_VEBT_low(Xb,Nb)) ) ).

% in_children_def
tff(fact_2611_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : ( bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),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),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_2612_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),Nb),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),Nb)))),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))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_2613_concat__bit__Suc,axiom,
    ! [Nb: nat,K: int,L: int] : ( bit_concat_bit(aa(nat,nat,suc,Nb),K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),bit_concat_bit(Nb,divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2))),L))) ) ).

% concat_bit_Suc
tff(fact_2614_arctan__inverse,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero(real) )
     => ( aa(real,real,arctan,divide_divide(real,one_one(real),Xb)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Xb)),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(real,real,arctan,Xb)) ) ) ).

% arctan_inverse
tff(fact_2615_option_Osize_I3_J,axiom,
    ! [A: $tType] : ( aa(option(A),nat,size_size(option(A)),none(A)) = aa(nat,nat,suc,zero_zero(nat)) ) ).

% option.size(3)
tff(fact_2616_binomial__Suc__n,axiom,
    ! [Nb: nat] : ( aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),Nb) = aa(nat,nat,suc,Nb) ) ).

% binomial_Suc_n
tff(fact_2617_binomial__n__n,axiom,
    ! [Nb: nat] : ( aa(nat,nat,binomial(Nb),Nb) = one_one(nat) ) ).

% binomial_n_n
tff(fact_2618_concat__bit__0,axiom,
    ! [K: int,L: int] : ( bit_concat_bit(zero_zero(nat),K,L) = L ) ).

% concat_bit_0
tff(fact_2619_binomial__0__Suc,axiom,
    ! [K: nat] : ( aa(nat,nat,binomial(zero_zero(nat)),aa(nat,nat,suc,K)) = zero_zero(nat) ) ).

% binomial_0_Suc
tff(fact_2620_binomial__1,axiom,
    ! [Nb: nat] : ( aa(nat,nat,binomial(Nb),aa(nat,nat,suc,zero_zero(nat))) = Nb ) ).

% binomial_1
tff(fact_2621_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa(nat,nat,binomial(Nb),K) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K) ) ).

% binomial_eq_0_iff
tff(fact_2622_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K: nat] : ( aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K))) ) ).

% binomial_Suc_Suc
tff(fact_2623_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_2624_binomial__n__0,axiom,
    ! [Nb: nat] : ( aa(nat,nat,binomial(Nb),zero_zero(nat)) = one_one(nat) ) ).

% binomial_n_0
tff(fact_2625_eq__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,Nb) )
    <=> ( pred_numeral(K) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_2626_Suc__eq__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( ( aa(nat,nat,suc,Nb) = aa(num,nat,numeral_numeral(nat),K) )
    <=> ( Nb = pred_numeral(K) ) ) ).

% Suc_eq_numeral
tff(fact_2627_concat__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_concat_bit(Nb,K,L))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_2628_concat__bit__negative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_concat_bit(Nb,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_2629_pred__numeral__inc,axiom,
    ! [K: num] : ( pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ) ).

% pred_numeral_inc
tff(fact_2630_zero__less__binomial__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Nb),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ).

% zero_less_binomial_iff
tff(fact_2631_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_2632_less__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pred_numeral(K)),Nb) ) ).

% less_numeral_Suc
tff(fact_2633_less__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),pred_numeral(K)) ) ).

% less_Suc_numeral
tff(fact_2634_le__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),pred_numeral(K)),Nb) ) ).

% le_numeral_Suc
tff(fact_2635_le__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),pred_numeral(K)) ) ).

% le_Suc_numeral
tff(fact_2636_diff__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),pred_numeral(K)) ) ).

% diff_Suc_numeral
tff(fact_2637_diff__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),pred_numeral(K)),Nb) ) ).

% diff_numeral_Suc
tff(fact_2638_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : ( bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_2639_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : ( bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(num,int,numeral_numeral(int),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),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_2640_choose__one,axiom,
    ! [Nb: nat] : ( aa(nat,nat,binomial(Nb),one_one(nat)) = Nb ) ).

% choose_one
tff(fact_2641_concat__bit__assoc,axiom,
    ! [Nb: nat,K: int,Ma: nat,L: int,R2: int] : ( bit_concat_bit(Nb,K,bit_concat_bit(Ma,L,R2)) = bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb),bit_concat_bit(Nb,K,L),R2) ) ).

% concat_bit_assoc
tff(fact_2642_binomial__eq__0,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K)
     => ( aa(nat,nat,binomial(Nb),K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_2643_Suc__times__binomial__eq,axiom,
    ! [Nb: nat,K: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,binomial(Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K))),aa(nat,nat,suc,K)) ) ).

% Suc_times_binomial_eq
tff(fact_2644_Suc__times__binomial,axiom,
    ! [K: nat,Nb: 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,Nb)),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,binomial(Nb),K)) ) ).

% Suc_times_binomial
tff(fact_2645_binomial__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( aa(nat,nat,binomial(Nb),K) = aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)) ) ) ).

% binomial_symmetric
tff(fact_2646_choose__mult__lemma,axiom,
    ! [Ma: nat,R2: 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),Ma),R2)),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),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),Ma),R2)),K)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),R2)),Ma)) ) ).

% choose_mult_lemma
tff(fact_2647_binomial__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Nb: nat,K: nat] : ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K)) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Nb)),K) ) ) ).

% binomial_gbinomial
tff(fact_2648_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_2649_zero__less__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Nb),K)) ) ).

% zero_less_binomial
tff(fact_2650_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_2651_choose__mult,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Nb),Ma)),aa(nat,nat,binomial(Ma),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),K))) ) ) ) ).

% choose_mult
tff(fact_2652_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K: nat] : ( aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,suc,K)) ) ).

% binomial_Suc_Suc_eq_times
tff(fact_2653_binomial__absorb__comp,axiom,
    ! [Nb: 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),Nb),K)),aa(nat,nat,binomial(Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K)) ) ).

% binomial_absorb_comp
tff(fact_2654_option_Osize__neq,axiom,
    ! [A: $tType,Xb: option(A)] : ( aa(option(A),nat,size_size(option(A)),Xb) != zero_zero(nat) ) ).

% option.size_neq
tff(fact_2655_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_2656_binomial__absorption,axiom,
    ! [K: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K)) ) ).

% binomial_absorption
tff(fact_2657_binomial__fact__lemma,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( 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),Nb),K)))),aa(nat,nat,binomial(Nb),K)) = semiring_char_0_fact(nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_2658_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Nb),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_2659_choose__reduce__nat,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( aa(nat,nat,binomial(Nb),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),Nb),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),Nb),one_one(nat))),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_2660_times__binomial__minus1__eq,axiom,
    ! [K: nat,Nb: 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(Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_2661_binomial__altdef__nat,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( aa(nat,nat,binomial(Nb),K) = divide_divide(nat,semiring_char_0_fact(nat,Nb),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),Nb),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_2662_binomial__strict__mono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K7)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),K7)) ) ) ).

% binomial_strict_mono
tff(fact_2663_binomial__strict__antimono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K7),Nb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Nb),K7)),aa(nat,nat,binomial(Nb),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_2664_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_2665_binomial__addition__formula,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,binomial(Nb),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),Nb),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K)) ) ) ).

% binomial_addition_formula
tff(fact_2666_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( 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(Nb),K))) = divide_divide(A,semiring_char_0_fact(A,Nb),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))) ) ) ) ).

% fact_binomial
tff(fact_2667_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K)) = divide_divide(A,semiring_char_0_fact(A,Nb),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),Nb),K)))) ) ) ) ).

% binomial_fact
tff(fact_2668_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_2669_arctan__one,axiom,
    aa(real,real,arctan,one_one(real)) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% arctan_one
tff(fact_2670_choose__two,axiom,
    ! [Nb: nat] : ( aa(nat,nat,binomial(Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% choose_two
tff(fact_2671_minus__pi__half__less__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),zero_zero(real)) ).

% minus_pi_half_less_zero
tff(fact_2672_arctan__lbound,axiom,
    ! [Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y)) ).

% arctan_lbound
tff(fact_2673_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% arctan_bounded
tff(fact_2674_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,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,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))))))))))) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% machin_Euler
tff(fact_2675_machin,axiom,
    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,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(one2))))))),aa(real,real,arctan,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_2676_binomial__code,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,nat,binomial(Nb),K) = $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K),
          zero_zero(nat),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)),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),Nb),K)),one_one(nat)),Nb,one_one(nat)),semiring_char_0_fact(nat,K))) ) ) ).

% binomial_code
tff(fact_2677_sin__cos__npi,axiom,
    ! [Nb: nat] : ( sin(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))),Nb)))),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))),Nb) ) ).

% sin_cos_npi
tff(fact_2678_cos__pi__eq__zero,axiom,
    ! [Ma: nat] : ( aa(real,real,cos(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))),Ma)))),aa(num,real,numeral_numeral(real),bit0(one2)))) = zero_zero(real) ) ).

% cos_pi_eq_zero
tff(fact_2679_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : ( bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),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),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_2680_cot__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cot(real),Xb)),zero_zero(real)) ) ) ).

% cot_less_zero
tff(fact_2681_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => ( 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))),Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),modulo_modulo(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_2682_nat__dvd__1__iff__1,axiom,
    ! [Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),one_one(nat))
    <=> ( Ma = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_2683_int__dvd__int__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(nat,int,semiring_1_of_nat(int),Ma)),aa(nat,int,semiring_1_of_nat(int),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),Nb) ) ).

% int_dvd_int_iff
tff(fact_2684_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),zero_zero(A)),A2)
        <=> ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_2685_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),zero_zero(A)) ) ).

% dvd_0_right
tff(fact_2686_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% dvd_add_triv_left_iff
tff(fact_2687_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% dvd_add_triv_right_iff
tff(fact_2688_dvd__1__left,axiom,
    ! [K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,suc,zero_zero(nat))),K) ).

% dvd_1_left
tff(fact_2689_dvd__1__iff__1,axiom,
    ! [Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( Ma = aa(nat,nat,suc,zero_zero(nat)) ) ) ).

% dvd_1_iff_1
tff(fact_2690_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),divide_divide(A,B2,A2)),divide_divide(A,C2,A2))
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2) ) ) ) ) ).

% div_dvd_div
tff(fact_2691_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,uminus_uminus(A),Xb)),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),Y) ) ) ).

% minus_dvd_iff
tff(fact_2692_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),Y) ) ) ).

% dvd_minus_iff
tff(fact_2693_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: A,K: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Ma),aa(A,A,abs_abs(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Ma),K) ) ) ).

% dvd_abs_iff
tff(fact_2694_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ma: A,K: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,abs_abs(A),Ma)),K)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Ma),K) ) ) ).

% abs_dvd_iff
tff(fact_2695_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : ( bit_se2584673776208193580ke_bit(A,Nb,zero_zero(A)) = zero_zero(A) ) ) ).

% take_bit_of_0
tff(fact_2696_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
    <=> ( ( K = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_2697_sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sin_zero
tff(fact_2698_cos__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : ( aa(A,A,cos(A),aa(A,A,uminus_uminus(A),Xb)) = aa(A,A,cos(A),Xb) ) ) ).

% cos_minus
tff(fact_2699_sin__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : ( sin(A,aa(A,A,uminus_uminus(A),Xb)) = aa(A,A,uminus_uminus(A),sin(A,Xb)) ) ) ).

% sin_minus
tff(fact_2700_zdvd1__eq,axiom,
    ! [Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Xb),one_one(int))
    <=> ( aa(int,int,abs_abs(int),Xb) = one_one(int) ) ) ).

% zdvd1_eq
tff(fact_2701_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_2702_cot__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( aa(A,A,cot(A),aa(A,A,uminus_uminus(A),Xb)) = aa(A,A,uminus_uminus(A),aa(A,A,cot(A),Xb)) ) ) ).

% cot_minus
tff(fact_2703_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( C2 = zero_zero(A) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_2704_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_2705_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_2706_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_2707_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A)) ) ) ) ).

% unit_prod
tff(fact_2708_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2709_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2710_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,A2)) = B2 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_2711_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,A2)),A2) = B2 ) ) ) ).

% dvd_div_mult_self
tff(fact_2712_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),divide_divide(A,A2,B2)),one_one(A)) ) ) ) ).

% unit_div
tff(fact_2713_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),divide_divide(A,one_one(A),A2)),one_one(A)) ) ) ).

% unit_div_1_unit
tff(fact_2714_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( divide_divide(A,one_one(A),divide_divide(A,one_one(A),A2)) = A2 ) ) ) ).

% unit_div_1_div_1
tff(fact_2715_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
           => ( 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),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ) ).

% div_add
tff(fact_2716_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
           => ( 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),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ) ).

% div_diff
tff(fact_2717_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_2718_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_2719_take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : ( bit_se2584673776208193580ke_bit(A,zero_zero(nat),A2) = zero_zero(A) ) ) ).

% take_bit_0
tff(fact_2720_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat] : ( bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb),one_one(A)) = one_one(A) ) ) ).

% take_bit_Suc_1
tff(fact_2721_take__bit__numeral__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num] : ( bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L),one_one(A)) = one_one(A) ) ) ).

% take_bit_numeral_1
tff(fact_2722_dvd__nat__abs__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),aa(int,nat,nat2,aa(int,int,abs_abs(int),K)))
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(nat,int,semiring_1_of_nat(int),Nb)),K) ) ).

% dvd_nat_abs_iff
tff(fact_2723_nat__abs__dvd__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),Nb)
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),aa(nat,int,semiring_1_of_nat(int),Nb)) ) ).

% nat_abs_dvd_iff
tff(fact_2724_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R2)),L)),K)
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
        & ( ( R2 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_2725_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),aa(int,int,sgn_sgn(int),R2))),K)
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
        & ( ( R2 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_2726_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R2)),K))
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
        | ( R2 = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_2727_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(int,int,sgn_sgn(int),R2)))
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
        | ( R2 = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_2728_concat__bit__of__zero__2,axiom,
    ! [Nb: nat,K: int] : ( bit_concat_bit(Nb,K,zero_zero(int)) = bit_se2584673776208193580ke_bit(int,Nb,K) ) ).

% concat_bit_of_zero_2
tff(fact_2729_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,one_one(A),A2)) = divide_divide(A,B2,A2) ) ) ) ).

% unit_mult_div_div
tff(fact_2730_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,A2)),A2) = B2 ) ) ) ).

% unit_div_mult_self
tff(fact_2731_even__Suc__Suc__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Nb)))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ) ).

% even_Suc_Suc_iff
tff(fact_2732_even__Suc,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,Nb))
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ) ).

% even_Suc
tff(fact_2733_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% pow_divides_pow_iff
tff(fact_2734_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat] :
          ( ( bit_se2584673776208193580ke_bit(A,Nb,one_one(A)) = zero_zero(A) )
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_2735_sin__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,real_Vector_of_real(A,pi)) = zero_zero(A) ) ) ).

% sin_of_real_pi
tff(fact_2736_cos__pi,axiom,
    aa(real,real,cos(real),pi) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% cos_pi
tff(fact_2737_of__nat__nat__take__bit__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat,K: int] : ( aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,bit_se2584673776208193580ke_bit(int,Nb,K))) = aa(int,A,ring_1_of_int(A),bit_se2584673776208193580ke_bit(int,Nb,K)) ) ) ).

% of_nat_nat_take_bit_eq
tff(fact_2738_cos__periodic__pi2,axiom,
    ! [Xb: real] : ( aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),pi),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,cos(real),Xb)) ) ).

% cos_periodic_pi2
tff(fact_2739_cos__periodic__pi,axiom,
    ! [Xb: real] : ( aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),pi)) = aa(real,real,uminus_uminus(real),aa(real,real,cos(real),Xb)) ) ).

% cos_periodic_pi
tff(fact_2740_sin__periodic__pi2,axiom,
    ! [Xb: real] : ( sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),pi),Xb)) = aa(real,real,uminus_uminus(real),sin(real,Xb)) ) ).

% sin_periodic_pi2
tff(fact_2741_sin__periodic__pi,axiom,
    ! [Xb: real] : ( sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),pi)) = aa(real,real,uminus_uminus(real),sin(real,Xb)) ) ).

% sin_periodic_pi
tff(fact_2742_cos__minus__pi,axiom,
    ! [Xb: real] : ( aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),pi)) = aa(real,real,uminus_uminus(real),aa(real,real,cos(real),Xb)) ) ).

% cos_minus_pi
tff(fact_2743_cos__pi__minus,axiom,
    ! [Xb: real] : ( aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,cos(real),Xb)) ) ).

% cos_pi_minus
tff(fact_2744_sin__minus__pi,axiom,
    ! [Xb: real] : ( sin(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),pi)) = aa(real,real,uminus_uminus(real),sin(real,Xb)) ) ).

% sin_minus_pi
tff(fact_2745_odd__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> ~ ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ) ).

% odd_add
tff(fact_2746_even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ) ).

% even_add
tff(fact_2747_power__minus__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat,A2: A] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ).

% power_minus_odd
tff(fact_2748_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) ) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_2749_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( divide_divide(nat,aa(nat,nat,suc,Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% odd_Suc_div_two
tff(fact_2750_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( divide_divide(nat,aa(nat,nat,suc,Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ).

% even_Suc_div_two
tff(fact_2751_power__int__minus__left__odd,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Nb: int,A2: A] :
          ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)
         => ( power_int(A,aa(A,A,uminus_uminus(A),A2),Nb) = aa(A,A,uminus_uminus(A),power_int(A,A2,Nb)) ) ) ) ).

% power_int_minus_left_odd
tff(fact_2752_power__int__minus__left__even,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Nb: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)
         => ( power_int(A,aa(A,A,uminus_uminus(A),A2),Nb) = power_int(A,A2,Nb) ) ) ) ).

% power_int_minus_left_even
tff(fact_2753_sin__cos__squared__add3,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: 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),Xb)),aa(A,A,cos(A),Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Xb))) = one_one(A) ) ) ).

% sin_cos_squared_add3
tff(fact_2754_cos__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,cos(A),real_Vector_of_real(A,pi)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% cos_of_real_pi
tff(fact_2755_sin__npi__int,axiom,
    ! [Nb: int] : ( sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),Nb))) = zero_zero(real) ) ).

% sin_npi_int
tff(fact_2756_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,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2757_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),zero_zero(A))
        <=> ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq
tff(fact_2758_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,aa(nat,fun(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_2759_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)))
        <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% even_plus_one_iff
tff(fact_2760_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ).

% even_diff
tff(fact_2761_neg__one__odd__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Nb) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% neg_one_odd_power
tff(fact_2762_neg__one__even__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Nb) = one_one(A) ) ) ) ).

% neg_one_even_power
tff(fact_2763_even__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ) ) ).

% even_of_nat
tff(fact_2764_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))) = Nb ) ) ).

% odd_Suc_minus_one
tff(fact_2765_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2584673776208193580ke_bit(A,Nb,A2))
        <=> ( ( Nb = zero_zero(nat) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% even_take_bit_eq
tff(fact_2766_even__diff__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)) ) ) ).

% even_diff_nat
tff(fact_2767_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,aa(nat,fun(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,aa(nat,fun(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_2768_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( 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))) = divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_2
tff(fact_2769_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( 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))) = divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_two
tff(fact_2770_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( 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),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_2771_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% even_power
tff(fact_2772_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : ( bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,zero_zero(nat)),A2) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% take_bit_Suc_0
tff(fact_2773_odd__two__times__div__two__nat,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)) ) ) ).

% odd_two_times_div_two_nat
tff(fact_2774_sin__pi__half,axiom,
    sin(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) = one_one(real) ).

% sin_pi_half
tff(fact_2775_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_2776_cos__npi,axiom,
    ! [Nb: 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),Nb)),pi)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Nb) ) ).

% cos_npi
tff(fact_2777_cos__npi2,axiom,
    ! [Nb: 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),Nb))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Nb) ) ).

% cos_npi2
tff(fact_2778_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))),one_one(A)) = A2 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_2779_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,aa(nat,fun(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,aa(nat,fun(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_2780_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),one_one(A)))
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_2781_sin__cos__squared__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cos(A),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ) ).

% sin_cos_squared_add
tff(fact_2782_sin__cos__squared__add2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: 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),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ) ).

% sin_cos_squared_add2
tff(fact_2783_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),divide_divide(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_2784_sin__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,divide_divide(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_2785_cos__2npi,axiom,
    ! [Nb: 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),Nb))),pi)) = one_one(real) ) ).

% cos_2npi
tff(fact_2786_sin__2pi__minus,axiom,
    ! [Xb: 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)),Xb)) = aa(real,real,uminus_uminus(real),sin(real,Xb)) ) ).

% sin_2pi_minus
tff(fact_2787_sin__int__2pin,axiom,
    ! [Nb: int] : ( sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),aa(int,real,ring_1_of_int(real),Nb))) = zero_zero(real) ) ).

% sin_int_2pin
tff(fact_2788_cos__int__2pin,axiom,
    ! [Nb: 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),Nb))) = one_one(real) ) ).

% cos_int_2pin
tff(fact_2789_cos__npi__int,axiom,
    ! [Nb: 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),Nb))) = $ite(aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ) ).

% cos_npi_int
tff(fact_2790_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => ( 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))),Nb)) = divide_divide(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_2791_sin__3over2__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(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_2792_of__nat__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Ma: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),Nb) ) ) ).

% of_nat_dvd_iff
tff(fact_2793_dvd__trans,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ) ).

% dvd_trans
tff(fact_2794_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),A2) ) ).

% dvd_refl
tff(fact_2795_take__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,Ma: nat] : ( bit_se2584673776208193580ke_bit(A,Nb,aa(nat,A,semiring_1_of_nat(A),Ma)) = aa(nat,A,semiring_1_of_nat(A),bit_se2584673776208193580ke_bit(nat,Nb,Ma)) ) ) ).

% take_bit_of_nat
tff(fact_2796_take__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K: int] : ( bit_se2584673776208193580ke_bit(A,Nb,aa(int,A,ring_1_of_int(A),K)) = aa(int,A,ring_1_of_int(A),bit_se2584673776208193580ke_bit(int,Nb,K)) ) ) ).

% take_bit_of_int
tff(fact_2797_sin__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : ( sin(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),aa(A,A,cos(A),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),Xb)),sin(A,Y))) ) ) ).

% sin_diff
tff(fact_2798_sin__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : ( sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),aa(A,A,cos(A),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),Xb)),sin(A,Y))) ) ) ).

% sin_add
tff(fact_2799_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( aa(A,A,cos(A),Xb) = one_one(A) )
         => ( sin(A,Xb) = zero_zero(A) ) ) ) ).

% cos_one_sin_zero
tff(fact_2800_take__bit__add,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,A2: A,B2: A] : ( bit_se2584673776208193580ke_bit(A,Nb,aa(A,A,aa(A,fun(A,A),plus_plus(A),bit_se2584673776208193580ke_bit(A,Nb,A2)),bit_se2584673776208193580ke_bit(A,Nb,B2))) = bit_se2584673776208193580ke_bit(A,Nb,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ).

% take_bit_add
tff(fact_2801_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),zero_zero(A)),A2)
         => ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_2802_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
        <=> ( ( A2 = zero_zero(A) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_2803_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) ) ).

% dvd_triv_right
tff(fact_2804_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2) ) ) ).

% dvd_mult_right
tff(fact_2805_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),D3)
           => aa(A,$o,aa(A,fun(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),D3)) ) ) ) ).

% mult_dvd_mono
tff(fact_2806_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% dvd_triv_left
tff(fact_2807_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ).

% dvd_mult_left
tff(fact_2808_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ).

% dvd_mult2
tff(fact_2809_division__decomp,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ? [B7: A,C5: A] :
              ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B7),C5) )
              & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B7),B2)
              & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C5),C2) ) ) ) ).

% division_decomp
tff(fact_2810_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ).

% dvd_mult
tff(fact_2811_dvd__productE,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [P3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),P3),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
         => ~ ! [X: A,Y3: A] :
                ( ( P3 = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y3) )
               => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X),A2)
                 => ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Y3),B2) ) ) ) ) ).

% dvd_productE
tff(fact_2812_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(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_2813_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,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ).

% dvdI
tff(fact_2814_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(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_2815_take__bit__minus,axiom,
    ! [Nb: nat,K: int] : ( bit_se2584673776208193580ke_bit(int,Nb,aa(int,int,uminus_uminus(int),bit_se2584673776208193580ke_bit(int,Nb,K))) = bit_se2584673776208193580ke_bit(int,Nb,aa(int,int,uminus_uminus(int),K)) ) ).

% take_bit_minus
tff(fact_2816_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A)) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_2817_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ).

% unit_imp_dvd
tff(fact_2818_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),one_one(A)),A2) ) ).

% one_dvd
tff(fact_2819_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ) ).

% dvd_add
tff(fact_2820_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% dvd_add_left_iff
tff(fact_2821_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ) ).

% dvd_add_right_iff
tff(fact_2822_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),Z)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z)) ) ) ) ).

% dvd_diff
tff(fact_2823_dvd__diff__commute,axiom,
    ! [A: $tType] :
      ( euclid5891614535332579305n_ring(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) ) ) ).

% dvd_diff_commute
tff(fact_2824_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
           => ( divide_divide(A,divide_divide(A,A2,D3),divide_divide(A,B2,D3)) = divide_divide(A,A2,B2) ) ) ) ) ).

% div_div_div_same
tff(fact_2825_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( divide_divide(A,A2,C2) = divide_divide(A,B2,C2) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2826_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
           => ( ( divide_divide(A,A2,C2) = divide_divide(A,B2,C2) )
            <=> ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_2827_take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : ( bit_se2584673776208193580ke_bit(int,Nb,aa(int,int,aa(int,fun(int,int),minus_minus(int),bit_se2584673776208193580ke_bit(int,Nb,K)),bit_se2584673776208193580ke_bit(int,Nb,L))) = bit_se2584673776208193580ke_bit(int,Nb,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)) ) ).

% take_bit_diff
tff(fact_2828_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),zero_zero(nat)),A2)
     => ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_2829_gcd__nat_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),zero_zero(nat))
        & ( A2 != zero_zero(nat) ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_2830_gcd__nat_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),zero_zero(nat)),A2)
    <=> ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_unique
tff(fact_2831_gcd__nat_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),zero_zero(nat)),A2)
        & ( zero_zero(nat) != A2 ) ) ).

% gcd_nat.extremum_strict
tff(fact_2832_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),zero_zero(nat)) ).

% gcd_nat.extremum
tff(fact_2833_dvd__if__abs__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [L: A,K: A] :
          ( ( aa(A,A,abs_abs(A),L) = aa(A,A,abs_abs(A),K) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),L),K) ) ) ).

% dvd_if_abs_eq
tff(fact_2834_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),modulo_modulo(A,A2,B2))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A2) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_2835_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),modulo_modulo(A,A2,B2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A2) ) ) ) ).

% dvd_mod_iff
tff(fact_2836_dvd__diff__nat,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_2837_uminus__dvd__conv_I2_J,axiom,
    ! [D3: int,Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),Ta)
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,uminus_uminus(int),Ta)) ) ).

% uminus_dvd_conv(2)
tff(fact_2838_uminus__dvd__conv_I1_J,axiom,
    ! [D3: int,Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),Ta)
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,uminus_uminus(int),D3)),Ta) ) ).

% uminus_dvd_conv(1)
tff(fact_2839_zdvd__zdiffD,axiom,
    ! [K: int,Ma: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),aa(int,int,aa(int,fun(int,int),minus_minus(int),Ma),Nb))
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),Nb)
       => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),Ma) ) ) ).

% zdvd_zdiffD
tff(fact_2840_zdvd__antisym__abs,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),B2)
     => ( aa(int,$o,aa(int,fun(int,$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_2841_cos__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : ( aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),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),Xb)),aa(A,A,cos(A),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Y))) ) ) ).

% cos_diff
tff(fact_2842_cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : ( aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),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),Xb)),aa(A,A,cos(A),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Y))) ) ) ).

% cos_add
tff(fact_2843_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sin(A,Xb) = zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,cos(A),Xb)) = one_one(real) ) ) ) ).

% sin_zero_norm_cos_one
tff(fact_2844_sin__zero__abs__cos__one,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
     => ( aa(real,real,abs_abs(real),aa(real,real,cos(real),Xb)) = one_one(real) ) ) ).

% sin_zero_abs_cos_one
tff(fact_2845_sincos__principal__value,axiom,
    ! [Xb: real] :
    ? [Y3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Y3)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),pi)
      & ( sin(real,Y3) = sin(real,Xb) )
      & ( aa(real,real,cos(real),Y3) = aa(real,real,cos(real),Xb) ) ) ).

% sincos_principal_value
tff(fact_2846_sin__le__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xb)),one_one(real)) ).

% sin_le_one
tff(fact_2847_take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),bit_se2584673776208193580ke_bit(int,Nb,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_2848_take__bit__nonnegative,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se2584673776208193580ke_bit(int,Nb,K)) ).

% take_bit_nonnegative
tff(fact_2849_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),bit_se2584673776208193580ke_bit(int,Nb,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_2850_not__take__bit__negative,axiom,
    ! [Nb: nat,K: int] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se2584673776208193580ke_bit(int,Nb,K)),zero_zero(int)) ).

% not_take_bit_negative
tff(fact_2851_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( ( bit_ri4674362597316999326ke_bit(A,Nb,A2) = bit_ri4674362597316999326ke_bit(A,Nb,B2) )
        <=> ( bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb),A2) = bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb),B2) ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_2852_cos__le__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,cos(real),Xb)),one_one(real)) ).

% cos_le_one
tff(fact_2853_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),zero_zero(A)),one_one(A)) ) ).

% not_is_unit_0
tff(fact_2854_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X3)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S)) ) ) ) ).

% pinf(9)
tff(fact_2855_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X3)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S))
          <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S)) ) ) ) ).

% pinf(10)
tff(fact_2856_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S)) ) ) ) ).

% minf(9)
tff(fact_2857_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z2)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S))
          <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S)) ) ) ) ).

% minf(10)
tff(fact_2858_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
         => ( ( divide_divide(A,A2,B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_2859_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
            & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A)) ) ) ) ).

% is_unit_mult_iff
tff(fact_2860_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_2861_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_2862_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_2863_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_2864_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(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_2865_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(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_2866_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,D3: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),C2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),divide_divide(A,C2,D3)) = 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),D3)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_2867_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),divide_divide(A,B2,C2)) ) ) ).

% dvd_mult_imp_div
tff(fact_2868_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A2)
         => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,divide_divide(A,A2,B2),C2) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_2869_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
           => ( divide_divide(A,A2,divide_divide(A,B2,C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),C2) ) ) ) ) ).

% div_div_eq_right
tff(fact_2870_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ) ) ).

% div_mult_swap
tff(fact_2871_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,C2)),A2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),C2) ) ) ) ).

% dvd_div_mult
tff(fact_2872_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),divide_divide(A,C2,B2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ) ).

% dvd_div_unit_iff
tff(fact_2873_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),divide_divide(A,A2,B2)),C2)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ) ).

% div_unit_dvd_iff
tff(fact_2874_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( ( divide_divide(A,B2,A2) = divide_divide(A,C2,A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_2875_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A2)
         => ( 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),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_2876_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
         => ( 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),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_2877_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
         => ( divide_divide(A,A2,aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) ) ) ) ).

% dvd_div_neg
tff(fact_2878_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
         => ( divide_divide(A,aa(A,A,uminus_uminus(A),A2),B2) = aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) ) ) ) ).

% dvd_neg_div
tff(fact_2879_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ).

% mod_0_imp_dvd
tff(fact_2880_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
        <=> ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_2881_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_2882_mod__eq__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,B2,C2) )
        <=> aa(A,$o,aa(A,fun(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_2883_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),modulo_modulo(A,A2,B2))) ) ).

% dvd_minus_mod
tff(fact_2884_dvd__pos__nat,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma) ) ) ).

% dvd_pos_nat
tff(fact_2885_nat__dvd__not__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
       => ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),Ma) ) ) ).

% nat_dvd_not_less
tff(fact_2886_dvd__minus__self,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma)
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),Nb) ) ) ).

% dvd_minus_self
tff(fact_2887_zdvd__antisym__nonneg,axiom,
    ! [Ma: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ma)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
       => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Ma),Nb)
         => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Nb),Ma)
           => ( Ma = Nb ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_2888_zdvd__not__zless,axiom,
    ! [Ma: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ma)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ma),Nb)
       => ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Nb),Ma) ) ) ).

% zdvd_not_zless
tff(fact_2889_less__eq__dvd__minus,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),Nb)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma)) ) ) ).

% less_eq_dvd_minus
tff(fact_2890_dvd__diffD1,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Ma)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_2891_dvd__diffD,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Ma) ) ) ) ).

% dvd_diffD
tff(fact_2892_zdvd__mono,axiom,
    ! [K: int,Ma: int,Ta: int] :
      ( ( K != zero_zero(int) )
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Ma),Ta)
      <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),Ma)),aa(int,int,aa(int,fun(int,int),times_times(int),K),Ta)) ) ) ).

% zdvd_mono
tff(fact_2893_zdvd__mult__cancel,axiom,
    ! [K: int,Ma: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),Ma)),aa(int,int,aa(int,fun(int,int),times_times(int),K),Nb))
     => ( ( K != zero_zero(int) )
       => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Ma),Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_2894_bezout__lemma__nat,axiom,
    ! [D3: nat,A2: nat,B2: nat,Xb: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),A2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),B2)
       => ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),D3) )
            | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y)),D3) ) )
         => ? [X: nat,Y3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),A2)
              & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),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)),D3) )
                | ( 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)),D3) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_2895_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D5: nat,X: nat,Y3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D5),A2)
      & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D5),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)),D5) )
        | ( 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)),D5) ) ) ) ).

% bezout_add_nat
tff(fact_2896_bezout1__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D5: nat,X: nat,Y3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D5),A2)
      & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D5),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)) = D5 )
        | ( 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)) = D5 ) ) ) ).

% bezout1_nat
tff(fact_2897_zdvd__reduce,axiom,
    ! [K: int,Nb: int,Ma: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),aa(int,int,aa(int,fun(int,int),times_times(int),K),Ma)))
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),Nb) ) ).

% zdvd_reduce
tff(fact_2898_zdvd__period,axiom,
    ! [A2: int,D3: int,Xb: int,Ta: int,C2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),D3)
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Ta))
      <=> aa(int,$o,aa(int,fun(int,$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),Xb),aa(int,int,aa(int,fun(int,int),times_times(int),C2),D3))),Ta)) ) ) ).

% zdvd_period
tff(fact_2899_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( ( bit_se2584673776208193580ke_bit(A,Nb,A2) = zero_zero(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),A2) ) ) ).

% take_bit_eq_0_iff
tff(fact_2900_cos__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Ma: int,Xb: 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),Ma)),real_Vector_of_real(A,Xb))) = 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),Ma)),Xb))) ) ) ).

% cos_int_times_real
tff(fact_2901_sin__cos__le1,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),sin(real,Xb)),sin(real,Y))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,cos(real),Xb)),aa(real,real,cos(real),Y))))),one_one(real)) ).

% sin_cos_le1
tff(fact_2902_even__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] :
          ( aa(A,$o,aa(A,fun(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,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K) ) ) ).

% even_of_int_iff
tff(fact_2903_sin__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Ma: int,Xb: real] : ( sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Ma)),real_Vector_of_real(A,Xb))) = real_Vector_of_real(A,sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),Ma)),Xb))) ) ) ).

% sin_int_times_real
tff(fact_2904_nat__dvd__iff,axiom,
    ! [Z: int,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(int,nat,nat2,Z)),Ma)
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z),aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Z),aa(nat,int,semiring_1_of_nat(int),Ma)),Ma = zero_zero(nat)) ) ).

% nat_dvd_iff
tff(fact_2905_sin__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xb)),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),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% sin_squared_eq
tff(fact_2906_cos__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,cos(A),Xb)),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),sin(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% cos_squared_eq
tff(fact_2907_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Ma: nat,Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,Nb))
         => ( bit_se2584673776208193580ke_bit(A,Ma,bit_ri4674362597316999326ke_bit(A,Nb,A2)) = bit_se2584673776208193580ke_bit(A,Ma,A2) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_2908_sin__x__ge__neg__x,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),Xb)),sin(real,Xb)) ) ).

% sin_x_ge_neg_x
tff(fact_2909_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [C3: A] : ( B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) ) ) ) ) ).

% unit_dvdE
tff(fact_2910_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,aa(A,fun(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_2911_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A,D3: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),D3)
               => ( ( divide_divide(A,B2,A2) = divide_divide(A,D3,C2) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),D3) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_2912_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),divide_divide(A,B2,C2))
            <=> aa(A,$o,aa(A,fun(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_2913_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),divide_divide(A,A2,B2)),C2)
            <=> aa(A,$o,aa(A,fun(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_2914_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
           => ( ( 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_2915_sin__ge__minus__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),sin(real,Xb)) ).

% sin_ge_minus_one
tff(fact_2916_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( ( divide_divide(A,A2,B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_2917_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D7: A,Ta: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),D7)
         => ! [X3: A,K4: A] :
              ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Ta))
            <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D7))),Ta)) ) ) ) ).

% inf_period(4)
tff(fact_2918_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D7: A,Ta: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),D7)
         => ! [X3: A,K4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Ta))
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D7))),Ta)) ) ) ) ).

% inf_period(3)
tff(fact_2919_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(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_2920_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( ( A2 = divide_divide(A,C2,B2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_2921_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,divide_divide(A,A2,B2),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_2922_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),C2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),B2) ) ) ) ).

% unit_div_commute
tff(fact_2923_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_2924_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),one_one(A))
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,divide_divide(A,A2,B2),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2925_cos__ge__minus__one,axiom,
    ! [Xb: 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),Xb)) ).

% cos_ge_minus_one
tff(fact_2926_abs__sin__le__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,Xb))),one_one(real)) ).

% abs_sin_le_one
tff(fact_2927_take__bit__decr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,Nb,K) != zero_zero(int) )
     => ( bit_se2584673776208193580ke_bit(int,Nb,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),bit_se2584673776208193580ke_bit(int,Nb,K)),one_one(int)) ) ) ).

% take_bit_decr_eq
tff(fact_2928_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( modulo_modulo(A,A2,B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_2929_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_2930_abs__cos__le__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,cos(real),Xb))),one_one(real)) ).

% abs_cos_le_one
tff(fact_2931_sin__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W)),sin(A,Z)) = 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_2932_sin__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W)),aa(A,A,cos(A),Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),sin(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% sin_times_cos
tff(fact_2933_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)),sin(A,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),sin(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% cos_times_sin
tff(fact_2934_sin__plus__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,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),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_2935_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),sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,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),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_2936_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))),sin(A,divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z),aa(num,A,numeral_numeral(A),bit0(one2)))))),sin(A,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_2937_cos__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: 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))),Xb)) = 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),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% cos_double
tff(fact_2938_dvd__imp__le,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ) ).

% dvd_imp_le
tff(fact_2939_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Nb: nat] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,Nb))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb))) ) ).

% fact_fact_dvd_fact
tff(fact_2940_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_2941_dvd__mult__cancel,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_2942_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [D5: nat,X: nat,Y3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D5),A2)
          & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D5),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)),D5) ) ) ) ).

% bezout_add_strong_nat
tff(fact_2943_sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( sin(A,Xb) = aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),bit0(one2)))),Xb)) ) ) ).

% sin_cos_eq
tff(fact_2944_cos__sin__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( aa(A,A,cos(A),Xb) = sin(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),bit0(one2)))),Xb)) ) ) ).

% cos_sin_eq
tff(fact_2945_zdvd__imp__le,axiom,
    ! [Z: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Z),Nb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),Nb) ) ) ).

% zdvd_imp_le
tff(fact_2946_mod__greater__zero__iff__not__dvd,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,Ma,Nb))
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),Ma) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_2947_dvd__imp__le__int,axiom,
    ! [I: int,D3: int] :
      ( ( I != zero_zero(int) )
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),I)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),D3)),aa(int,int,abs_abs(int),I)) ) ) ).

% dvd_imp_le_int
tff(fact_2948_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,Ma: nat,Q4: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
     => ( ( modulo_modulo(nat,Ma,Q4) = modulo_modulo(nat,Nb,Q4) )
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Q4),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_2949_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: int,A2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),B2),A2)
         => member(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_2950_real__of__nat__div,axiom,
    ! [D3: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),Nb)
     => ( aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Nb,D3)) = divide_divide(real,aa(nat,real,semiring_1_of_nat(real),Nb),aa(nat,real,semiring_1_of_nat(real),D3)) ) ) ).

% real_of_nat_div
tff(fact_2951_real__of__int__div,axiom,
    ! [D3: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),Nb)
     => ( aa(int,real,ring_1_of_int(real),divide_divide(int,Nb,D3)) = divide_divide(real,aa(int,real,ring_1_of_int(real),Nb),aa(int,real,ring_1_of_int(real),D3)) ) ) ).

% real_of_int_div
tff(fact_2952_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
       => ( aa(int,int,sgn_sgn(int),modulo_modulo(int,K,L)) = aa(int,int,sgn_sgn(int),L) ) ) ) ).

% sgn_mod
tff(fact_2953_dvd__fact,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),semiring_char_0_fact(nat,Nb)) ) ) ).

% dvd_fact
tff(fact_2954_even__nat__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(int,nat,nat2,K))
      <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K) ) ) ).

% even_nat_iff
tff(fact_2955_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),sin(A,W)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% cos_double_sin
tff(fact_2956_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( bit_se2584673776208193580ke_bit(A,Nb,A2) = $ite(aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2),zero_zero(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),one_one(A))) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_2957_minus__sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( aa(A,A,uminus_uminus(A),sin(A,Xb)) = aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ) ).

% minus_sin_cos_eq
tff(fact_2958_even__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),zero_zero(A)) ) ).

% even_zero
tff(fact_2959_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [B4: A] :
                  ( ( B4 != zero_zero(A) )
                 => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B4),one_one(A))
                   => ( ( divide_divide(A,one_one(A),A2) = B4 )
                     => ( ( divide_divide(A,one_one(A),B4) = A2 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B4) = one_one(A) )
                         => ( divide_divide(A,C2,A2) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B4) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_2960_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = divide_divide(A,one_one(A),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_2961_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) = divide_divide(A,one_one(A),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_2962_sin__eq__0__pi,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),pi)
       => ( ( sin(real,Xb) = zero_zero(real) )
         => ( Xb = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_2963_odd__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),one_one(A)) ) ).

% odd_one
tff(fact_2964_odd__even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% odd_even_add
tff(fact_2965_even__minus,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% even_minus
tff(fact_2966_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xb: 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),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),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),Xb)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_2967_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Xb: A,Ma: nat,Nb: nat] :
          ( ( Xb != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Ma)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb))
          <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),one_one(A))
              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ) ) ) ).

% dvd_power_iff
tff(fact_2968_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: nat,Xb: A] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
            | ( Xb = one_one(A) ) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)) ) ) ).

% dvd_power
tff(fact_2969_sin__zero__iff__int2,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
    <=> ? [I3: int] : ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I3)),pi) ) ) ).

% sin_zero_iff_int2
tff(fact_2970_div2__even__ext__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( divide_divide(nat,Xb,aa(num,nat,numeral_numeral(nat),bit0(one2))) = divide_divide(nat,Y,aa(num,nat,numeral_numeral(nat),bit0(one2))) )
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xb)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Y) )
       => ( Xb = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_2971_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)))),semiring_char_0_fact(A,Nb)) ) ) ).

% choose_dvd
tff(fact_2972_dvd__mult__cancel2,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Ma)),Ma)
      <=> ( Nb = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_2973_dvd__mult__cancel1,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb)),Ma)
      <=> ( Nb = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_2974_dvd__minus__add,axiom,
    ! [Q4: nat,Nb: nat,R2: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q4),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q4),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),Ma))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Q4))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),Ma)),Q4))) ) ) ) ).

% dvd_minus_add
tff(fact_2975_sincos__total__pi,axiom,
    ! [Y: real,Xb: 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),Xb),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) )
       => ? [T2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),pi)
            & ( Xb = aa(real,real,cos(real),T2) )
            & ( Y = sin(real,T2) ) ) ) ) ).

% sincos_total_pi
tff(fact_2976_power__dvd__imp__le,axiom,
    ! [I: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Nb))
     => ( 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),Ma),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_2977_mod__nat__eqI,axiom,
    ! [R2: nat,Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R2),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),Ma)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),R2))
         => ( modulo_modulo(nat,Ma,Nb) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_2978_sin__expansion__lemma,axiom,
    ! [Xb: real,Ma: nat] : ( sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),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,Ma))),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),Xb),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Ma)),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ).

% sin_expansion_lemma
tff(fact_2979_zdvd__mult__cancel1,axiom,
    ! [Ma: int,Nb: int] :
      ( ( Ma != zero_zero(int) )
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),Ma),Nb)),Ma)
      <=> ( aa(int,int,abs_abs(int),Nb) = one_one(int) ) ) ) ).

% zdvd_mult_cancel1
tff(fact_2980_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
        | ( ( L = zero_zero(int) )
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) )
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L) ) ) ).

% mod_int_pos_iff
tff(fact_2981_sin__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
    <=> ? [I3: int] :
          ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),I3)
          & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I3)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_2982_cos__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa(real,real,cos(real),Xb) = zero_zero(real) )
    <=> ? [I3: int] :
          ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),I3)
          & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I3)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_2983_aset_I10_J,axiom,
    ! [D3: int,D7: int,A3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),D7)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A3)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
         => ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),Ta))
           => ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D7)),Ta)) ) ) ) ).

% aset(10)
tff(fact_2984_aset_I9_J,axiom,
    ! [D3: int,D7: int,A3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),D7)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A3)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),Ta))
           => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D7)),Ta)) ) ) ) ).

% aset(9)
tff(fact_2985_bset_I10_J,axiom,
    ! [D3: int,D7: int,B3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),D7)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B3)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
         => ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),Ta))
           => ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D7)),Ta)) ) ) ) ).

% bset(10)
tff(fact_2986_bset_I9_J,axiom,
    ! [D3: int,D7: int,B3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),D7)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D7))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B3)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),Ta))
           => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D7)),Ta)) ) ) ) ).

% bset(9)
tff(fact_2987_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : ( bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,Nb,aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% take_bit_Suc_bit0
tff(fact_2988_cos__expansion__lemma,axiom,
    ! [Xb: real,Ma: nat] : ( aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),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,Ma))),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) = aa(real,real,uminus_uminus(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Ma)),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))) ) ).

% cos_expansion_lemma
tff(fact_2989_sin__zero__iff,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
    <=> ( ? [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)
            & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N4)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)
            & ( Xb = 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),N4)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_2990_cos__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa(real,real,cos(real),Xb) = zero_zero(real) )
    <=> ( ? [N4: nat] :
            ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)
            & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N4)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N4: nat] :
            ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)
            & ( Xb = 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),N4)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_2991_take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se2584673776208193580ke_bit(int,Nb,K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ).

% take_bit_int_less_exp
tff(fact_2992_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_2993_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,Xb: fun(nat,fun(A,A)),Xaa: nat,Xba: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xb,Xaa,Xba,Xc) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xba),Xaa),Xc,set_fo6178422350223883121st_nat(A,Xb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xaa),one_one(nat)),Xba,aa(A,A,aa(nat,fun(A,A),Xb,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_2994_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xb: 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),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),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),Xb)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_2995_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 )
            & ! [Y4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),pi)
                  & ( aa(real,real,cos(real),Y4) = Y ) )
               => ( Y4 = X ) ) ) ) ) ).

% cos_total
tff(fact_2996_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_2997_odd__iff__mod__2__eq__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_2998_uminus__power__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))) ) ) ).

% uminus_power_if
tff(fact_2999_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% odd_pos
tff(fact_3000_power__int__minus__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Nb: int] :
          ( power_int(A,aa(A,A,uminus_uminus(A),A2),Nb) = $ite(aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb),power_int(A,A2,Nb),aa(A,A,uminus_uminus(A),power_int(A,A2,Nb))) ) ) ).

% power_int_minus_left
tff(fact_3001_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2638667681897837118et_bit(A,Ma,A2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            | ( Ma = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_3002_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se5668285175392031749et_bit(A,Ma,A2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            & ( Ma != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_3003_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,Ma,A2))
        <=> ~ ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            <=> ( Ma = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_3004_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$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,$o,aa(int,fun(int,$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_3005_sincos__total__pi__half,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( 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),Xb),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) )
         => ? [T2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T2)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
              & ( Xb = aa(real,real,cos(real),T2) )
              & ( Y = sin(real,T2) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_3006_sincos__total__2pi__le,axiom,
    ! [Xb: 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),Xb),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) )
     => ? [T2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T2)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
          & ( Xb = aa(real,real,cos(real),T2) )
          & ( Y = sin(real,T2) ) ) ) ).

% sincos_total_2pi_le
tff(fact_3007_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$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,$o,aa(int,fun(int,$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_3008_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$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,$o,aa(int,fun(int,$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_3009_sincos__total__2pi,axiom,
    ! [Xb: 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),Xb),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) )
     => ~ ! [T2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( ( Xb = aa(real,real,cos(real),T2) )
               => ( Y != sin(real,T2) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_3010_take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : ( bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_se2584673776208193580ke_bit(int,Nb,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_3011_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),bit_se2584673776208193580ke_bit(int,Nb,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))),Nb)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_3012_take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se2584673776208193580ke_bit(int,Nb,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))),Nb)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_3013_cos__integer__2pi,axiom,
    ! [Nb: real] :
      ( member(real,Nb,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)),Nb)) = one_one(real) ) ) ).

% cos_integer_2pi
tff(fact_3014_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ~ ! [B4: 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))),B4)),one_one(A)) ) ) ) ).

% oddE
tff(fact_3015_sin__pi__divide__n__ge__0,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,divide_divide(real,pi,aa(nat,real,semiring_1_of_nat(real),Nb)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_3016_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = $ite(aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2),zero_zero(A),one_one(A)) ) ) ).

% mod2_eq_if
tff(fact_3017_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
           => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) ) )
         => ~ ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
             => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_3018_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq
tff(fact_3019_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ).

% zero_le_odd_power
tff(fact_3020_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => aa(A,$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),Nb)) ) ) ).

% zero_le_even_power
tff(fact_3021_minus__one__power__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Nb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% minus_one_power_iff
tff(fact_3022_central__binomial__odd,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(nat,nat,binomial(Nb),aa(nat,nat,suc,divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))) = aa(nat,nat,binomial(Nb),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% central_binomial_odd
tff(fact_3023_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)) = 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_3024_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),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),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_3025_cos__double__less__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),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))),Xb))),one_one(real)) ) ) ).

% cos_double_less_one
tff(fact_3026_sin__30,axiom,
    sin(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2))))) = divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sin_30
tff(fact_3027_sin__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),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),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),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( ( sin(real,Xb) = sin(real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_3028_sin__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),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),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),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xb)),sin(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_3029_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(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),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Y)),sin(real,Xb)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_3030_take__bit__int__eq__self,axiom,
    ! [K: int,Nb: 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))),Nb))
       => ( bit_se2584673776208193580ke_bit(int,Nb,K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_3031_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,Nb,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))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_3032_cos__60,axiom,
    aa(real,real,cos(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% cos_60
tff(fact_3033_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : ( bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_se2584673776208193580ke_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ).

% take_bit_numeral_minus_bit0
tff(fact_3034_take__bit__incr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,Nb,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))),Nb)),one_one(int)) )
     => ( bit_se2584673776208193580ke_bit(int,Nb,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)),bit_se2584673776208193580ke_bit(int,Nb,K)) ) ) ).

% take_bit_incr_eq
tff(fact_3035_cos__one__2pi__int,axiom,
    ! [Xb: real] :
      ( ( aa(real,real,cos(real),Xb) = one_one(real) )
    <=> ? [X4: int] : ( Xb = 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_3036_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
        <=> ( ( Nb = zero_zero(nat) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
              & ( A2 != zero_zero(A) ) )
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_less_power_eq
tff(fact_3037_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_3038_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : ( bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb),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,Nb))),one_one(A)) ) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_3039_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : ( bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,Nb,aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ) ).

% take_bit_Suc_bit1
tff(fact_3040_take__bit__numeral__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : ( bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),K),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,nat,numeral_numeral(nat),K))),one_one(A)) ) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_3041_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : ( bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,Nb,divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% take_bit_Suc
tff(fact_3042_cos__treble__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: 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))),Xb)) = 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),Xb)),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),Xb))) ) ) ).

% cos_treble_cos
tff(fact_3043_sin__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xb)),zero_zero(real)) ) ) ).

% sin_less_zero
tff(fact_3044_sin__monotone__2pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(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),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Y)),sin(real,Xb)) ) ) ) ).

% sin_monotone_2pi
tff(fact_3045_sin__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),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),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),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xb)),sin(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_3046_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),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),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( sin(real,X) = Y )
            & ! [Y4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
                  & ( sin(real,Y4) = Y ) )
               => ( Y4 = X ) ) ) ) ) ).

% sin_total
tff(fact_3047_take__bit__int__less__eq,axiom,
    ! [Nb: 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))),Nb)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),bit_se2584673776208193580ke_bit(int,Nb,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))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_3048_cos__gt__zero__pi,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),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),Xb)) ) ) ).

% cos_gt_zero_pi
tff(fact_3049_cos__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),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),Xb)) ) ) ).

% cos_ge_zero
tff(fact_3050_take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: 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))),Nb))),bit_se2584673776208193580ke_bit(int,Nb,K)) ) ).

% take_bit_int_greater_eq
tff(fact_3051_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Ma: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ma)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ) ).

% even_mask_div_iff'
tff(fact_3052_cos__one__2pi,axiom,
    ! [Xb: real] :
      ( ( aa(real,real,cos(real),Xb) = one_one(real) )
    <=> ( ? [X4: nat] : ( Xb = 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] : ( Xb = 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_3053_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K: int] : ( bit_ri4674362597316999326ke_bit(int,Nb,K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb),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))),Nb)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_3054_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
            & ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
              | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3055_even__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% even_mod_4_div_2
tff(fact_3056_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : ( bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ) ).

% take_bit_numeral_bit1
tff(fact_3057_take__bit__minus__small__eq,axiom,
    ! [K: int,Nb: 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))),Nb))
       => ( bit_se2584673776208193580ke_bit(int,Nb,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))),Nb)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_3058_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ma: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ma)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) = zero_zero(A) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ) ) ).

% even_mask_div_iff
tff(fact_3059_odd__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
     => ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% odd_mod_4_div_2
tff(fact_3060_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => 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),Nb)),Xb))),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)),Xb)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_3061_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : ( bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb),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),bit_se2584673776208193580ke_bit(int,Nb,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_3062_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Ma: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ma)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma)
            | ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) = zero_zero(A) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
              & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),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),Nb),Ma)))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3063_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( bit_se2584673776208193580ke_bit(A,Nb,A2) = $ite(Nb = 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),bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ) ).

% take_bit_rec
tff(fact_3064_fact__code,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : ( semiring_char_0_fact(A,Nb) = aa(nat,A,semiring_1_of_nat(A),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb,one_one(nat))) ) ) ).

% fact_code
tff(fact_3065_sin__coeff__def,axiom,
    ! [X3: nat] :
      ( sin_coeff(X3) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),X3),zero_zero(real),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X3),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),semiring_char_0_fact(real,X3))) ) ).

% sin_coeff_def
tff(fact_3066_cos__coeff__def,axiom,
    ! [X3: nat] :
      ( aa(nat,real,cos_coeff,X3) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),X3),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),divide_divide(nat,X3,aa(num,nat,numeral_numeral(nat),bit0(one2)))),semiring_char_0_fact(real,X3)),zero_zero(real)) ) ).

% cos_coeff_def
tff(fact_3067_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( aa(A,A,cos(A),Xb) != 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))),Xb)) != 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))),Xb)) = 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),Xb)),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),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% tan_double
tff(fact_3068_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( Z != complex2(aa(real,real,cos(real),T2),sin(real,T2)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_3069_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_3070_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),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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_3071_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_3072_tan__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( aa(A,A,tan(A),aa(A,A,uminus_uminus(A),Xb)) = aa(A,A,uminus_uminus(A),aa(A,A,tan(A),Xb)) ) ) ).

% tan_minus
tff(fact_3073_arccos__1,axiom,
    aa(real,real,arccos,one_one(real)) = zero_zero(real) ).

% arccos_1
tff(fact_3074_sin__coeff__0,axiom,
    sin_coeff(zero_zero(nat)) = zero_zero(real) ).

% sin_coeff_0
tff(fact_3075_cos__coeff__0,axiom,
    aa(nat,real,cos_coeff,zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_3076_tan__periodic__int,axiom,
    ! [Xb: real,I: int] : ( aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),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),Xb) ) ).

% tan_periodic_int
tff(fact_3077_arccos__minus__1,axiom,
    aa(real,real,arccos,aa(real,real,uminus_uminus(real),one_one(real))) = pi ).

% arccos_minus_1
tff(fact_3078_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_3079_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_3080_norm__cos__sin,axiom,
    ! [Ta: real] : ( real_V7770717601297561774m_norm(complex,complex2(aa(real,real,cos(real),Ta),sin(real,Ta))) = one_one(real) ) ).

% norm_cos_sin
tff(fact_3081_arcsin__1,axiom,
    aa(real,real,arcsin,one_one(real)) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))) ).

% arcsin_1
tff(fact_3082_arcsin__minus__1,axiom,
    aa(real,real,arcsin,aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% arcsin_minus_1
tff(fact_3083_dvd__antisym,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),Ma)
       => ( Ma = Nb ) ) ) ).

% dvd_antisym
tff(fact_3084_gcd__nat_Onot__eq__order__implies__strict,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != B2 )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
          & ( A2 != B2 ) ) ) ) ).

% gcd_nat.not_eq_order_implies_strict
tff(fact_3085_gcd__nat_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
     => ( A2 != B2 ) ) ).

% gcd_nat.strict_implies_not_eq
tff(fact_3086_gcd__nat_Ostrict__implies__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2) ) ).

% gcd_nat.strict_implies_order
tff(fact_3087_gcd__nat_Ostrict__iff__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) ) ) ).

% gcd_nat.strict_iff_order
tff(fact_3088_gcd__nat_Oorder__iff__strict,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
          & ( A2 != B2 ) )
        | ( A2 = B2 ) ) ) ).

% gcd_nat.order_iff_strict
tff(fact_3089_gcd__nat_Ostrict__iff__not,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),A2) ) ) ).

% gcd_nat.strict_iff_not
tff(fact_3090_gcd__nat_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),C2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2)
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans2
tff(fact_3091_gcd__nat_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),C2)
          & ( B2 != C2 ) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2)
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans1
tff(fact_3092_gcd__nat_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),C2)
          & ( B2 != C2 ) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2)
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans
tff(fact_3093_gcd__nat_Oantisym,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),A2)
       => ( A2 = B2 ) ) ) ).

% gcd_nat.antisym
tff(fact_3094_gcd__nat_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),A2)
        & ( A2 != A2 ) ) ).

% gcd_nat.irrefl
tff(fact_3095_gcd__nat_Oeq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),A2) ) ) ).

% gcd_nat.eq_iff
tff(fact_3096_gcd__nat_Otrans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),C2)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2) ) ) ).

% gcd_nat.trans
tff(fact_3097_gcd__nat_Orefl,axiom,
    ! [A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),A2) ).

% gcd_nat.refl
tff(fact_3098_gcd__nat_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
     => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),A2)
          & ( B2 != A2 ) ) ) ).

% gcd_nat.asym
tff(fact_3099_complex__minus,axiom,
    ! [A2: real,B2: real] : ( aa(complex,complex,uminus_uminus(complex),complex2(A2,B2)) = complex2(aa(real,real,uminus_uminus(real),A2),aa(real,real,uminus_uminus(real),B2)) ) ).

% complex_minus
tff(fact_3100_one__complex_Ocode,axiom,
    one_one(complex) = complex2(one_one(real),zero_zero(real)) ).

% one_complex.code
tff(fact_3101_Complex__eq__1,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = one_one(complex) )
    <=> ( ( A2 = one_one(real) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_1
tff(fact_3102_sin__coeff__Suc,axiom,
    ! [Nb: nat] : ( sin_coeff(aa(nat,nat,suc,Nb)) = divide_divide(real,aa(nat,real,cos_coeff,Nb),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb))) ) ).

% sin_coeff_Suc
tff(fact_3103_cot__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( aa(A,A,cot(A),Xb) = aa(A,A,inverse_inverse(A),aa(A,A,tan(A),Xb)) ) ) ).

% cot_altdef
tff(fact_3104_tan__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( aa(A,A,tan(A),Xb) = aa(A,A,inverse_inverse(A),aa(A,A,cot(A),Xb)) ) ) ).

% tan_altdef
tff(fact_3105_Complex__eq__neg__1,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) )
    <=> ( ( A2 = aa(real,real,uminus_uminus(real),one_one(real)) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_1
tff(fact_3106_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_3107_cos__coeff__Suc,axiom,
    ! [Nb: nat] : ( aa(nat,real,cos_coeff,aa(nat,nat,suc,Nb)) = divide_divide(real,aa(real,real,uminus_uminus(real),sin_coeff(Nb)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb))) ) ).

% cos_coeff_Suc
tff(fact_3108_arccos__le__arccos,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),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,Xb)) ) ) ) ).

% arccos_le_arccos
tff(fact_3109_arccos__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),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,Xb) = aa(real,real,arccos,Y) )
      <=> ( Xb = Y ) ) ) ).

% arccos_eq_iff
tff(fact_3110_arccos__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),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,Xb)),aa(real,real,arccos,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xb) ) ) ) ).

% arccos_le_mono
tff(fact_3111_arcsin__le__arcsin,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),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,Xb)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_3112_arcsin__minus,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( aa(real,real,arcsin,aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,arcsin,Xb)) ) ) ) ).

% arcsin_minus
tff(fact_3113_arcsin__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),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,Xb) = aa(real,real,arcsin,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_3114_arcsin__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),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,Xb)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_3115_take__bit__nat__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( bit_se2584673776208193580ke_bit(nat,Nb,aa(int,nat,nat2,K)) = aa(int,nat,nat2,bit_se2584673776208193580ke_bit(int,Nb,K)) ) ) ).

% take_bit_nat_eq
tff(fact_3116_nat__take__bit__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,nat,nat2,bit_se2584673776208193580ke_bit(int,Nb,K)) = bit_se2584673776208193580ke_bit(nat,Nb,aa(int,nat,nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_3117_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_3118_arccos__less__arccos,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),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,Xb)) ) ) ) ).

% arccos_less_arccos
tff(fact_3119_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,Ma: nat] :
      ( ( bit_se2584673776208193580ke_bit(nat,Nb,Ma) = 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))),Nb)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_3120_take__bit__nat__less__exp,axiom,
    ! [Nb: nat,Ma: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),bit_se2584673776208193580ke_bit(nat,Nb,Ma)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% take_bit_nat_less_exp
tff(fact_3121_take__bit__nat__eq__self,axiom,
    ! [Ma: nat,Nb: 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))),Nb))
     => ( bit_se2584673776208193580ke_bit(nat,Nb,Ma) = Ma ) ) ).

% take_bit_nat_eq_self
tff(fact_3122_arccos__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),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,Xb)),aa(real,real,arccos,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb) ) ) ) ).

% arccos_less_mono
tff(fact_3123_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_3124_arcsin__less__arcsin,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),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,Xb)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_3125_arcsin__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),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,Xb)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_3126_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_3127_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),bit_se2584673776208193580ke_bit(nat,Nb,Ma)),Ma)
    <=> 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))),Nb)),Ma) ) ).

% take_bit_nat_less_self_iff
tff(fact_3128_tan__45,axiom,
    aa(real,real,tan(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) = one_one(real) ).

% tan_45
tff(fact_3129_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_3130_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_3131_sin__arccos__nonzero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => ( sin(real,aa(real,real,arccos,Xb)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_3132_arccos__cos2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Xb)
       => ( aa(real,real,arccos,aa(real,real,cos(real),Xb)) = aa(real,real,uminus_uminus(real),Xb) ) ) ) ).

% arccos_cos2
tff(fact_3133_arccos__minus,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),aa(real,real,arccos,Xb)) ) ) ) ).

% arccos_minus
tff(fact_3134_cos__arcsin__nonzero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => ( aa(real,real,cos(real),aa(real,real,arcsin,Xb)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_3135_tan__total,axiom,
    ! [Y: real] :
    ? [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(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),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),X) = Y )
      & ! [Y4: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y4),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( aa(real,real,tan(real),Y4) = Y ) )
         => ( Y4 = X ) ) ) ).

% tan_total
tff(fact_3136_tan__monotone,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(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),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),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),Xb)) ) ) ) ).

% tan_monotone
tff(fact_3137_tan__monotone_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(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),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),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),Xb)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_3138_tan__mono__lt__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),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),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),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),Xb)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_3139_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),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),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),X) = Y ) ) ).

% lemma_tan_total1
tff(fact_3140_tan__minus__45,axiom,
    aa(real,real,tan(real),aa(real,real,uminus_uminus(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_3141_tan__inverse,axiom,
    ! [Y: 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),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))),Y)) ) ).

% tan_inverse
tff(fact_3142_tan__cot,axiom,
    ! [Xb: real] : ( aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))),Xb)) = aa(real,real,inverse_inverse(real),aa(real,real,tan(real),Xb)) ) ).

% tan_cot
tff(fact_3143_complex__inverse,axiom,
    ! [A2: real,B2: real] : ( aa(complex,complex,inverse_inverse(complex),complex2(A2,B2)) = complex2(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))))),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_3144_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_3145_arccos__minus__abs,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),aa(real,real,arccos,Xb)) ) ) ).

% arccos_minus_abs
tff(fact_3146_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,cos(A),Xb) != 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),Xb)),aa(A,A,tan(A),Y)) = divide_divide(A,sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),Xb)),aa(A,A,cos(A),Y))) ) ) ) ) ).

% add_tan_eq
tff(fact_3147_tan__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Xb)),zero_zero(real)) ) ) ).

% tan_less_zero
tff(fact_3148_tan__mono__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),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),Xb)),aa(real,real,tan(real),Y)) ) ) ) ).

% tan_mono_le
tff(fact_3149_tan__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),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),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),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),Xb)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_3150_tan__bound__pi2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),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),Xb))),one_one(real)) ) ).

% tan_bound_pi2
tff(fact_3151_arctan__unique,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( ( aa(real,real,tan(real),Xb) = Y )
         => ( aa(real,real,arctan,Y) = Xb ) ) ) ) ).

% arctan_unique
tff(fact_3152_arctan__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,real,arctan,aa(real,real,tan(real),Xb)) = Xb ) ) ) ).

% arctan_tan
tff(fact_3153_arctan,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(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)),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_3154_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,cos(A),Xb) != 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),Xb)),aa(A,A,tan(A),Y))) = divide_divide(A,aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),Xb)),aa(A,A,cos(A),Y))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_3155_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,cos(A),Xb) != 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),Xb),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,tan(A),Xb)),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),Xb)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3156_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,cos(A),Xb) != 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),Xb),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),Xb)),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),Xb)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_add
tff(fact_3157_tan__total__pi4,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ? [Z2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))),Z2)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
          & ( aa(real,real,tan(real),Z2) = Xb ) ) ) ).

% tan_total_pi4
tff(fact_3158_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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arccos_le_pi2
tff(fact_3159_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),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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_3160_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),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_3161_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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arcsin_ubound
tff(fact_3162_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),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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_bounded
tff(fact_3163_arcsin__sin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,real,arcsin,sin(real,Xb)) = Xb ) ) ) ).

% arcsin_sin
tff(fact_3164_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( aa(A,A,cos(A),Xb) != 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),Xb)),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),Xb))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ) ).

% tan_sec
tff(fact_3165_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( aa(A,A,tan(A),Xb) = divide_divide(A,sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)),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))),Xb))),one_one(A))) ) ) ).

% tan_half
tff(fact_3166_le__arcsin__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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),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,Xb))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Y)),Xb) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_3167_arcsin__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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),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,Xb)),Y)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),sin(real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_3168_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),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),pi)
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_3169_sin__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( sin(real,Xb) = divide_divide(real,aa(real,real,tan(real),Xb),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),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% sin_tan
tff(fact_3170_cos__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( aa(real,real,cos(real),Xb) = 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),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% cos_tan
tff(fact_3171_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : ( bit_se8732182000553998342ip_bit(A,zero_zero(nat),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ) ).

% flip_bit_0
tff(fact_3172_set__decode__0,axiom,
    ! [Xb: nat] :
      ( member(nat,zero_zero(nat),nat_set_decode(Xb))
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xb) ) ).

% set_decode_0
tff(fact_3173_set__decode__Suc,axiom,
    ! [Nb: nat,Xb: nat] :
      ( member(nat,aa(nat,nat,suc,Nb),nat_set_decode(Xb))
    <=> member(nat,Nb,nat_set_decode(divide_divide(nat,Xb,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% set_decode_Suc
tff(fact_3174_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_3175_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q)))
        <=> ( (P)
           => (Q) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_3176_of__bool__eq__0__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P)) = zero_zero(A) )
        <=> ~ (P) ) ) ).

% of_bool_eq_0_iff
tff(fact_3177_of__bool__eq_I1_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa($o,A,zero_neq_one_of_bool(A),$false) = zero_zero(A) ) ) ).

% of_bool_eq(1)
tff(fact_3178_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_3179_of__bool__eq__1__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P)) = one_one(A) )
        <=> (P) ) ) ).

% of_bool_eq_1_iff
tff(fact_3180_of__bool__eq_I2_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa($o,A,zero_neq_one_of_bool(A),$true) = one_one(A) ) ) ).

% of_bool_eq(2)
tff(fact_3181_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_3182_real__sqrt__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa(real,real,sqrt,Xb) = one_one(real) )
    <=> ( Xb = one_one(real) ) ) ).

% real_sqrt_eq_1_iff
tff(fact_3183_real__sqrt__one,axiom,
    aa(real,real,sqrt,one_one(real)) = one_one(real) ).

% real_sqrt_one
tff(fact_3184_abs__bool__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: $o] : ( aa(A,A,abs_abs(A),aa($o,A,zero_neq_one_of_bool(A),(P))) = aa($o,A,zero_neq_one_of_bool(A),(P)) ) ) ).

% abs_bool_eq
tff(fact_3185_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_3186_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_3187_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_3188_of__bool__not__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P: $o] : ( aa($o,A,zero_neq_one_of_bool(A),~ (P)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ) ).

% of_bool_not_iff
tff(fact_3189_Suc__0__mod__eq,axiom,
    ! [Nb: nat] : ( modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),Nb) = aa($o,nat,zero_neq_one_of_bool(nat),Nb != aa(nat,nat,suc,zero_zero(nat))) ) ).

% Suc_0_mod_eq
tff(fact_3190_real__sqrt__lt__1__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_3191_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_3192_real__sqrt__le__1__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xb)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real)) ) ).

% real_sqrt_le_1_iff
tff(fact_3193_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_3194_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,sgn_sgn(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ) ).

% sgn_mult_self_eq
tff(fact_3195_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ) ).

% sgn_abs
tff(fact_3196_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : ( aa(A,A,sgn_sgn(A),aa(A,A,abs_abs(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_3197_take__bit__of__Suc__0,axiom,
    ! [Nb: nat] : ( bit_se2584673776208193580ke_bit(nat,Nb,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)),Nb)) ) ).

% take_bit_of_Suc_0
tff(fact_3198_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : ( bit_se2584673776208193580ke_bit(A,Nb,one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)) ) ) ).

% take_bit_of_1
tff(fact_3199_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : ( aa(A,A,sgn_sgn(A),aa(nat,A,semiring_1_of_nat(A),Nb)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)) ) ) ).

% sgn_of_nat
tff(fact_3200_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: $o] : ( divide_divide(A,aa($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_3201_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] : ( divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ) ).

% bits_1_div_exp
tff(fact_3202_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : ( divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ) ).

% one_div_2_pow_eq
tff(fact_3203_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Ma: nat,Nb: nat] : ( bit_se2584673776208193580ke_bit(A,Ma,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) ) ) ).

% take_bit_of_exp
tff(fact_3204_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : ( modulo_modulo(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)) ) ) ).

% one_mod_2_pow_eq
tff(fact_3205_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_3206_real__sqrt__minus,axiom,
    ! [Xb: real] : ( aa(real,real,sqrt,aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,sqrt,Xb)) ) ).

% real_sqrt_minus
tff(fact_3207_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P3: $o,Q4: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P3)) = aa($o,A,zero_neq_one_of_bool(A),(Q4)) )
        <=> ( (P3)
          <=> (Q4) ) ) ) ).

% of_bool_eq_iff
tff(fact_3208_divide__complex__def,axiom,
    ! [Xb: complex,Y: complex] : ( divide_divide(complex,Xb,Y) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Xb),aa(complex,complex,inverse_inverse(complex),Y)) ) ).

% divide_complex_def
tff(fact_3209_real__sqrt__inverse,axiom,
    ! [Xb: real] : ( aa(real,real,sqrt,aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb)) ) ).

% real_sqrt_inverse
tff(fact_3210_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).

% zero_less_eq_of_bool
tff(fact_3211_real__sqrt__ge__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,Xb)) ) ).

% real_sqrt_ge_one
tff(fact_3212_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A)) ) ).

% of_bool_less_eq_one
tff(fact_3213_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,$o),P3: $o] :
          ( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P3)))
        <=> ~ ( ( (P3)
                & ~ aa(A,$o,P,one_one(A)) )
              | ( ~ (P3)
                & ~ aa(A,$o,P,zero_zero(A)) ) ) ) ) ).

% split_of_bool_asm
tff(fact_3214_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,$o),P3: $o] :
          ( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P3)))
        <=> ( ( (P3)
             => aa(A,$o,P,one_one(A)) )
            & ( ~ (P3)
             => aa(A,$o,P,zero_zero(A)) ) ) ) ) ).

% split_of_bool
tff(fact_3215_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P3: $o] :
          ( aa($o,A,zero_neq_one_of_bool(A),(P3)) = $ite((P3),one_one(A),zero_zero(A)) ) ) ).

% of_bool_def
tff(fact_3216_sqrt__divide__self__eq,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( divide_divide(real,aa(real,real,sqrt,Xb),Xb) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb)) ) ) ).

% sqrt_divide_self_eq
tff(fact_3217_subset__decode__imp__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),nat_set_decode(Ma)),nat_set_decode(Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ).

% subset_decode_imp_le
tff(fact_3218_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [A4: A] :
              ( ( divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))) = A4 )
             => aa(A,$o,P,A4) )
         => ( ! [A4: A,B4: $o] :
                ( aa(A,$o,P,A4)
               => ( ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B4))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4)),aa(num,A,numeral_numeral(A),bit0(one2))) = A4 )
                 => aa(A,$o,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B4))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4))) ) )
           => aa(A,$o,P,A2) ) ) ) ).

% bits_induct
tff(fact_3219_real__inv__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,inverse_inverse(real),Xb) ) ) ).

% real_inv_sqrt_pow2
tff(fact_3220_tan__30,axiom,
    aa(real,real,tan(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2))))) = 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_3221_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Ma: nat,Nb: nat] : ( modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ma),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ma)) ) ) ).

% exp_mod_exp
tff(fact_3222_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
       => ( divide_divide(int,K,L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),divide_divide(int,aa(int,int,abs_abs(int),K),aa(int,int,abs_abs(int),L)))),aa($o,int,zero_neq_one_of_bool(int),~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_3223_arsinh__real__aux,axiom,
    ! [Xb: 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),Xb),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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ).

% arsinh_real_aux
tff(fact_3224_powr__half__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(real,real,sqrt,Xb) ) ) ).

% powr_half_sqrt
tff(fact_3225_arsinh__real__def,axiom,
    ! [Xb: real] : ( aa(real,real,arsinh(real),Xb) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ) ).

% arsinh_real_def
tff(fact_3226_cos__x__y__le__one,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),divide_divide(real,Xb,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),Xb),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_3227_arcosh__real__def,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => ( aa(real,real,arcosh(real),Xb) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ) ) ).

% arcosh_real_def
tff(fact_3228_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ma: nat,Nb: nat] :
          ( divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ma),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,
              aa(A,fun(A,A),times_times(A),
                aa($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))),Ma) != zero_zero(A) )
                  & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma) ))),
              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),Ma),Nb))) ) ) ).

% exp_div_exp_eq
tff(fact_3229_cos__arctan,axiom,
    ! [Xb: real] : ( aa(real,real,cos(real),aa(real,real,arctan,Xb)) = 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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% cos_arctan
tff(fact_3230_sin__arctan,axiom,
    ! [Xb: real] : ( sin(real,aa(real,real,arctan,Xb)) = divide_divide(real,Xb,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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% sin_arctan
tff(fact_3231_sin__cos__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,Xb))
     => ( sin(real,Xb) = 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),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_cos_sqrt
tff(fact_3232_arctan__half,axiom,
    ! [Xb: real] : ( aa(real,real,arctan,Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,divide_divide(real,Xb,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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) ).

% arctan_half
tff(fact_3233_sin__arccos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( sin(real,aa(real,real,arccos,Xb)) = 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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% sin_arccos
tff(fact_3234_cos__arcsin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( aa(real,real,cos(real),aa(real,real,arcsin,Xb)) = 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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% cos_arcsin
tff(fact_3235_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
     => ( sin(real,aa(real,real,arccos,Y)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_arccos_abs
tff(fact_3236_divide__int__unfold,axiom,
    ! [K: int,Ma: nat,L: int,Nb: nat] :
      ( divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),Ma)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),Nb))) = $ite(
          ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
          | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
          | ( Nb = zero_zero(nat) ) ),
          zero_zero(int),
          $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Ma,Nb)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,Ma,Nb)),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),Ma)))))) ) ) ).

% divide_int_unfold
tff(fact_3237_modulo__int__def,axiom,
    ! [K: int,L: int] :
      ( modulo_modulo(int,K,L) = $ite(
          L = zero_zero(int),
          K,
          $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),L)),aa($o,int,zero_neq_one_of_bool(int),~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)))),aa(nat,int,semiring_1_of_nat(int),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_3238_modulo__int__unfold,axiom,
    ! [K: int,Ma: nat,L: int,Nb: nat] :
      ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),Ma)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),Nb))) = $ite(
          ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
          | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
          | ( Nb = zero_zero(nat) ) ),
          aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),Ma)),
          $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,Ma,Nb))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),Ma))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,Ma,Nb))))) ) ) ).

% modulo_int_unfold
tff(fact_3239_divide__int__def,axiom,
    ! [K: int,L: int] :
      ( divide_divide(int,K,L) = $ite(
          L = zero_zero(int),
          zero_zero(int),
          $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)))))) ) ) ).

% divide_int_def
tff(fact_3240_cis__multiple__2pi,axiom,
    ! [Nb: real] :
      ( member(real,Nb,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)),Nb)) = one_one(complex) ) ) ).

% cis_multiple_2pi
tff(fact_3241_buildup__gives__empty,axiom,
    ! [Nb: nat] : ( vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot(set(nat)) ) ).

% buildup_gives_empty
tff(fact_3242_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: num] : ( bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,pred_numeral(Nb)))) ) ) ).

% mask_numeral
tff(fact_3243_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_3244_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
     => ( ( ? [A4: $o,B4: $o] : ( A1 = vEBT_Leaf((A4),(B4)) )
         => ( A22 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) )
             => ( ( A22 = Deg )
               => ( ! [X3: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => vEBT_invar_vebt(X3,N) )
                 => ( vEBT_invar_vebt(Summary,M)
                   => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
                     => ( ( M = N )
                       => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
                         => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_1)
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_1) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X3: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => vEBT_invar_vebt(X3,N) )
                   => ( vEBT_invar_vebt(Summary,M)
                     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
                       => ( ( M = aa(nat,nat,suc,N) )
                         => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
                           => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_1)
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_1) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma2: nat] :
                  ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),Deg,TreeList,Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X3: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                         => vEBT_invar_vebt(X3,N) )
                     => ( vEBT_invar_vebt(Summary,M)
                       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
                         => ( ( M = N )
                           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
                             => ( ! [I4: nat] :
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),X_13)
                                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I4) ) )
                               => ( ( ( Mi = Ma2 )
                                   => ! [X3: vEBT_VEBT] :
                                        ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_1) ) )
                                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),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))),Deg))
                                     => ~ ( ( Mi != 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))),M))
                                             => ( ( ( vEBT_VEBT_high(Ma2,N) = I4 )
                                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(Ma2,N)) )
                                                & ! [X3: nat] :
                                                    ( ( ( vEBT_VEBT_high(X3,N) = I4 )
                                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(X3,N)) )
                                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X3)
                                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma2: nat] :
                    ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),Deg,TreeList,Summary) )
                   => ( ( A22 = Deg )
                     => ( ! [X3: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                           => vEBT_invar_vebt(X3,N) )
                       => ( vEBT_invar_vebt(Summary,M)
                         => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
                           => ( ( M = aa(nat,nat,suc,N) )
                             => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
                               => ( ! [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))),M))
                                     => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),X_13)
                                      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I4) ) )
                                 => ( ( ( Mi = Ma2 )
                                     => ! [X3: vEBT_VEBT] :
                                          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                         => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_1) ) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma2)
                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                                       => ~ ( ( Mi != Ma2 )
                                           => ! [I4: nat] :
                                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                                               => ( ( ( vEBT_VEBT_high(Ma2,N) = I4 )
                                                   => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(Ma2,N)) )
                                                  & ! [X3: nat] :
                                                      ( ( ( vEBT_VEBT_high(X3,N) = I4 )
                                                        & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(X3,N)) )
                                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X3)
                                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3245_mask__nat__positive__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),bit_se2239418461657761734s_mask(nat,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% mask_nat_positive_iff
tff(fact_3246_mi__ma__2__deg,axiom,
    ! [Mi2: nat,Ma: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma)),Dega,TreeLista,Summarya),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),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))),Dega)) ) ) ).

% mi_ma_2_deg
tff(fact_3247_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_3248_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_3249_mask__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] :
          ( ( bit_se2239418461657761734s_mask(A,Nb) = zero_zero(A) )
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_3250_norm__cis,axiom,
    ! [A2: real] : ( real_V7770717601297561774m_norm(complex,cis(A2)) = one_one(real) ) ).

% norm_cis
tff(fact_3251_cis__zero,axiom,
    cis(zero_zero(real)) = one_one(complex) ).

% cis_zero
tff(fact_3252_both__member__options__from__complete__tree__to__child,axiom,
    ! [Dega: nat,Mi2: nat,Ma: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Dega)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma)),Dega,TreeLista,Summarya)),Xb)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
          | ( Xb = Mi2 )
          | ( Xb = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_3253_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_3254_cis__inverse,axiom,
    ! [A2: real] : ( aa(complex,complex,inverse_inverse(complex),cis(A2)) = cis(aa(real,real,uminus_uminus(real),A2)) ) ).

% cis_inverse
tff(fact_3255_member__inv,axiom,
    ! [Mi2: nat,Ma: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma)),Dega,TreeLista,Summarya)),Xb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Dega)
        & ( ( Xb = Mi2 )
          | ( Xb = Ma )
          | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Ma)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
            & aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% member_inv
tff(fact_3256_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Mi2: nat,Ma: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Dega)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma)),Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_3257_mask__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,zero_zero(nat))) = one_one(A) ) ) ).

% mask_Suc_0
tff(fact_3258_take__bit__minus__one__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : ( bit_se2584673776208193580ke_bit(A,Nb,aa(A,A,uminus_uminus(A),one_one(A))) = bit_se2239418461657761734s_mask(A,Nb) ) ) ).

% take_bit_minus_one_eq_mask
tff(fact_3259_cis__pi,axiom,
    cis(pi) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% cis_pi
tff(fact_3260_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_3261_of__int__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : ( aa(int,A,ring_1_of_int(A),bit_se2239418461657761734s_mask(int,Nb)) = bit_se2239418461657761734s_mask(A,Nb) ) ) ).

% of_int_mask_eq
tff(fact_3262_of__nat__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : ( aa(nat,A,semiring_1_of_nat(A),bit_se2239418461657761734s_mask(nat,Nb)) = bit_se2239418461657761734s_mask(A,Nb) ) ) ).

% of_nat_mask_eq
tff(fact_3263_nat__mask__eq,axiom,
    ! [Nb: nat] : ( aa(int,nat,nat2,bit_se2239418461657761734s_mask(int,Nb)) = bit_se2239418461657761734s_mask(nat,Nb) ) ).

% nat_mask_eq
tff(fact_3264_prod__decode__aux_Ocases,axiom,
    ! [Xb: product_prod(nat,nat)] :
      ~ ! [K2: nat,M: nat] : ( Xb != aa(nat,product_prod(nat,nat),product_Pair(nat,nat,K2),M) ) ).

% prod_decode_aux.cases
tff(fact_3265_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_3266_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_3267_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi2: nat,Ma: nat,Va2: list(vEBT_VEBT),Vb: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma)),zero_zero(nat),Va2,Vb),Xb)
    <=> ( ( Xb = Mi2 )
        | ( Xb = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_3268_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% diff_shunt_var
tff(fact_3269_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_3270_mask__nonnegative__int,axiom,
    ! [Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se2239418461657761734s_mask(int,Nb)) ).

% mask_nonnegative_int
tff(fact_3271_not__mask__negative__int,axiom,
    ! [Nb: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se2239418461657761734s_mask(int,Nb)),zero_zero(int)) ).

% not_mask_negative_int
tff(fact_3272_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod(nat,nat),Va2: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] : ~ vEBT_VEBT_minNull(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va2,Vb,Vc)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_3273_vebt__member_Osimps_I3_J,axiom,
    ! [V2: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Xb: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy,Uz)),Xb) ).

% vebt_member.simps(3)
tff(fact_3274_VEBT__internal_OminNull_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ( Xb != vEBT_Leaf($false,$false) )
     => ( ! [Uv2: $o] : ( Xb != vEBT_Leaf($true,(Uv2)) )
       => ( ! [Uu2: $o] : ( Xb != vEBT_Leaf((Uu2),$true) )
         => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : ( Xb != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : ( Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) ) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_3275_less__mask,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),bit_se2239418461657761734s_mask(nat,Nb)) ) ).

% less_mask
tff(fact_3276_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xb)
     => ( ! [Uv2: $o] : ( Xb != vEBT_Leaf($true,(Uv2)) )
       => ( ! [Uu2: $o] : ( Xb != vEBT_Leaf((Uu2),$true) )
         => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : ( Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_3277_option_Osize_I4_J,axiom,
    ! [A: $tType,X2: A] : ( aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X2)) = aa(nat,nat,suc,zero_zero(nat)) ) ).

% option.size(4)
tff(fact_3278_vebt__member_Osimps_I4_J,axiom,
    ! [V2: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Xb: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc)),Xb) ).

% vebt_member.simps(4)
tff(fact_3279_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xb)
      <=> (Y) )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv2: $o] : ( Xb = vEBT_Leaf($true,(Uv2)) )
           => (Y) )
         => ( ( ? [Uu2: $o] : ( Xb = vEBT_Leaf((Uu2),$true) )
             => (Y) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) )
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_3280_take__bit__eq__mask__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,Nb,K) = bit_se2239418461657761734s_mask(int,Nb) )
    <=> ( bit_se2584673776208193580ke_bit(int,Nb,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_3281_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_3282_Suc__mask__eq__exp,axiom,
    ! [Nb: nat] : ( aa(nat,nat,suc,bit_se2239418461657761734s_mask(nat,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ) ).

% Suc_mask_eq_exp
tff(fact_3283_mask__nat__less__exp,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),bit_se2239418461657761734s_mask(nat,Nb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% mask_nat_less_exp
tff(fact_3284_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,Nb))
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_3285_mask__nat__def,axiom,
    ! [Nb: nat] : ( bit_se2239418461657761734s_mask(nat,Nb) = 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))),Nb)),one_one(nat)) ) ).

% mask_nat_def
tff(fact_3286_mask__half__int,axiom,
    ! [Nb: nat] : ( divide_divide(int,bit_se2239418461657761734s_mask(int,Nb),aa(num,int,numeral_numeral(int),bit0(one2))) = bit_se2239418461657761734s_mask(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) ) ).

% mask_half_int
tff(fact_3287_mask__int__def,axiom,
    ! [Nb: nat] : ( bit_se2239418461657761734s_mask(int,Nb) = 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))),Nb)),one_one(int)) ) ).

% mask_int_def
tff(fact_3288_mask__eq__exp__minus__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : ( bit_se2239418461657761734s_mask(A,Nb) = 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))),Nb)),one_one(A)) ) ) ).

% mask_eq_exp_minus_1
tff(fact_3289_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,Ma: nat,Dega: nat,Mi2: nat,Maa: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X,Nb) )
     => ( vEBT_invar_vebt(Summarya,Ma)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma) )
         => ( ( Ma = Nb )
           => ( ( Dega = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Ma) )
             => ( ! [I2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma))
                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),X_13)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mi2 = Maa )
                   => ! [X: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                       => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_12) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Maa)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Dega))
                     => ( ( ( Mi2 != Maa )
                         => ! [I2: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma))
                             => ( ( ( vEBT_VEBT_high(Maa,Nb) = I2 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),vEBT_VEBT_low(Maa,Nb)) )
                                & ! [X: nat] :
                                    ( ( ( vEBT_VEBT_high(X,Nb) = I2 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),vEBT_VEBT_low(X,Nb)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Maa) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Maa)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_3290_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,Ma: nat,Dega: nat,Mi2: nat,Maa: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X,Nb) )
     => ( vEBT_invar_vebt(Summarya,Ma)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma) )
         => ( ( Ma = aa(nat,nat,suc,Nb) )
           => ( ( Dega = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Ma) )
             => ( ! [I2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma))
                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),X_13)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mi2 = Maa )
                   => ! [X: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                       => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_12) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Maa)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Dega))
                     => ( ( ( Mi2 != Maa )
                         => ! [I2: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma))
                             => ( ( ( vEBT_VEBT_high(Maa,Nb) = I2 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),vEBT_VEBT_low(Maa,Nb)) )
                                & ! [X: nat] :
                                    ( ( ( vEBT_VEBT_high(X,Nb) = I2 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),vEBT_VEBT_low(X,Nb)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Maa) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Maa)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_3291_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Nb: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,Nb,K) = bit_se2239418461657761734s_mask(int,Nb) )
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_3292_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : ( size_num(bit0(X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X2)),aa(nat,nat,suc,zero_zero(nat))) ) ).

% num.size_gen(2)
tff(fact_3293_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
    <=> ( ( ? [A5: $o,B5: $o] : ( A1 = vEBT_Leaf((A5),(B5)) )
          & ( A22 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList2: list(vEBT_VEBT),N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X4,N4) )
            & vEBT_invar_vebt(Summary2,N4)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),N4) )
            & ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_13) ) )
        | ? [TreeList2: list(vEBT_VEBT),N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X4,N4) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N4))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N4)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),aa(nat,nat,suc,N4)) )
            & ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_13) ) )
        | ? [TreeList2: list(vEBT_VEBT),N4: nat,Summary2: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X4,N4) )
            & vEBT_invar_vebt(Summary2,N4)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),N4) )
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4))
               => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),X_13)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I3) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                 => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_13) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi3),Ma3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A22))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4))
                 => ( ( ( vEBT_VEBT_high(Ma3,N4) = I3 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(Ma3,N4)) )
                    & ! [X4: nat] :
                        ( ( ( vEBT_VEBT_high(X4,N4) = I3 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(X4,N4)) )
                       => ( 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) ) ) ) ) ) )
        | ? [TreeList2: list(vEBT_VEBT),N4: nat,Summary2: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X4,N4) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N4))
            & ( 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))),aa(nat,nat,suc,N4)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),aa(nat,nat,suc,N4)) )
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N4)))
               => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),X_13)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I3) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                 => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_13) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi3),Ma3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A22))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N4)))
                 => ( ( ( vEBT_VEBT_high(Ma3,N4) = I3 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(Ma3,N4)) )
                    & ! [X4: nat] :
                        ( ( ( vEBT_VEBT_high(X4,N4) = I3 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(X4,N4)) )
                       => ( aa(nat,$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_3294_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q4: A,R2: A] :
          ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),product_Pair(A,A,Q4),R2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R2),aa(A,product_prod(A,A),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))),Q4)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R2),aa(num,A,numeral_numeral(A),L))),aa(A,product_prod(A,A),product_Pair(A,A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q4)),R2)) ) ) ).

% divmod_step_eq
tff(fact_3295_card_Oempty,axiom,
    ! [A: $tType] : ( aa(set(A),nat,finite_card(A),bot_bot(set(A))) = zero_zero(nat) ) ).

% card.empty
tff(fact_3296_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q4: A,R2: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),product_Pair(A,A,Q4),R2))
        <=> ( R2 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_3297_product__nth,axiom,
    ! [A: $tType,B: $tType,Nb: nat,Xs: list(A),Ys2: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),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)),Nb) = aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Xs),divide_divide(nat,Nb,aa(list(B),nat,size_size(list(B)),Ys2)))),aa(nat,B,nth(B,Ys2),modulo_modulo(nat,Nb,aa(list(B),nat,size_size(list(B)),Ys2)))) ) ) ).

% product_nth
tff(fact_3298_cis__minus__pi__half,axiom,
    cis(aa(real,real,uminus_uminus(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_3299_norm__ii,axiom,
    real_V7770717601297561774m_norm(complex,imaginary_unit) = one_one(real) ).

% norm_ii
tff(fact_3300_complex__i__mult__minus,axiom,
    ! [Xb: 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),imaginary_unit),Xb)) = aa(complex,complex,uminus_uminus(complex),Xb) ) ).

% complex_i_mult_minus
tff(fact_3301_inverse__i,axiom,
    aa(complex,complex,inverse_inverse(complex),imaginary_unit) = aa(complex,complex,uminus_uminus(complex),imaginary_unit) ).

% inverse_i
tff(fact_3302_divide__i,axiom,
    ! [Xb: complex] : ( divide_divide(complex,Xb,imaginary_unit) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,uminus_uminus(complex),imaginary_unit)),Xb) ) ).

% divide_i
tff(fact_3303_i__squared,axiom,
    aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),imaginary_unit) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% i_squared
tff(fact_3304_exp__pi__i_H,axiom,
    exp(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,pi))) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% exp_pi_i'
tff(fact_3305_exp__pi__i,axiom,
    exp(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,pi)),imaginary_unit)) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% exp_pi_i
tff(fact_3306_divide__numeral__i,axiom,
    ! [Z: complex,Nb: num] : ( divide_divide(complex,Z,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(num,complex,numeral_numeral(complex),Nb)),imaginary_unit)) = 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),Nb)) ) ).

% divide_numeral_i
tff(fact_3307_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_3308_i__even__power,axiom,
    ! [Nb: 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),Nb),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))),Nb) ) ).

% i_even_power
tff(fact_3309_exp__two__pi__i_H,axiom,
    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),real_Vector_of_real(complex,pi)),aa(num,complex,numeral_numeral(complex),bit0(one2))))) = one_one(complex) ).

% exp_two_pi_i'
tff(fact_3310_exp__two__pi__i,axiom,
    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))),real_Vector_of_real(complex,pi))),imaginary_unit)) = one_one(complex) ).

% exp_two_pi_i
tff(fact_3311_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_3312_complex__i__not__one,axiom,
    imaginary_unit != one_one(complex) ).

% complex_i_not_one
tff(fact_3313_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o,D5: nat] : ( Xb != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),D5) )
     => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Deg2: nat] : ( Xb != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Mima,Deg,TreeList,Summary)),Deg2) ) ) ).

% VEBT_internal.valid'.cases
tff(fact_3314_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),W) = Z )
    <=> ( W = aa(complex,complex,uminus_uminus(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Z)) ) ) ).

% i_times_eq_iff
tff(fact_3315_complex__i__not__neg__numeral,axiom,
    ! [W: num] : ( imaginary_unit != aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W)) ) ).

% complex_i_not_neg_numeral
tff(fact_3316_imaginary__unit_Ocode,axiom,
    imaginary_unit = complex2(zero_zero(real),one_one(real)) ).

% imaginary_unit.code
tff(fact_3317_Complex__eq__i,axiom,
    ! [Xb: real,Y: real] :
      ( ( complex2(Xb,Y) = imaginary_unit )
    <=> ( ( Xb = zero_zero(real) )
        & ( Y = one_one(real) ) ) ) ).

% Complex_eq_i
tff(fact_3318_Complex__mult__i,axiom,
    ! [A2: real,B2: real] : ( aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(A2,B2)),imaginary_unit) = complex2(aa(real,real,uminus_uminus(real),B2),A2) ) ).

% Complex_mult_i
tff(fact_3319_i__mult__Complex,axiom,
    ! [A2: real,B2: real] : ( aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),complex2(A2,B2)) = complex2(aa(real,real,uminus_uminus(real),B2),A2) ) ).

% i_mult_Complex
tff(fact_3320_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X: nat] : ( Xb != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A4),(B4))),X) )
     => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Ux2: nat] : ( Xb != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Ux2) )
       => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList: list(vEBT_VEBT),S2: vEBT_VEBT,X: nat] : ( Xb != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList,S2)),X) ) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_3321_cmod__unit__one,axiom,
    ! [A2: real] : ( real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,aa(real,real,cos(real),A2))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,sin(real,A2))))) = one_one(real) ) ).

% cmod_unit_one
tff(fact_3322_vebt__member_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X: nat] : ( Xb != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A4),(B4))),X) )
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,X: nat] : ( Xb != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),X) )
       => ( ! [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,X: nat] : ( Xb != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy2,Uz2)),X) )
         => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X: nat] : ( Xb != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),X) )
           => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : ( Xb != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary)),X) ) ) ) ) ) ).

% vebt_member.cases
tff(fact_3323_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : ( Xb != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),Uw2) )
     => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT,Uz2: nat] : ( Xb != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Uz2) )
       => ( ! [Mi: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT,X: nat] : ( Xb != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)),X) )
         => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT,X: nat] : ( Xb != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList,Vc2)),X) )
           => ~ ! [V4: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT,X: nat] : ( Xb != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList,Vd)),X) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_3324_Arg__minus__ii,axiom,
    arg(aa(complex,complex,uminus_uminus(complex),imaginary_unit)) = divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% Arg_minus_ii
tff(fact_3325_csqrt__ii,axiom,
    csqrt(imaginary_unit) = divide_divide(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),one_one(complex)),imaginary_unit),real_Vector_of_real(complex,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% csqrt_ii
tff(fact_3326_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Ma: num,Nb: num] :
          ( unique8689654367752047608divmod(A,bit0(Ma),aa(num,num,bit1,Nb)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Ma),Nb),aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),bit0(Ma))),unique1321980374590559556d_step(A,aa(num,num,bit1,Nb),unique8689654367752047608divmod(A,bit0(Ma),bit0(aa(num,num,bit1,Nb))))) ) ) ).

% divmod_algorithm_code(7)
tff(fact_3327_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Ma: num,Nb: num] :
          ( unique8689654367752047608divmod(A,aa(num,num,bit1,Ma),aa(num,num,bit1,Nb)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),Ma),Nb),aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Ma))),unique1321980374590559556d_step(A,aa(num,num,bit1,Nb),unique8689654367752047608divmod(A,aa(num,num,bit1,Ma),bit0(aa(num,num,bit1,Nb))))) ) ) ).

% divmod_algorithm_code(8)
tff(fact_3328_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q4: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),zero_zero(int))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),one_one(int)),B2,aa(int,product_prod(int,int),product_Pair(int,int,Q4),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),aa(int,product_prod(int,int),product_Pair(int,int,Q4),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R2)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_3329_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( csqrt(Z) = one_one(complex) )
    <=> ( Z = one_one(complex) ) ) ).

% csqrt_eq_1
tff(fact_3330_csqrt__1,axiom,
    csqrt(one_one(complex)) = one_one(complex) ).

% csqrt_1
tff(fact_3331_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Ma: num] : ( unique8689654367752047608divmod(A,Ma,one2) = aa(A,product_prod(A,A),product_Pair(A,A,aa(num,A,numeral_numeral(A),Ma)),zero_zero(A)) ) ) ).

% divmod_algorithm_code(2)
tff(fact_3332_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : ( unique8689654367752047608divmod(A,one2,bit0(Nb)) = aa(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_3333_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : ( unique8689654367752047608divmod(A,one2,aa(num,num,bit1,Nb)) = aa(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_3334_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero(int),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),K)) ).

% eucl_rel_int_by0
tff(fact_3335_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q4: int] :
      ( ( L != zero_zero(int) )
     => ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L) )
       => eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q4),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_3336_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q4: int,R2: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q4),R2))
     => ( ( B2 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A2),B2,
            aa(int,product_prod(int,int),
              product_Pair(int,int,
                $ite(R2 = zero_zero(int),aa(int,int,uminus_uminus(int),Q4),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q4)),one_one(int)))),
              $ite(R2 = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),R2)))) ) ) ).

% zminus1_lemma
tff(fact_3337_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_3338_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q4),R2))
    <=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q4)),R2) )
        & $ite(
            aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L),
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),L) ),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)),
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),R2)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int)) ),
              Q4 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_3339_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q4: int] :
      ( ( aa(int,int,sgn_sgn(int),R2) = aa(int,int,sgn_sgn(int),L) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R2)),aa(int,int,abs_abs(int),L))
       => ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L)),R2) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q4),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_3340_cis__Arg__unique,axiom,
    ! [Z: complex,Xb: real] :
      ( ( aa(complex,complex,sgn_sgn(complex),Z) = cis(Xb) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
         => ( arg(Z) = Xb ) ) ) ) ).

% cis_Arg_unique
tff(fact_3341_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( ( aa(complex,complex,sgn_sgn(complex),Z) = cis(arg(Z)) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z))
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arg(Z)),pi) ) ) ).

% Arg_correct
tff(fact_3342_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),product_Pair(int,int,zero_zero(int)),K3) ) )
        | ? [L3: int,K3: int,Q6: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q6),zero_zero(int)) )
            & ( L3 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q6),L3) ) )
        | ? [R5: int,L3: int,K3: int,Q6: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q6),R5) )
            & ( aa(int,int,sgn_sgn(int),R5) = aa(int,int,sgn_sgn(int),L3) )
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L3))
            & ( K3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q6),L3)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_3343_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),product_Pair(int,int,zero_zero(int)),A1) ) )
       => ( ! [Q2: int] :
              ( ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q2),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q2),A22) ) ) )
         => ~ ! [R4: int,Q2: int] :
                ( ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q2),R4) )
               => ( ( aa(int,int,sgn_sgn(int),R4) = aa(int,int,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),Q2),A22)),R4) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_3344_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Ma: num,Nb: num] :
          ( unique8689654367752047608divmod(A,Ma,Nb) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),Ma),Nb),aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),Ma)),unique1321980374590559556d_step(A,Nb,unique8689654367752047608divmod(A,Ma,bit0(Nb)))) ) ) ).

% divmod_divmod_step
tff(fact_3345_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q4: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q4),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),aa(int,product_prod(int,int),product_Pair(int,int,Q4),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_3346_one__div__minus__numeral,axiom,
    ! [Nb: num] : ( divide_divide(int,one_one(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,Nb))) ) ).

% one_div_minus_numeral
tff(fact_3347_minus__one__div__numeral,axiom,
    ! [Nb: num] : ( divide_divide(int,aa(int,int,uminus_uminus(int),one_one(int)),aa(num,int,numeral_numeral(int),Nb)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,Nb))) ) ).

% minus_one_div_numeral
tff(fact_3348_Divides_Oadjust__div__eq,axiom,
    ! [Q4: int,R2: int] : ( adjust_div(aa(int,product_prod(int,int),product_Pair(int,int,Q4),R2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q4),aa($o,int,zero_neq_one_of_bool(int),R2 != zero_zero(int))) ) ).

% Divides.adjust_div_eq
tff(fact_3349_numeral__div__minus__numeral,axiom,
    ! [Ma: num,Nb: num] : ( divide_divide(int,aa(num,int,numeral_numeral(int),Ma),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,Ma,Nb))) ) ).

% numeral_div_minus_numeral
tff(fact_3350_minus__numeral__div__numeral,axiom,
    ! [Ma: num,Nb: num] : ( divide_divide(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma)),aa(num,int,numeral_numeral(int),Nb)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,Ma,Nb))) ) ).

% minus_numeral_div_numeral
tff(fact_3351_divmod__BitM__2__eq,axiom,
    ! [Ma: num] : ( unique8689654367752047608divmod(int,bitM(Ma),bit0(one2)) = aa(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),Ma)),one_one(int))),one_one(int)) ) ).

% divmod_BitM_2_eq
tff(fact_3352_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,nat),X2: A] : ( size_option(A,Xb,aa(A,option(A),some(A),X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xb,X2)),aa(nat,nat,suc,zero_zero(nat))) ) ).

% option.size_gen(2)
tff(fact_3353_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : ( bit_ri4674362597316999326ke_bit(int,Nb,K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb),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,Nb))),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)))) ) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_3354_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),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ) ).

% and_int_unfold
tff(fact_3355_sinh__ln__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( sinh(real,aa(real,real,ln_ln(real),Xb)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),aa(real,real,inverse_inverse(real),Xb)),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% sinh_ln_real
tff(fact_3356_bit_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),zero_zero(A)) = zero_zero(A) ) ) ).

% bit.conj_zero_right
tff(fact_3357_bit_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),Xb) = zero_zero(A) ) ) ).

% bit.conj_zero_left
tff(fact_3358_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_3359_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_3360_bit__0__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( bit_se5641148757651400278ts_bit(A,zero_zero(A)) = bot_bot(fun(nat,$o)) ) ) ).

% bit_0_eq
tff(fact_3361_sinh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sinh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sinh_0
tff(fact_3362_sinh__minus,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : ( sinh(A,aa(A,A,uminus_uminus(A),Xb)) = aa(A,A,uminus_uminus(A),sinh(A,Xb)) ) ) ).

% sinh_minus
tff(fact_3363_bit_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),aa(A,A,uminus_uminus(A),one_one(A))) = Xb ) ) ).

% bit.conj_one_right
tff(fact_3364_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_3365_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_3366_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_3367_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_3368_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_3369_bit__numeral__Bit0__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Ma: num,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),bit0(Ma))),aa(nat,nat,suc,Nb))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),Ma)),Nb) ) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_3370_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_3371_and__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb))),one_one(A)) = one_one(A) ) ) ).

% and_numerals(8)
tff(fact_3372_bit__numeral__Bit1__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Ma: num,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Ma))),aa(nat,nat,suc,Nb))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),Ma)),Nb) ) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_3373_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_3374_signed__take__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_ri4674362597316999326ke_bit(int,Nb,K))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_3375_signed__take__bit__negative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_ri4674362597316999326ke_bit(int,Nb,K)),zero_zero(int))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% signed_take_bit_negative_iff
tff(fact_3376_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_3377_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(Xb))),one_one(A)) = zero_zero(A) ) ) ).

% and_numerals(5)
tff(fact_3378_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(W)))),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),Nb) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_3379_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W)))),aa(nat,nat,suc,Nb))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),Nb) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_3380_and__minus__numerals_I2_J,axiom,
    ! [Nb: 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,Nb)))) = one_one(int) ) ).

% and_minus_numerals(2)
tff(fact_3381_and__minus__numerals_I6_J,axiom,
    ! [Nb: 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,Nb)))),one_one(int)) = one_one(int) ) ).

% and_minus_numerals(6)
tff(fact_3382_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),zero_zero(nat))
        <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% bit_0
tff(fact_3383_and__minus__numerals_I5_J,axiom,
    ! [Nb: 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(Nb)))),one_one(int)) = zero_zero(int) ) ).

% and_minus_numerals(5)
tff(fact_3384_and__minus__numerals_I1_J,axiom,
    ! [Nb: 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(Nb)))) = zero_zero(int) ) ).

% and_minus_numerals(1)
tff(fact_3385_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Nb: num] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(W)))),aa(num,nat,numeral_numeral(nat),Nb))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),pred_numeral(Nb)) ) ).

% bit_minus_numeral_int(1)
tff(fact_3386_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W)))),aa(num,nat,numeral_numeral(nat),Nb))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),pred_numeral(Nb)) ) ).

% bit_minus_numeral_int(2)
tff(fact_3387_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),Nb)
        <=> ( ( Nb = zero_zero(nat) )
            & ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% bit_mod_2_iff
tff(fact_3388_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: 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,Xb))),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),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ) ).

% and_numerals(7)
tff(fact_3389_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_3390_bit__of__nat__iff__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Ma: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(nat,A,semiring_1_of_nat(A),Ma)),Nb)
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(nat,Ma),Nb) ) ) ).

% bit_of_nat_iff_bit
tff(fact_3391_of__nat__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,Nb: nat] : ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Ma),Nb)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ).

% of_nat_and_eq
tff(fact_3392_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( ! [N: nat] :
              ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N)
              | ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N) )
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),Nb)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
              | aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Nb) ) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_3393_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_3394_not__bit__1__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(nat,nat,suc,Nb)) ) ).

% not_bit_1_Suc
tff(fact_3395_bit__1__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),Nb)
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_3396_bit__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: num] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(num,nat,numeral_numeral(nat),Nb)) ) ).

% bit_numeral_simps(1)
tff(fact_3397_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,bit_se2584673776208193580ke_bit(A,Ma,A2)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ) ).

% bit_take_bit_iff
tff(fact_3398_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [B2: $o,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa($o,A,zero_neq_one_of_bool(A),(B2))),Nb)
        <=> ( (B2)
            & ( Nb = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_3399_AND__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => 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),Xb),Y)) ) ).

% AND_lower
tff(fact_3400_AND__upper1,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Y)),Xb) ) ).

% AND_upper1
tff(fact_3401_AND__upper2,axiom,
    ! [Y: int,Xb: 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),Xb),Y)),Y) ) ).

% AND_upper2
tff(fact_3402_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_3403_AND__upper2_H,axiom,
    ! [Y: int,Z: int,Xb: 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),Xb),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_3404_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = zero_zero(A) )
        <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_3405_inc__BitM__eq,axiom,
    ! [Nb: num] : ( inc(bitM(Nb)) = bit0(Nb) ) ).

% inc_BitM_eq
tff(fact_3406_BitM__inc__eq,axiom,
    ! [Nb: num] : ( bitM(inc(Nb)) = aa(num,num,bit1,Nb) ) ).

% BitM_inc_eq
tff(fact_3407_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_3408_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_3409_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xb: 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),Xb),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_3410_bit__not__int__iff_H,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),K)),one_one(int))),Nb)
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% bit_not_int_iff'
tff(fact_3411_eval__nat__numeral_I2_J,axiom,
    ! [Nb: num] : ( aa(num,nat,numeral_numeral(nat),bit0(Nb)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bitM(Nb))) ) ).

% eval_nat_numeral(2)
tff(fact_3412_sinh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : ( sinh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xb)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xb)),sinh(A,Y))) ) ) ).

% sinh_add
tff(fact_3413_cosh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : ( cosh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xb)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xb)),sinh(A,Y))) ) ) ).

% cosh_add
tff(fact_3414_sinh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : ( sinh(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xb)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xb)),sinh(A,Y))) ) ) ).

% sinh_diff
tff(fact_3415_cosh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : ( cosh(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xb)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xb)),sinh(A,Y))) ) ) ).

% cosh_diff
tff(fact_3416_one__plus__BitM,axiom,
    ! [Nb: num] : ( aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bitM(Nb)) = bit0(Nb) ) ).

% one_plus_BitM
tff(fact_3417_BitM__plus__one,axiom,
    ! [Nb: num] : ( aa(num,num,aa(num,fun(num,num),plus_plus(num),bitM(Nb)),one2) = bit0(Nb) ) ).

% BitM_plus_one
tff(fact_3418_cosh__plus__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),cosh(A,Xb)),sinh(A,Xb)) = exp(A,Xb) ) ) ).

% cosh_plus_sinh
tff(fact_3419_sinh__plus__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),sinh(A,Xb)),cosh(A,Xb)) = exp(A,Xb) ) ) ).

% sinh_plus_cosh
tff(fact_3420_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,Ma: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma)
     => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),bit_se2584673776208193580ke_bit(int,Ma,K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_3421_bit__concat__bit__iff,axiom,
    ! [Ma: nat,K: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,bit_concat_bit(Ma,K,L)),Nb)
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma)
          & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) )
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
          & aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_3422_signed__take__bit__eq__concat__bit,axiom,
    ! [Nb: nat,K: int] : ( bit_ri4674362597316999326ke_bit(int,Nb,K) = bit_concat_bit(Nb,K,aa(int,int,uminus_uminus(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)))) ) ).

% signed_take_bit_eq_concat_bit
tff(fact_3423_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat,A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) = zero_zero(A) )
         => ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_3424_bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,suc,Nb))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),Nb) ) ) ).

% bit_Suc
tff(fact_3425_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) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% one_and_eq
tff(fact_3426_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)) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% and_one_eq
tff(fact_3427_numeral__BitM,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : ( aa(num,A,numeral_numeral(A),bitM(Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),bit0(Nb))),one_one(A)) ) ) ).

% numeral_BitM
tff(fact_3428_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M4)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),M4)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N) ) )
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))
              <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N) ) ) ) ).

% int_bit_bound
tff(fact_3429_sinh__minus__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),sinh(A,Xb)),cosh(A,Xb)) = aa(A,A,uminus_uminus(A),exp(A,aa(A,A,uminus_uminus(A),Xb))) ) ) ).

% sinh_minus_cosh
tff(fact_3430_cosh__minus__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),cosh(A,Xb)),sinh(A,Xb)) = exp(A,aa(A,A,uminus_uminus(A),Xb)) ) ) ).

% cosh_minus_sinh
tff(fact_3431_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),Nb)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
              | ( Nb = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_3432_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),Nb)
              | ( Nb = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_3433_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( ! [J2: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,suc,J2))
         => ( aa(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))),Nb)
          <=> $ite(Nb = zero_zero(nat),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2),aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)),Nb)) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_3434_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> $ite(Nb = zero_zero(nat),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2),aa(nat,$o,bit_se5641148757651400278ts_bit(A,divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ) ) ).

% bit_rec
tff(fact_3435_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,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)
              & ~ aa(int,$o,aa(int,fun(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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ).

% and_int_rec
tff(fact_3436_sinh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : ( sinh(A,Z) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),exp(A,Z)),exp(A,aa(A,A,uminus_uminus(A),Z))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% sinh_field_def
tff(fact_3437_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xb: fun(A,nat)] : ( size_option(A,Xb,none(A)) = aa(nat,nat,suc,zero_zero(nat)) ) ).

% option.size_gen(1)
tff(fact_3438_cosh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,Xb)),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),sinh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ) ).

% cosh_square_eq
tff(fact_3439_hyperbolic__pythagoras,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ) ).

% hyperbolic_pythagoras
tff(fact_3440_sinh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,Xb)),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),cosh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ) ).

% sinh_square_eq
tff(fact_3441_set__bit__eq,axiom,
    ! [Nb: nat,K: int] : ( bit_se5668285175392031749et_bit(int,Nb,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,bit_se5641148757651400278ts_bit(int,K),Nb))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))) ) ).

% set_bit_eq
tff(fact_3442_unset__bit__eq,axiom,
    ! [Nb: nat,K: int] : ( bit_se2638667681897837118et_bit(int,Nb,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,bit_se5641148757651400278ts_bit(int,K),Nb))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))) ) ).

% unset_bit_eq
tff(fact_3443_take__bit__Suc__from__most,axiom,
    ! [Nb: nat,K: int] : ( bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb),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))),Nb)),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)))),bit_se2584673776208193580ke_bit(int,Nb,K)) ) ).

% take_bit_Suc_from_most
tff(fact_3444_cosh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : ( cosh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% cosh_double
tff(fact_3445_and__int_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Xaa) = Y )
     => ( Y = $ite(
            ( member(int,Xb,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & member(int,Xaa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xb)
                & ~ aa(int,$o,aa(int,fun(int,$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,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xb)
                  & ~ aa(int,$o,aa(int,fun(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),divide_divide(int,Xb,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,Xaa,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.elims
tff(fact_3446_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
          ( member(int,K,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
          & member(int,L,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
          aa(int,int,uminus_uminus(int),
            aa($o,int,zero_neq_one_of_bool(int),
              ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)
              & ~ aa(int,$o,aa(int,fun(int,$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,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)
                & ~ aa(int,$o,aa(int,fun(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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ).

% and_int.simps
tff(fact_3447_xor__Suc__0__eq,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Nb),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),Nb),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ) ).

% xor_Suc_0_eq
tff(fact_3448_Suc__0__xor__eq,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ) ).

% Suc_0_xor_eq
tff(fact_3449_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,aa(int,fun(list($o),int),aa(fun($o,int),fun(int,fun(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_3450_bit_Oxor__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),Xb) = zero_zero(A) ) ) ).

% bit.xor_self
tff(fact_3451_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_3452_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_3453_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_3454_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)),insert2(A),B2),bot_bot(set(A)))))
    <=> ~ member(A,B2,A3) ) ).

% subset_Compl_singleton
tff(fact_3455_card__Diff__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( member(A,A2,A3)
     => ( ~ member(A,A2,B3)
       => ( aa(set(A),nat,finite_card(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)),insert2(A),A2),B3))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_3456_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_3457_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_3458_xor__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(Xb))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb)) ) ) ).

% xor_numerals(5)
tff(fact_3459_xor__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb))),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(Xb)) ) ) ).

% xor_numerals(8)
tff(fact_3460_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_3461_and__nat__numerals_I3_J,axiom,
    ! [Xb: num] : ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit0(Xb))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ) ).

% and_nat_numerals(3)
tff(fact_3462_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_3463_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_3464_xor__nat__numerals_I3_J,axiom,
    ! [Xb: num] : ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit0(Xb))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb)) ) ).

% xor_nat_numerals(3)
tff(fact_3465_xor__nat__numerals_I4_J,axiom,
    ! [Xb: num] : ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(Xb)) ) ).

% xor_nat_numerals(4)
tff(fact_3466_and__nat__numerals_I4_J,axiom,
    ! [Xb: num] : ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ) ).

% and_nat_numerals(4)
tff(fact_3467_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_3468_Suc__0__and__eq,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) = modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% Suc_0_and_eq
tff(fact_3469_and__Suc__0__eq,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Nb),aa(nat,nat,suc,zero_zero(nat))) = modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% and_Suc_0_eq
tff(fact_3470_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: 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,Xb))),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),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ) ).

% xor_numerals(6)
tff(fact_3471_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(Xb))),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),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ) ).

% xor_numerals(4)
tff(fact_3472_of__nat__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,Nb: nat] : ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Ma),Nb)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ).

% of_nat_xor_eq
tff(fact_3473_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_3474_bit__Suc__0__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),Nb)
    <=> ( Nb = zero_zero(nat) ) ) ).

% bit_Suc_0_iff
tff(fact_3475_not__bit__Suc__0__Suc,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,suc,Nb)) ).

% not_bit_Suc_0_Suc
tff(fact_3476_Compl__insert,axiom,
    ! [A: $tType,Xb: 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)),insert2(A),Xb),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)),insert2(A),Xb),bot_bot(set(A)))) ) ).

% Compl_insert
tff(fact_3477_not__bit__Suc__0__numeral,axiom,
    ! [Nb: num] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),Nb)) ).

% not_bit_Suc_0_numeral
tff(fact_3478_and__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Ma),Nb) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(nat,int,semiring_1_of_nat(int),Ma)),aa(nat,int,semiring_1_of_nat(int),Nb))) ) ).

% and_nat_def
tff(fact_3479_card__1__singletonE,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = one_one(nat) )
     => ~ ! [X: A] : ( A3 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ).

% card_1_singletonE
tff(fact_3480_bit__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(int,nat,nat2,K)),Nb)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ) ).

% bit_nat_iff
tff(fact_3481_card__1__singleton__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X4: A] : ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A))) ) ) ).

% card_1_singleton_iff
tff(fact_3482_card__eq__SucD,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
     => ? [B4: A,B8: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B4),B8) )
          & ~ member(A,B4,B8)
          & ( aa(set(A),nat,finite_card(A),B8) = K )
          & ( ( K = zero_zero(nat) )
           => ( B8 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_3483_card__Suc__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
    <=> ? [B5: A,B9: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B5),B9) )
          & ~ member(A,B5,B9)
          & ( aa(set(A),nat,finite_card(A),B9) = K )
          & ( ( K = zero_zero(nat) )
           => ( B9 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_3484_card__Diff__singleton__if,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))) = $ite(member(A,Xb,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),one_one(nat)),aa(set(A),nat,finite_card(A),A3)) ) ).

% card_Diff_singleton_if
tff(fact_3485_card__Diff__singleton,axiom,
    ! [A: $tType,Xb: A,A3: set(A)] :
      ( member(A,Xb,A3)
     => ( aa(set(A),nat,finite_card(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)),insert2(A),Xb),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_3486_atLeastAtMostPlus1__int__conv,axiom,
    ! [Ma: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ma),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Nb))
     => ( set_or1337092689740270186AtMost(int,Ma,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Nb)) = aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Nb)),set_or1337092689740270186AtMost(int,Ma,Nb)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_3487_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)),insert2(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_3488_card__insert__le__m1,axiom,
    ! [A: $tType,Nb: nat,Y: set(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),Y))),Nb) ) ) ).

% card_insert_le_m1
tff(fact_3489_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sinh(A,Xb) = zero_zero(A) )
        <=> member(A,exp(A,Xb),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),one_one(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(A,A,uminus_uminus(A),one_one(A))),bot_bot(set(A))))) ) ) ).

% sinh_zero_iff
tff(fact_3490_and__nat__unfold,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Ma),Nb) = $ite(
          ( ( Ma = zero_zero(nat) )
          | ( Nb = 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),modulo_modulo(nat,Ma,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,Nb,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),divide_divide(nat,Ma,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% and_nat_unfold
tff(fact_3491_and__nat__rec,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Ma),Nb) = aa(nat,nat,
          aa(nat,fun(nat,nat),plus_plus(nat),
            aa($o,nat,zero_neq_one_of_bool(nat),
              ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma)
              & ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ))),
          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),divide_divide(nat,Ma,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% and_nat_rec
tff(fact_3492_xor__nat__unfold,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Ma),Nb) = $ite(
          Ma = zero_zero(nat),
          Nb,
          $ite(Nb = zero_zero(nat),Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,Ma,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,Nb,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),divide_divide(nat,Ma,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ) ).

% xor_nat_unfold
tff(fact_3493_xor__nat__rec,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Ma),Nb) = aa(nat,nat,
          aa(nat,fun(nat,nat),plus_plus(nat),
            aa($o,nat,zero_neq_one_of_bool(nat),
              ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma) != ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ))),
          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),divide_divide(nat,Ma,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% xor_nat_rec
tff(fact_3494_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,$o,aa(A,fun(A,$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,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ) ).

% xor_one_eq
tff(fact_3495_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,$o,aa(A,fun(A,$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,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ) ).

% one_xor_eq
tff(fact_3496_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list($o),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(list($o),A,aa(A,fun(list($o),A),aa(fun($o,A),fun(A,fun(list($o),A)),groups4207007520872428315er_sum($o,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),bit0(one2))),Bs)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list($o),nat,size_size(list($o)),Bs))
            & aa(nat,$o,nth($o,Bs),Nb) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_3497_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(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) = $ite(
            ( member(int,K,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & member(int,L,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)
                & ~ aa(int,$o,aa(int,fun(int,$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,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)
                  & ~ aa(int,$o,aa(int,fun(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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.psimps
tff(fact_3498_and__int_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Xaa) = Y )
     => ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,Xb),Xaa))
       => ~ ( ( Y = $ite(
                  ( member(int,Xb,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                  & member(int,Xaa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
                  aa(int,int,uminus_uminus(int),
                    aa($o,int,zero_neq_one_of_bool(int),
                      ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xb)
                      & ~ aa(int,$o,aa(int,fun(int,$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,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xb)
                        & ~ aa(int,$o,aa(int,fun(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),divide_divide(int,Xb,aa(num,int,numeral_numeral(int),bit0(one2)))),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),product_Pair(int,int,Xb),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_3499_push__bit__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : ( aa(A,A,bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),Nb)),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),Nb))) ) ) ).

% push_bit_numeral_minus_1
tff(fact_3500_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A4),
                $ite(Xaa = one_one(nat),(B4),$false) ) )
       => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : ( Xb != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
         => ( ! [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : ( Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy2,Uz2) )
           => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : ( Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary) )
                   => $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,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(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_3501_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B4),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
           => (Y) )
         => ( ( ? [V4: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy2,Uz2) )
             => (Y) )
           => ( ( ? [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
               => (Y) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary) )
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                $false,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,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(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_3502_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : ( vEBT_VEBT_set_vebt(Ta) = collect(nat,vEBT_vebt_member(Ta)) ) ).

% set_vebt'_def
tff(fact_3503_push__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_3504_push__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),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_3505_push__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : ( aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),zero_zero(A)) = zero_zero(A) ) ) ).

% push_bit_of_0
tff(fact_3506_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,A2: A] :
          ( ( aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_3507_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,Nb: nat,A2: A] : ( aa(A,A,bit_se4730199178511100633sh_bit(A,Ma),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2)) = aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)),A2) ) ) ).

% push_bit_push_bit
tff(fact_3508_concat__bit__of__zero__1,axiom,
    ! [Nb: nat,L: int] : ( bit_concat_bit(Nb,zero_zero(int),L) = aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),L) ) ).

% concat_bit_of_zero_1
tff(fact_3509_card__Collect__less__nat,axiom,
    ! [Nb: nat] : ( aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aa(nat,fun(nat,$o)),Nb))) = Nb ) ).

% card_Collect_less_nat
tff(fact_3510_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_3511_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_3512_card__Collect__le__nat,axiom,
    ! [Nb: nat] : ( aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ab(nat,fun(nat,$o)),Nb))) = aa(nat,nat,suc,Nb) ) ).

% card_Collect_le_nat
tff(fact_3513_push__bit__Suc__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,K: num] : ( aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),K)) = aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(num,A,numeral_numeral(A),bit0(K))) ) ) ).

% push_bit_Suc_numeral
tff(fact_3514_push__bit__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K: num] : ( aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(K)))) ) ) ).

% push_bit_Suc_minus_numeral
tff(fact_3515_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : ( aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb)),A2) = aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),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_3516_push__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : ( aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),one_one(A)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) ) ) ).

% push_bit_of_1
tff(fact_3517_push__bit__of__Suc__0,axiom,
    ! [Nb: nat] : ( aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Nb),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))),Nb) ) ).

% push_bit_of_Suc_0
tff(fact_3518_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2))
        <=> ( ( Nb != zero_zero(nat) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% even_push_bit_iff
tff(fact_3519_push__bit__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [L: num,K: num] : ( aa(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,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_3520_flip__bit__int__def,axiom,
    ! [Nb: nat,K: int] : ( bit_se8732182000553998342ip_bit(int,Nb,K) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),one_one(int))) ) ).

% flip_bit_int_def
tff(fact_3521_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_ac(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_3522_Compl__eq,axiom,
    ! [A: $tType,A3: set(A)] : ( aa(set(A),set(A),uminus_uminus(set(A)),A3) = collect(A,aTP_Lamp_ad(set(A),fun(A,$o),A3)) ) ).

% Compl_eq
tff(fact_3523_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] : ( collect(A,aTP_Lamp_ae(fun(A,$o),fun(A,$o),P)) = aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P)) ) ).

% Collect_neg_eq
tff(fact_3524_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_3525_push__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K: int] : ( aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(int,A,ring_1_of_int(A),K)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K)) ) ) ).

% push_bit_of_int
tff(fact_3526_push__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,Ma: nat] : ( aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(nat,A,semiring_1_of_nat(A),Ma)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Nb),Ma)) ) ) ).

% push_bit_of_nat
tff(fact_3527_of__nat__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,Nb: nat] : ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Ma),Nb)) = aa(A,A,bit_se4730199178511100633sh_bit(A,Ma),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ).

% of_nat_push_bit
tff(fact_3528_push__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : ( aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Nb),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K)) ) ).

% push_bit_nat_eq
tff(fact_3529_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),$o),ord_less(set(A)),collect(A,aTP_Lamp_af(A,fun(A,$o),A2))),collect(A,aTP_Lamp_af(A,fun(A,$o),B2)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
            & ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3530_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aTP_Lamp_af(A,fun(A,$o),A2))),collect(A,aTP_Lamp_af(A,fun(A,$o),B2)))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% subset_divisors_dvd
tff(fact_3531_push__bit__minus,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : ( aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2)) ) ) ).

% push_bit_minus
tff(fact_3532_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_ag(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_3533_push__bit__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : ( aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),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,bit_se4730199178511100633sh_bit(A,Nb),A2)),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),B2)) ) ) ).

% push_bit_add
tff(fact_3534_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] :
          ( aa(num,A,numeral_numeral(A),bit0(Nb)) = $let(
              m2: A,
              m2:= aa(num,A,numeral_numeral(A),Nb),
              aa(A,A,aa(A,fun(A,A),plus_plus(A),m2),m2) ) ) ) ).

% numeral_code(2)
tff(fact_3535_nat__less__as__int,axiom,
    ! [X3: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_less_as_int
tff(fact_3536_nat__leq__as__int,axiom,
    ! [X3: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_leq_as_int
tff(fact_3537_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] :
          ( aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)) = $let(
              m2: A,
              m2:= aa(num,A,numeral_numeral(A),Nb),
              aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),m2),m2)),one_one(A)) ) ) ) ).

% numeral_code(3)
tff(fact_3538_nat__plus__as__int,axiom,
    ! [X3: nat,Xa: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X3),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa))) ) ).

% nat_plus_as_int
tff(fact_3539_nat__times__as__int,axiom,
    ! [X3: nat,Xa: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X3),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa))) ) ).

% nat_times_as_int
tff(fact_3540_nat__minus__as__int,axiom,
    ! [X3: nat,Xa: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X3),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa))) ) ).

% nat_minus_as_int
tff(fact_3541_card__less__Suc2,axiom,
    ! [M9: set(nat),I: nat] :
      ( ~ member(nat,zero_zero(nat),M9)
     => ( aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ah(set(nat),fun(nat,fun(nat,$o)),M9),I))) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ai(set(nat),fun(nat,fun(nat,$o)),M9),I))) ) ) ).

% card_less_Suc2
tff(fact_3542_card__less__Suc,axiom,
    ! [M9: set(nat),I: nat] :
      ( member(nat,zero_zero(nat),M9)
     => ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ah(set(nat),fun(nat,fun(nat,$o)),M9),I)))) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ai(set(nat),fun(nat,fun(nat,$o)),M9),I))) ) ) ).

% card_less_Suc
tff(fact_3543_card__less,axiom,
    ! [M9: set(nat),I: nat] :
      ( member(nat,zero_zero(nat),M9)
     => ( aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ai(set(nat),fun(nat,fun(nat,$o)),M9),I))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_3544_nat__div__as__int,axiom,
    ! [X3: nat,Xa: nat] : ( divide_divide(nat,X3,Xa) = aa(int,nat,nat2,divide_divide(int,aa(nat,int,semiring_1_of_nat(int),X3),aa(nat,int,semiring_1_of_nat(int),Xa))) ) ).

% nat_div_as_int
tff(fact_3545_nat__mod__as__int,axiom,
    ! [X3: nat,Xa: nat] : ( modulo_modulo(nat,X3,Xa) = aa(int,nat,nat2,modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),X3),aa(nat,int,semiring_1_of_nat(int),Xa))) ) ).

% nat_mod_as_int
tff(fact_3546_card__nth__roots,axiom,
    ! [C2: complex,Nb: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( aa(set(complex),nat,finite_card(complex),collect(complex,aa(nat,fun(complex,$o),aTP_Lamp_aj(complex,fun(nat,fun(complex,$o)),C2),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_3547_card__roots__unity__eq,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(set(complex),nat,finite_card(complex),collect(complex,aTP_Lamp_ak(nat,fun(complex,$o),Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_3548_XOR__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( 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),Xb),Y)) ) ) ).

% XOR_lower
tff(fact_3549_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] :
          ( bit_ri4674362597316999326ke_bit(A,Nb,A2) = $let(
              l: A,
              l:= bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb),A2),
              $ite(aa(nat,$o,bit_se5641148757651400278ts_bit(A,l),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),l),aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb)),aa(A,A,uminus_uminus(A),one_one(A)))),l) ) ) ) ).

% signed_take_bit_code
tff(fact_3550_card__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),collect(A,aTP_Lamp_al(nat,fun(A,$o),Nb)))),Nb) ) ) ).

% card_roots_unity
tff(fact_3551_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,Nb: nat,A2: A] : ( aa(A,A,bit_se4730199178511100633sh_bit(A,Ma),bit_se2584673776208193580ke_bit(A,Nb,A2)) = bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb),aa(A,A,bit_se4730199178511100633sh_bit(A,Ma),A2)) ) ) ).

% push_bit_take_bit
tff(fact_3552_take__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,Nb: nat,A2: A] : ( bit_se2584673776208193580ke_bit(A,Ma,aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2)) = aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb),A2)) ) ) ).

% take_bit_push_bit
tff(fact_3553_flip__bit__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : ( bit_se8732182000553998342ip_bit(nat,Ma,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Nb),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Ma),one_one(nat))) ) ).

% flip_bit_nat_def
tff(fact_3554_diff__nat__eq__if,axiom,
    ! [Z: int,Z3: int] :
      ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z3)) = $ite(
          aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),zero_zero(int)),
          aa(int,nat,nat2,Z),
          $let(
            d: int,
            d:= aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z3),
            $ite(aa(int,$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_3555_atLeast0__atMost__Suc,axiom,
    ! [Nb: nat] : ( set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(nat,nat,suc,Nb)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% atLeast0_atMost_Suc
tff(fact_3556_atLeastAtMost__insertL,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => ( aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),Ma),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),Nb)) = set_or1337092689740270186AtMost(nat,Ma,Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_3557_atLeastAtMostSuc__conv,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,Nb))
     => ( set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(nat,nat,suc,Nb)),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_3558_Icc__eq__insert__lb__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => ( set_or1337092689740270186AtMost(nat,Ma,Nb) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),Ma),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_3559_bit__push__bit__iff__int,axiom,
    ! [Ma: nat,K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_se4730199178511100633sh_bit(int,Ma),K)),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma)) ) ) ).

% bit_push_bit_iff_int
tff(fact_3560_xor__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Ma),Nb) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(nat,int,semiring_1_of_nat(int),Ma)),aa(nat,int,semiring_1_of_nat(int),Nb))) ) ).

% xor_nat_def
tff(fact_3561_set__decode__def,axiom,
    ! [Xb: nat] : ( nat_set_decode(Xb) = collect(nat,aTP_Lamp_am(nat,fun(nat,$o),Xb)) ) ).

% set_decode_def
tff(fact_3562_bit__push__bit__iff__nat,axiom,
    ! [Ma: nat,Q4: nat,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Ma),Q4)),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(nat,Q4),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_3563_concat__bit__eq,axiom,
    ! [Nb: nat,K: int,L: int] : ( bit_concat_bit(Nb,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),bit_se2584673776208193580ke_bit(int,Nb,K)),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),L)) ) ).

% concat_bit_eq
tff(fact_3564_flip__bit__eq__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : ( bit_se8732182000553998342ip_bit(A,Nb,A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),one_one(A))) ) ) ).

% flip_bit_eq_xor
tff(fact_3565_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] :
          ( comm_s3205402744901411588hammer(A,A2,Nb) = $ite(Nb = zero_zero(nat),one_one(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_an(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),one_one(A))) ) ) ).

% pochhammer_code
tff(fact_3566_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_3567_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),divide_divide(A,set_fo6178422350223883121st_nat(A,aTP_Lamp_ao(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_3568_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option(product_prod(nat,nat)),V2: nat,TreeLista: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy,aa(nat,nat,suc,V2),TreeLista,S),Xb)
    <=> $let(
          pos2: nat,
          pos2:= vEBT_VEBT_high(Xb,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos2),vEBT_VEBT_low(Xb,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_3569_push__bit__minus__one,axiom,
    ! [Nb: nat] : ( aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),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))),Nb)) ) ).

% push_bit_minus_one
tff(fact_3570_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V2: nat,TreeLista: list(vEBT_VEBT),Vd2: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeLista,Vd2),Xb)
    <=> $let(
          pos2: nat,
          pos2:= vEBT_VEBT_high(Xb,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos2),vEBT_VEBT_low(Xb,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_3571_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z: nat] :
      ( ~ member(nat,Nb,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))),Nb)),Z)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),Nb),nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_3572_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi2: nat,Ma: nat,V2: nat,TreeLista: list(vEBT_VEBT),Vc: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma)),aa(nat,nat,suc,V2),TreeLista,Vc),Xb)
    <=> ( ( Xb = Mi2 )
        | ( Xb = Ma )
        | $let(
            pos2: nat,
            pos2:= vEBT_VEBT_high(Xb,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos2),vEBT_VEBT_low(Xb,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_3573_vebt__member_Osimps_I5_J,axiom,
    ! [Mi2: nat,Ma: nat,Va2: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeLista,Summarya)),Xb)
    <=> $ite(
          Xb = Mi2,
          $true,
          $ite(
            Xb = Ma,
            $true,
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mi2),
              $false,
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xb),
                $false,
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xb,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,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),vEBT_VEBT_low(Xb,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_3574_XOR__upper,axiom,
    ! [Xb: int,Nb: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
       => ( 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))),Nb))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xb),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ) ) ).

% XOR_upper
tff(fact_3575_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xb,Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A4),
                $ite(Xaa = one_one(nat),(B4),$false) ) )
       => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : ( Xb != vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [S2: vEBT_VEBT] : ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList,S2) )
               => $let(
                    pos2: nat,
                    pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_3576_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xb,Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A4),
                  $ite(Xaa = one_one(nat),(B4),$false) ) )
       => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [S2: vEBT_VEBT] : ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList,S2) )
             => ~ $let(
                    pos2: nat,
                    pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_3577_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xb,Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B4),$false) ) ) )
       => ( ( ? [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : ( Xb = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
           => (Y) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [S2: vEBT_VEBT] : ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList,S2) )
               => ( (Y)
                <=> ~ $let(
                        pos2: nat,
                        pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_3578_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xb,Xaa)
     => ( ! [Mi: nat,Ma2: nat] :
            ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2) )
           => ~ ( ( Xaa = Mi )
                | ( Xaa = Ma2 ) ) )
       => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [Vc2: vEBT_VEBT] : ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList,Vc2) )
             => ~ ( ( Xaa = Mi )
                  | ( Xaa = Ma2 )
                  | $let(
                      pos2: nat,
                      pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
         => ~ ! [V4: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [Vd: vEBT_VEBT] : ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList,Vd) )
               => ~ $let(
                      pos2: nat,
                      pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_3579_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,A0),A1))
     => ( ! [K2: int,L2: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,K2),L2))
           => ( ( ~ ( member(int,K2,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                    & member(int,L2,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),P,divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L2,aa(num,int,numeral_numeral(int),bit0(one2)))) )
             => aa(int,$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_3580_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,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K) != ~ aa(int,$o,aa(int,fun(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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ).

% xor_int_rec
tff(fact_3581_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A4),
                  $ite(Xaa = one_one(nat),(B4),$false) ) )
       => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [Summary: vEBT_VEBT] : ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary) )
             => ~ $ite(
                    Xaa = Mi,
                    $true,
                    $ite(
                      Xaa = Ma2,
                      $true,
                      $ite(
                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                        $false,
                        $ite(
                          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(Xaa,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(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_3582_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xb,Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
         => (Y) )
       => ( ( ? [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : ( Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
           => (Y) )
         => ( ! [Mi: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2) )
               => ( (Y)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma2 ) ) ) )
           => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList,Vc2) )
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos2: nat,
                            pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
             => ~ ! [V4: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Vd: vEBT_VEBT] : ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList,Vd) )
                   => ( (Y)
                    <=> ~ $let(
                            pos2: nat,
                            pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_3583_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xb,Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : ( Xb != vEBT_Leaf((Uu2),(Uv2)) )
       => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : ( Xb != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
         => ( ! [Mi: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2) )
               => ( ( Xaa = Mi )
                  | ( Xaa = Ma2 ) ) )
           => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList,Vc2) )
                 => ( ( Xaa = Mi )
                    | ( Xaa = Ma2 )
                    | $let(
                        pos2: nat,
                        pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
             => ~ ! [V4: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Vd: vEBT_VEBT] : ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList,Vd) )
                   => $let(
                        pos2: nat,
                        pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_3584_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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2))))),
                  $ite(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_3585_monoseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => topological_monoseq(real,aTP_Lamp_ap(real,fun(nat,real),Xb)) ) ).

% monoseq_arctan_series
tff(fact_3586_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,Nb: nat] : ( aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,Nb))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_aq(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))),Nb)),one_one(nat)))) ) ) ).

% pochhammer_times_pochhammer_half
tff(fact_3587_ln__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(num,real,numeral_numeral(real),bit0(one2)))
       => ( aa(real,real,ln_ln(real),Xb) = suminf(real,aTP_Lamp_ar(real,fun(nat,real),Xb)) ) ) ) ).

% ln_series
tff(fact_3588_arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,real,arctan,Xb) = suminf(real,aTP_Lamp_as(real,fun(nat,real),Xb)) ) ) ).

% arctan_series
tff(fact_3589_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,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3),ring_1_Ints(B)) ) ) ).

% Ints_prod
tff(fact_3590_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),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7121269368397514597t_prod(B,nat),F2),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_at(fun(B,nat),fun(B,A),F2)),A3) ) ) ).

% of_nat_prod
tff(fact_3591_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),aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7121269368397514597t_prod(B,int),F2),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_au(fun(B,int),fun(B,A),F2)),A3) ) ) ).

% of_int_prod
tff(fact_3592_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(nat,A)] : ( suminf(A,aTP_Lamp_av(fun(nat,A),fun(nat,A),F2)) = aa(nat,A,F2,zero_zero(nat)) ) ) ).

% powser_zero
tff(fact_3593_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),Ma),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb)))) ) ) ).

% prod.cl_ivl_Suc
tff(fact_3594_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_aw(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_3595_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Ma: nat,K: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ax(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_3596_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,Ma: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Nb,Ma)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ay(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Ma)),set_or1337092689740270186AtMost(nat,Nb,Ma)) ) ) ).

% prod.atLeastAtMost_rev
tff(fact_3597_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_3598_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Ma: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_3599_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Ma: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Ma)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),Nb))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_3600_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Ma: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Ma)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_aw(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Ma,Nb))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_3601_fact__prod,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : ( semiring_char_0_fact(A,Nb) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_az(nat,nat)),set_or1337092689740270186AtMost(nat,one_one(nat),Nb))) ) ) ).

% fact_prod
tff(fact_3602_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F2: fun(nat,A),A2: nat,B2: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F2),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ba(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,one_one(A)) ) ) ).

% prod_atLeastAtMost_code
tff(fact_3603_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Ma: nat,Nb: nat,G: fun(nat,A),P3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P3))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P3)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_3604_fact__eq__fact__times,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
     => ( semiring_char_0_fact(nat,Ma) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,Nb)),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_az(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Ma))) ) ) ).

% fact_eq_fact_times
tff(fact_3605_monoseq__realpow,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => topological_monoseq(real,aa(real,fun(nat,real),power_power(real),Xb)) ) ) ).

% monoseq_realpow
tff(fact_3606_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : ( comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_bb(A,fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ) ).

% pochhammer_Suc_prod
tff(fact_3607_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : ( comm_s3205402744901411588hammer(A,A2,Nb) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_bc(A,fun(nat,fun(nat,A)),A2),Nb)),set_or1337092689740270186AtMost(nat,one_one(nat),Nb)) ) ) ).

% pochhammer_prod_rev
tff(fact_3608_fact__div__fact,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
     => ( divide_divide(nat,semiring_char_0_fact(nat,Ma),semiring_char_0_fact(nat,Nb)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_az(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),Ma)) ) ) ).

% fact_div_fact
tff(fact_3609_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_bd(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% prod.in_pairs
tff(fact_3610_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : ( comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_bc(A,fun(nat,fun(nat,A)),A2),Nb)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ) ).

% pochhammer_Suc_prod_rev
tff(fact_3611_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)) = divide_divide(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_be(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_3612_pi__series,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = suminf(real,aTP_Lamp_bf(nat,real)) ).

% pi_series
tff(fact_3613_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)) = 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_3614_prod_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),bot_bot(set(B))) = one_one(A) ) ) ).

% prod.empty
tff(fact_3615_prod__le__power,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B),Nb: B,K: nat] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),Nb) ) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),K)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),Nb)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Nb),K)) ) ) ) ) ).

% prod_le_power
tff(fact_3616_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_bg(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_3617_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_bh(B,A)),A3) = one_one(A) ) ) ).

% prod.neutral_const
tff(fact_3618_int__prod,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A)] : ( aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F2),A3)) = aa(set(A),int,aa(fun(A,int),fun(set(A),int),groups7121269368397514597t_prod(A,int),aTP_Lamp_bi(fun(A,nat),fun(A,int),F2)),A3) ) ).

% int_prod
tff(fact_3619_prod__int__eq,axiom,
    ! [I: nat,J: nat] : ( aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I,J)) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_bj(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_3620_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : ( aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J))) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_bj(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_3621_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A3: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) != one_one(A) )
         => ~ ! [A4: B] :
                ( member(B,A4,A3)
               => ( aa(B,A,G,A4) = one_one(A) ) ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_3622_prod_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => ( aa(A,B,G,X) = one_one(B) ) )
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = one_one(B) ) ) ) ).

% prod.neutral
tff(fact_3623_prod__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),aa(A,B,G,I2)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)) ) ) ).

% prod_mono
tff(fact_3624_prod__nonneg,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,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ).

% prod_nonneg
tff(fact_3625_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)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ).

% prod_pos
tff(fact_3626_prod__ge__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,X)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ).

% prod_ge_1
tff(fact_3627_prod__le__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => ( aa(B,$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,$o),ord_less_eq(B),aa(A,B,F2,X)),one_one(B)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),one_one(B)) ) ) ).

% prod_le_1
tff(fact_3628_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,A0),A1))
     => ( ! [I2: int,J2: int] :
            ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,I2),J2))
           => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J2)
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))),J2) )
             => aa(int,$o,aa(int,fun(int,$o),P,I2),J2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_3629_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,Nb: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => bij_betw(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,aa(real,real,root(Nb),real_V7770717601297561774m_norm(complex,C2)))),cis(divide_divide(real,arg(C2),aa(nat,real,semiring_1_of_nat(real),Nb))))),collect(complex,aTP_Lamp_ak(nat,fun(complex,$o),Nb)),collect(complex,aa(nat,fun(complex,$o),aTP_Lamp_aj(complex,fun(nat,fun(complex,$o)),C2),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_3630_summable__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => summable(real,aTP_Lamp_as(real,fun(nat,real),Xb)) ) ).

% summable_arctan_series
tff(fact_3631_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),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ) ) ).

% xor_int_unfold
tff(fact_3632_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_bk(nat,fun(fun(nat,A),fun(nat,A)),I),F2)) ) ).

% summable_single
tff(fact_3633_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_bl(nat,A)) ) ).

% summable_zero
tff(fact_3634_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,aa(nat,fun(nat,A),aTP_Lamp_bm(fun(nat,A),fun(nat,fun(nat,A)),F2),K))
        <=> summable(A,F2) ) ) ).

% summable_iff_shift
tff(fact_3635_bit_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)) = zero_zero(A) ) ) ).

% bit.conj_cancel_right
tff(fact_3636_bit_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),Xb) = zero_zero(A) ) ) ).

% bit.conj_cancel_left
tff(fact_3637_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_bn(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_cmult_iff
tff(fact_3638_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_bo(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_divide_iff
tff(fact_3639_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_3640_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_3641_bit_Oxor__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,uminus_uminus(A),one_one(A))),Xb) = aa(A,A,bit_ri4277139882892585799ns_not(A),Xb) ) ) ).

% bit.xor_one_left
tff(fact_3642_bit_Oxor__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),Xb) ) ) ).

% bit.xor_one_right
tff(fact_3643_bit_Oxor__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),Xb) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% bit.xor_cancel_left
tff(fact_3644_bit_Oxor__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% bit.xor_cancel_right
tff(fact_3645_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_3646_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_3647_minus__not__numeral__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : ( aa(A,A,uminus_uminus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),Nb))) = aa(num,A,numeral_numeral(A),inc(Nb)) ) ) ).

% minus_not_numeral_eq
tff(fact_3648_push__bit__minus__one__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : ( aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb)) ) ) ).

% push_bit_minus_one_eq_not_mask
tff(fact_3649_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_3650_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_3651_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_bp(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_add
tff(fact_3652_summable__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_diff
tff(fact_3653_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: A] :
          ( summable(A,aTP_Lamp_br(A,fun(nat,A),C2))
        <=> ( C2 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_3654_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_3655_summable__minus__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_minus_iff
tff(fact_3656_summable__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => summable(A,aTP_Lamp_bs(fun(nat,A),fun(nat,A),F2)) ) ) ).

% summable_minus
tff(fact_3657_summable__Suc__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_Suc_iff
tff(fact_3658_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_bm(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) ) ) ).

% summable_ignore_initial_segment
tff(fact_3659_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_3660_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_3661_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_3662_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_bn(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
         => ( ( C2 != zero_zero(A) )
           => summable(A,F2) ) ) ) ).

% summable_mult_D
tff(fact_3663_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_3664_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_bp(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_add
tff(fact_3665_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_bq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_diff
tff(fact_3666_suminf__minus,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,uminus_uminus(A),suminf(A,F2)) ) ) ) ).

% suminf_minus
tff(fact_3667_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => ( ( suminf(A,F2) = zero_zero(A) )
            <=> ! [N4: nat] : ( aa(nat,A,F2,N4) = zero_zero(A) ) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_3668_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),suminf(A,F2)) ) ) ) ).

% suminf_nonneg
tff(fact_3669_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_3670_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_3671_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_3672_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_3673_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_bu(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_0_powser
tff(fact_3674_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_bv(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_zero_power'
tff(fact_3675_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_bw(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_bx(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_split_head
tff(fact_3676_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_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)) ) ) ).

% powser_split_head(3)
tff(fact_3677_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_3678_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),Ma: nat,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_ca(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F2),Ma),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_bx(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_3679_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_3680_disjunctive__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [B2: A,A2: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N)
             => aa(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_3681_take__bit__not__eq__mask__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : ( bit_se2584673776208193580ke_bit(A,Nb,aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),bit_se2239418461657761734s_mask(A,Nb)),bit_se2584673776208193580ke_bit(A,Nb,A2)) ) ) ).

% take_bit_not_eq_mask_diff
tff(fact_3682_minus__numeral__inc__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Nb))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),Nb)) ) ) ).

% minus_numeral_inc_eq
tff(fact_3683_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,$o),ord_less(A),zero_zero(A)),suminf(A,F2))
            <=> ? [I3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I3)) ) ) ) ) ).

% suminf_pos_iff
tff(fact_3684_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_3685_and__not__numerals_I2_J,axiom,
    ! [Nb: 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(Nb)))) = one_one(int) ) ).

% and_not_numerals(2)
tff(fact_3686_and__not__numerals_I4_J,axiom,
    ! [Ma: num] : ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(Ma))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),bit0(Ma)) ) ).

% and_not_numerals(4)
tff(fact_3687_not__numeral__Bit0__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : ( aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bit0(Nb))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb))) ) ) ).

% not_numeral_Bit0_eq
tff(fact_3688_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),one_one(real))
         => summable(A,aa(A,fun(nat,A),power_power(A),Xb)) ) ) ).

% complete_algebra_summable_geometric
tff(fact_3689_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_3690_suminf__split__head,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,aa(A,fun(A,A),minus_minus(A),suminf(A,F2)),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_3691_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : summable(A,aTP_Lamp_cb(A,fun(nat,A),Xb)) ) ).

% summable_exp
tff(fact_3692_bit__minus__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),K)),Nb)
    <=> aa(nat,$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)))),Nb) ) ).

% bit_minus_int_iff
tff(fact_3693_not__numeral__BitM__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : ( aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bitM(Nb))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(Nb))) ) ) ).

% not_numeral_BitM_eq
tff(fact_3694_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Ma: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( bit_se2584673776208193580ke_bit(A,Ma,aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_3695_push__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Ma: nat,Nb: nat] : ( aa(A,A,bit_se4730199178511100633sh_bit(A,Ma),bit_se2239418461657761734s_mask(A,Nb)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Ma))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Ma))) ) ) ).

% push_bit_mask_eq
tff(fact_3696_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : ( bit_se2638667681897837118et_bit(A,Nb,A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),one_one(A)))) ) ) ).

% unset_bit_eq_and_not
tff(fact_3697_unset__bit__int__def,axiom,
    ! [Nb: nat,K: int] : ( bit_se2638667681897837118et_bit(int,Nb,K) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),one_one(int)))) ) ).

% unset_bit_int_def
tff(fact_3698_and__not__numerals_I7_J,axiom,
    ! [Ma: num] : ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Ma))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),bit0(Ma)) ) ).

% and_not_numerals(7)
tff(fact_3699_and__not__numerals_I3_J,axiom,
    ! [Nb: 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,Nb)))) = zero_zero(int) ) ).

% and_not_numerals(3)
tff(fact_3700_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_by(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_by(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_bz(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_3701_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_by(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_bz(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_by(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_3702_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,F2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
         => ( summable(A,F2)
           => ? [N7: nat] :
              ! [N3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N3)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_bm(fun(nat,A),fun(nat,fun(nat,A)),F2),N3)))),R2) ) ) ) ) ).

% suminf_exist_split
tff(fact_3703_summable__power__series,axiom,
    ! [F2: fun(nat,real),Z: real] :
      ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,I2)),one_one(real))
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,I2))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Z)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z),one_one(real))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_cc(fun(nat,real),fun(real,fun(nat,real)),F2),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_3704_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),Nb)
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) != zero_zero(A) )
            & ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ) ).

% bit_not_iff_eq
tff(fact_3705_minus__exp__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: 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))),Nb)) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb)) ) ) ).

% minus_exp_eq_not_mask
tff(fact_3706_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C2: real,N5: nat,F2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),one_one(real))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),N)
               => aa(real,$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_3707_and__not__numerals_I8_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Ma))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Nb)))) = 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),Ma)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb))))) ) ).

% and_not_numerals(8)
tff(fact_3708_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,aa(int,fun(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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ).

% not_int_rec
tff(fact_3709_vebt__buildup_Oelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( ( ( Xb = zero_zero(nat) )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
               => ( Y != $ite(
                      aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                      $let(
                        half: nat,
                        half:= 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:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_3710_sin__paired,axiom,
    ! [Xb: real] : sums(real,aTP_Lamp_cd(real,fun(nat,real),Xb),sin(real,Xb)) ).

% sin_paired
tff(fact_3711_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,Ma: nat,Nb: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xb)),set_or1337092689740270186AtMost(nat,Ma,Nb)) = $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma),
              zero_zero(A),
              $ite(Xb = 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),Nb),one_one(nat))),Ma)),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),Xb),Ma)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,Nb))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))) ) ) ) ).

% sum_gp
tff(fact_3712_vebt__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A4),(B4))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B4),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = 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),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] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy2,Uz2)),Xaa)) )
             => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary)),Xaa))
                       => $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                $false,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,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(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_3713_intind,axiom,
    ! [A: $tType,I: nat,Nb: nat,P: fun(A,$o),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Nb)
     => ( aa(A,$o,P,Xb)
       => aa(A,$o,P,aa(nat,A,nth(A,replicate(A,Nb,Xb)),I)) ) ) ).

% intind
tff(fact_3714_replicate__eq__replicate,axiom,
    ! [A: $tType,Ma: nat,Xb: A,Nb: nat,Y: A] :
      ( ( replicate(A,Ma,Xb) = replicate(A,Nb,Y) )
    <=> ( ( Ma = Nb )
        & ( ( Ma != zero_zero(nat) )
         => ( Xb = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_3715_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,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3),ring_1_Ints(B)) ) ) ).

% Ints_sum
tff(fact_3716_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ce(B,A)),A3) = zero_zero(A) ) ) ).

% sum.neutral_const
tff(fact_3717_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,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F2),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cf(fun(B,nat),fun(B,A),F2)),A3) ) ) ).

% of_nat_sum
tff(fact_3718_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,aa(fun(B,int),fun(set(B),int),groups7311177749621191930dd_sum(B,int),F2),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cg(fun(B,int),fun(B,A),F2)),A3) ) ) ).

% of_int_sum
tff(fact_3719_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => sums(A,aTP_Lamp_bl(nat,A),zero_zero(A)) ) ).

% sums_zero
tff(fact_3720_sum_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),bot_bot(set(B))) = zero_zero(A) ) ) ).

% sum.empty
tff(fact_3721_Ball__set__replicate,axiom,
    ! [A: $tType,Nb: nat,A2: A,P: fun(A,$o)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),replicate(A,Nb,A2)))
         => aa(A,$o,P,X4) )
    <=> ( aa(A,$o,P,A2)
        | ( Nb = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_3722_Bex__set__replicate,axiom,
    ! [A: $tType,Nb: nat,A2: A,P: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),replicate(A,Nb,A2)))
          & aa(A,$o,P,X4) )
    <=> ( aa(A,$o,P,A2)
        & ( Nb != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_3723_in__set__replicate,axiom,
    ! [A: $tType,Xb: A,Nb: nat,Y: A] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),replicate(A,Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_3724_nth__replicate,axiom,
    ! [A: $tType,I: nat,Nb: nat,Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Nb)
     => ( aa(nat,A,nth(A,replicate(A,Nb,Xb)),I) = Xb ) ) ).

% nth_replicate
tff(fact_3725_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,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ch(fun(B,A),fun(B,A),F2)),A3)) ) ).

% sum_abs_ge_zero
tff(fact_3726_sum__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y: A,A3: set(B)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ci(A,fun(B,A),Y)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A3))),Y) ) ) ).

% sum_constant
tff(fact_3727_set__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( Nb != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),replicate(A,Nb,Xb)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_3728_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A),Xb: A] :
          ( sums(A,aTP_Lamp_bu(fun(nat,A),fun(nat,A),A2),Xb)
        <=> ( aa(nat,A,A2,zero_zero(nat)) = Xb ) ) ) ).

% powser_sums_zero_iff
tff(fact_3729_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),Ma),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb)))) ) ) ).

% sum.cl_ivl_Suc
tff(fact_3730_sums__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F2: fun(nat,A)] : sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bk(nat,fun(fun(nat,A),fun(nat,A)),I),F2),aa(nat,A,F2,I)) ) ).

% sums_single
tff(fact_3731_sums__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( sums(A,F2,A2)
         => ( sums(A,G,B2)
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% sums_diff
tff(fact_3732_sums__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( sums(A,F2,A2)
         => ( sums(A,G,B2)
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bp(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_3733_sums__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] :
          ( ! [N: nat] : ( aa(nat,A,F2,N) = zero_zero(A) )
         => sums(A,F2,zero_zero(A)) ) ) ).

% sums_0
tff(fact_3734_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) != zero_zero(A) )
         => ~ ! [A4: B] :
                ( member(B,A4,A3)
               => ( aa(B,A,G,A4) = zero_zero(A) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_3735_sum_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => ( aa(A,B,G,X) = zero_zero(B) ) )
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = zero_zero(B) ) ) ) ).

% sum.neutral
tff(fact_3736_sums__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),A2: A] :
          ( sums(A,F2,A2)
         => sums(A,aTP_Lamp_bs(fun(nat,A),fun(nat,A),F2),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% sums_minus
tff(fact_3737_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,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_cj(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,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),A3)) ) ) ).

% sum.distrib
tff(fact_3738_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,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_ck(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,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3)) ) ) ).

% sum_subtractf
tff(fact_3739_sum__negf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),A3: set(B)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cl(fun(B,A),fun(B,A),F2)),A3) = aa(A,A,uminus_uminus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)) ) ) ).

% sum_negf
tff(fact_3740_sum__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => aa(B,$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_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)) ) ) ).

% sum_nonneg
tff(fact_3741_sum__nonpos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),zero_zero(B)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),zero_zero(B)) ) ) ).

% sum_nonpos
tff(fact_3742_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(nat),F2: fun(nat,A),G: fun(nat,A)] :
          ( ~ member(nat,zero_zero(nat),A3)
         => ( ! [X: nat] :
                ( member(nat,aa(nat,nat,suc,X),A3)
               => ( aa(nat,A,F2,aa(nat,nat,suc,X)) = aa(nat,A,G,aa(nat,nat,suc,X)) ) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),A3) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),A3) ) ) ) ) ).

% sum_cong_Suc
tff(fact_3743_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D3: A] :
          ( ( C2 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cm(A,fun(fun(nat,A),fun(nat,A)),C2),F2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3))
          <=> sums(A,F2,D3) ) ) ) ).

% sums_mult_iff
tff(fact_3744_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D3: A] :
          ( ( C2 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cn(A,fun(fun(nat,A),fun(nat,A)),C2),F2),aa(A,A,aa(A,fun(A,A),times_times(A),D3),C2))
          <=> sums(A,F2,D3) ) ) ) ).

% sums_mult2_iff
tff(fact_3745_replicate__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = Nb )
     => ( ! [Y3: A] :
            ( member(A,Y3,aa(list(A),set(A),set2(A),Xs))
           => ( Y3 = Xb ) )
       => ( Xs = replicate(A,Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_3746_replicate__length__same,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ( X = Xb ) )
     => ( replicate(A,aa(list(A),nat,size_size(list(A)),Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_3747_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_co(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_3748_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Ma: nat,K: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_cp(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_3749_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A),A2: A] :
          ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bn(A,fun(fun(nat,A),fun(nat,A)),C2),F2),A2)
         => ( ( C2 != zero_zero(A) )
           => sums(A,F2,divide_divide(A,A2,C2)) ) ) ) ).

% sums_mult_D
tff(fact_3750_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( sums(A,aTP_Lamp_bt(fun(nat,A),fun(nat,A),F2),S)
           => sums(A,F2,S) ) ) ) ).

% sums_Suc_imp
tff(fact_3751_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),L: A] :
          ( sums(A,aTP_Lamp_cq(fun(nat,A),fun(nat,A),F2),L)
         => 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_3752_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A] :
          ( sums(A,aTP_Lamp_bt(fun(nat,A),fun(nat,A),F2),S)
        <=> sums(A,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_3753_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Nb: nat,F2: fun(nat,A),S: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
             => ( aa(nat,A,F2,I2) = zero_zero(A) ) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cr(nat,fun(fun(nat,A),fun(nat,A)),Nb),F2),S)
          <=> sums(A,F2,S) ) ) ) ).

% sums_zero_iff_shift
tff(fact_3754_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Ma: nat,I5: set(nat)] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_cs(A,fun(nat,fun(nat,A)),Xb),Ma)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Ma)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xb)),I5)) ) ) ).

% sum_power_add
tff(fact_3755_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,Ma: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Nb,Ma)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ct(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Ma)),set_or1337092689740270186AtMost(nat,Nb,Ma)) ) ) ).

% sum.atLeastAtMost_rev
tff(fact_3756_sum__bounded__above,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),F2: fun(A,B),K6: B] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),K6) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A3))),K6)) ) ) ).

% sum_bounded_above
tff(fact_3757_sum__bounded__below,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),K6: B,F2: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K6),aa(A,B,F2,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A3))),K6)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)) ) ) ).

% sum_bounded_below
tff(fact_3758_powser__sums__if,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Ma: nat,Z: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_cu(nat,fun(A,fun(nat,A)),Ma),Z),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Ma)) ) ).

% powser_sums_if
tff(fact_3759_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A)] : sums(A,aTP_Lamp_bu(fun(nat,A),fun(nat,A),A2),aa(nat,A,A2,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_3760_sum__shift__lb__Suc0__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_3761_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_3762_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Ma: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Ma)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),Nb))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_3763_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Ma: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,aa(nat,nat,suc,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_3764_set__replicate__Suc,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : ( aa(list(A),set(A),set2(A),replicate(A,aa(nat,nat,suc,Nb),Xb)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))) ) ).

% set_replicate_Suc
tff(fact_3765_set__replicate__conv__if,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( aa(list(A),set(A),set2(A),replicate(A,Nb,Xb)) = $ite(Nb = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))) ) ).

% set_replicate_conv_if
tff(fact_3766_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Ma: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Ma)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_co(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Ma,Nb))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_3767_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Ma: nat,Nb: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cv(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,Ma,Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,aa(nat,nat,suc,Nb))),aa(nat,A,F2,Ma)) ) ) ) ).

% sum_Suc_diff
tff(fact_3768_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),A2: nat,B2: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_cw(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,zero_zero(A)) ) ) ).

% sum_atLeastAtMost_code
tff(fact_3769_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Ma: nat,Nb: nat,G: fun(nat,A),P3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P3)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_3770_sum__bounded__above__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere8940638589300402666id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),F2: fun(A,B),K6: B] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I2)),K6) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A3))),K6)) ) ) ) ).

% sum_bounded_above_strict
tff(fact_3771_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),Xb: fun(A,B),A2: fun(A,B),B2: B,Delta: B] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,Xb,I2)) )
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),Xb),I5) = one_one(B) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A2,I2)),B2))),Delta) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_cx(fun(A,B),fun(fun(A,B),fun(A,B)),Xb),A2)),I5)),B2))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_3772_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),Ma: nat,Nb: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cy(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,Ma,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,Ma)),aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))),zero_zero(A)) ) ) ).

% sum_natinterval_diff
tff(fact_3773_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Ma: nat,Nb: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cz(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,Ma)) ) ) ) ).

% sum_telescope''
tff(fact_3774_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))
         => sums(A,aa(A,fun(nat,A),power_power(A),C2),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_3775_power__half__series,axiom,
    sums(real,aTP_Lamp_da(nat,real),one_one(real)) ).

% power_half_series
tff(fact_3776_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),one_one(A)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2)))),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aa(nat,fun(nat,$o)),Nb))) ) ) ).

% mask_eq_sum_exp
tff(fact_3777_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Ma: nat,Nb: nat,Xb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( 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)),Xb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xb)),set_or1337092689740270186AtMost(nat,Ma,Nb))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Ma)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,Nb))) ) ) ) ).

% sum_gp_multiplied
tff(fact_3778_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_db(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% sum.in_pairs
tff(fact_3779_sums__if_H,axiom,
    ! [G: fun(nat,real),Xb: real] :
      ( sums(real,G,Xb)
     => sums(real,aTP_Lamp_dc(fun(nat,real),fun(nat,real),G),Xb) ) ).

% sums_if'
tff(fact_3780_sums__if,axiom,
    ! [G: fun(nat,real),Xb: real,F2: fun(nat,real),Y: real] :
      ( sums(real,G,Xb)
     => ( sums(real,F2,Y)
       => sums(real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_dd(fun(nat,real),fun(fun(nat,real),fun(nat,real)),G),F2),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)) ) ) ).

% sums_if
tff(fact_3781_mask__eq__sum__exp__nat,axiom,
    ! [Nb: 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))),Nb)),aa(nat,nat,suc,zero_zero(nat))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aa(nat,fun(nat,$o)),Nb))) ) ).

% mask_eq_sum_exp_nat
tff(fact_3782_gauss__sum__nat,axiom,
    ! [Nb: nat] : ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_az(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% gauss_sum_nat
tff(fact_3783_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_de(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ) ).

% gbinomial_sum_up_index
tff(fact_3784_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,D3: A,Nb: nat] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_df(A,fun(A,fun(nat,A)),A2),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) = 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),Nb)),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),Nb)),D3))) ) ) ).

% double_arith_series
tff(fact_3785_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: nat] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))) ) ) ).

% double_gauss_sum
tff(fact_3786_arith__series__nat,axiom,
    ! [A2: nat,D3: nat,Nb: nat] : ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_dg(nat,fun(nat,fun(nat,nat)),A2),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),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),Nb),D3))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% arith_series_nat
tff(fact_3787_Sum__Icc__nat,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_az(nat,nat)),set_or1337092689740270186AtMost(nat,Ma,Nb)) = 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),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% Sum_Icc_nat
tff(fact_3788_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,D3: A,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dh(A,fun(A,fun(nat,A)),A2),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = 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),Nb)),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),Nb)),D3))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% arith_series
tff(fact_3789_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% gauss_sum
tff(fact_3790_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: nat] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))) ) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_3791_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,Ma: nat,Nb: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xb)),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb))) = $ite(Xb = one_one(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(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),Xb),Ma)),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),Xb),aa(nat,nat,suc,Nb)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))) ) ) ).

% sum_gp_offset
tff(fact_3792_cos__paired,axiom,
    ! [Xb: real] : sums(real,aTP_Lamp_di(real,fun(nat,real),Xb),aa(real,real,cos(real),Xb)) ).

% cos_paired
tff(fact_3793_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
          $let(
            half: nat,
            half:= 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:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_3794_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% gauss_sum_from_Suc_0
tff(fact_3795_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R2: A,Ma: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dj(A,fun(nat,A),R2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Ma)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Ma)),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,R2),aa(nat,nat,suc,Ma))) ) ) ).

% gchoose_row_sum_weighted
tff(fact_3796_vebt__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A4),(B4))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B4),$false) ) ) )
         => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary)),Xaa))
                 => ~ $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,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(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_3797_geometric__deriv__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => sums(A,aTP_Lamp_dk(A,fun(nat,A),Z),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_3798_vebt__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B4),$false) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A4),(B4))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = 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),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] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V4: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V4),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi,
                              $true,
                              $ite(
                                Xaa = Ma2,
                                $true,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                  $false,
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,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(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_3799_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A4),(B4))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B4),$false) ) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(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,TreeList: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList,S2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList,S2)),Xaa))
                   => $let(
                        pos2: nat,
                        pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_3800_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A4),(B4))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B4),$false) ) ) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList,S2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList,S2)),Xaa))
                 => ~ $let(
                        pos2: nat,
                        pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_3801_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xb,Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B4),$false) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A4),(B4))),Xaa)) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = 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),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xaa)) ) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V4: nat,TreeList: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList,S2) )
                 => ( ( (Y)
                    <=> $let(
                          pos2: nat,
                          pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy2,aa(nat,nat,suc,V4),TreeList,S2)),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_3802_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),Xaa)) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xaa)) )
           => ( ! [Mi: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa))
                   => ( ( Xaa = Mi )
                      | ( Xaa = Ma2 ) ) ) )
             => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList,Vc2) )
                   => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList,Vc2)),Xaa))
                     => ( ( Xaa = Mi )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos2: nat,
                            pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
               => ~ ! [V4: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList,Vd) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList,Vd)),Xaa))
                       => $let(
                            pos2: nat,
                            pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_3803_int__sum,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A)] : ( aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)) = aa(set(A),int,aa(fun(A,int),fun(set(A),int),groups7311177749621191930dd_sum(A,int),aTP_Lamp_bi(fun(A,nat),fun(A,int),F2)),A3) ) ).

% int_sum
tff(fact_3804_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,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_dl(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F2)),A3) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),G),A3)) ) ) ).

% sum_subtractf_nat
tff(fact_3805_card__eq__sum,axiom,
    ! [A: $tType,A3: set(A)] : ( aa(set(A),nat,finite_card(A),A3) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_dm(A,nat)),A3) ) ).

% card_eq_sum
tff(fact_3806_sum__SucD,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A),Nb: nat] :
      ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3) = aa(nat,nat,suc,Nb) )
     => ? [X: A] :
          ( member(A,X,A3)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X)) ) ) ).

% sum_SucD
tff(fact_3807_sum__Suc,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A)] : ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_dn(fun(A,nat),fun(A,nat),F2)),A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(set(A),nat,finite_card(A),A3)) ) ).

% sum_Suc
tff(fact_3808_real__of__card,axiom,
    ! [A: $tType,A3: set(A)] : ( aa(nat,real,semiring_1_of_nat(real),aa(set(A),nat,finite_card(A),A3)) = aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aTP_Lamp_do(A,real)),A3) ) ).

% real_of_card
tff(fact_3809_sum__diff1__nat,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A),A2: A] :
      ( aa(set(A),nat,aa(fun(A,nat),fun(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)),insert2(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,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(A,nat,F2,A2)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)) ) ).

% sum_diff1_nat
tff(fact_3810_sum__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Nb)
     => ( aa(set(complex),complex,aa(fun(complex,complex),fun(set(complex),complex),groups7311177749621191930dd_sum(complex,complex),aTP_Lamp_dp(complex,complex)),collect(complex,aa(complex,fun(complex,$o),aTP_Lamp_dq(nat,fun(complex,fun(complex,$o)),Nb),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_3811_norm__prod__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),Z: fun(A,B),W: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Z,I2))),one_one(real)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,W,I2))),one_one(real)) )
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Z),I5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),W),I5)))),aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aa(fun(A,B),fun(A,real),aTP_Lamp_dr(fun(A,B),fun(fun(A,B),fun(A,real)),Z),W)),I5)) ) ) ) ).

% norm_prod_diff
tff(fact_3812_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Nb)
     => ( aa(set(complex),complex,aa(fun(complex,complex),fun(set(complex),complex),groups7311177749621191930dd_sum(complex,complex),aTP_Lamp_dp(complex,complex)),collect(complex,aTP_Lamp_ak(nat,fun(complex,$o),Nb))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_3813_Sum__Icc__int,axiom,
    ! [Ma: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ma),Nb)
     => ( aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7311177749621191930dd_sum(int,int),aTP_Lamp_bj(int,int)),set_or1337092689740270186AtMost(int,Ma,Nb)) = 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),Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),Ma),aa(int,int,aa(int,fun(int,int),minus_minus(int),Ma),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ) ).

% Sum_Icc_int
tff(fact_3814_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [Mi: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa))
               => ~ ( ( Xaa = Mi )
                    | ( Xaa = Ma2 ) ) ) )
         => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList,Vc2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList,Vc2)),Xaa))
                 => ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma2 )
                      | $let(
                          pos2: nat,
                          pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
           => ~ ! [V4: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList,Vd) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList,Vd)),Xaa))
                   => ~ $let(
                          pos2: nat,
                          pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_3815_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xb,Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y)
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),Xaa)) ) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ( ~ (Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xaa)) ) )
           => ( ! [Mi: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma2 ) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa)) ) )
             => ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList,Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi )
                          | ( Xaa = Ma2 )
                          | $let(
                              pos2: nat,
                              pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma2)),aa(nat,nat,suc,V4),TreeList,Vc2)),Xaa)) ) )
               => ~ ! [V4: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList,Vd) )
                     => ( ( (Y)
                        <=> $let(
                              pos2: nat,
                              pos2:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos2),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V4),TreeList,Vd)),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_3816_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
       => ? [T2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),T2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),zero_zero(real))
            & ( aa(real,real,cos(real),Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ds(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_3817_Maclaurin__cos__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ? [T2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),Xb)
            & ( aa(real,real,cos(real),Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ds(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_3818_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ? [T2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),Xb)
            & ( sin(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_dt(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_3819_Maclaurin__sin__expansion4,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ? [T2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T2)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),Xb)
          & ( sin(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_dt(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_3820_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_3821_lessThan__0,axiom,
    set_ord_lessThan(nat,zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_3822_sum_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_lessThan(nat,Nb))),aa(nat,A,G,Nb)) ) ) ).

% sum.lessThan_Suc
tff(fact_3823_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_lessThan(nat,Nb))),aa(nat,A,G,Nb)) ) ) ).

% prod.lessThan_Suc
tff(fact_3824_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : ( aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_du(nat,real)),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = one_one(real) ) ).

% sumr_cos_zero_one
tff(fact_3825_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : ( set_ord_lessThan(A,U) = collect(A,aTP_Lamp_dv(A,fun(A,$o),U)) ) ) ).

% lessThan_def
tff(fact_3826_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ma: A,Nb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_ord_lessThan(A,Ma)),set_ord_lessThan(A,Nb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ma),Nb) ) ) ).

% lessThan_strict_subset_iff
tff(fact_3827_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)),insert2(nat),K),set_ord_lessThan(nat,K)) ) ).

% lessThan_Suc
tff(fact_3828_lessThan__empty__iff,axiom,
    ! [Nb: nat] :
      ( ( set_ord_lessThan(nat,Nb) = bot_bot(set(nat)) )
    <=> ( Nb = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_3829_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_dw(fun(nat,A),fun(nat,fun(nat,A)),G),Nb)),set_ord_lessThan(nat,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_lessThan(nat,Nb)) ) ) ).

% sum.nat_diff_reindex
tff(fact_3830_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_dx(fun(nat,A),fun(nat,fun(nat,A)),G),Nb)),set_ord_lessThan(nat,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_lessThan(nat,Nb)) ) ) ).

% prod.nat_diff_reindex
tff(fact_3831_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),Nb: A] :
          ( ! [X: A] : aa(nat,$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,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),P),set_ord_lessThan(A,Nb))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),Q),set_ord_lessThan(A,Nb))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_dy(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),set_ord_lessThan(A,Nb)) ) ) ) ).

% sum_diff_distrib
tff(fact_3832_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_co(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb))) ) ) ).

% sum.lessThan_Suc_shift
tff(fact_3833_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [F2: fun(nat,A),Nb: nat,R2: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,Nb))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),R2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dz(fun(nat,A),fun(A,fun(nat,A)),F2),R2)),set_ord_lessThan(nat,Nb)) ) ) ).

% sumr_diff_mult_const2
tff(fact_3834_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),Ma: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ea(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,Ma)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,Ma)) ) ) ).

% sum_lessThan_telescope'
tff(fact_3835_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),Ma: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cv(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,Ma)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,Ma)),aa(nat,A,F2,zero_zero(nat))) ) ) ).

% sum_lessThan_telescope
tff(fact_3836_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Xb: A] :
          ( ! [N: nat] : aa(A,$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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,N))),Xb)
           => summable(A,F2) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_3837_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_aw(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb))) ) ) ).

% prod.lessThan_Suc_shift
tff(fact_3838_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_co(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb)) ) ) ).

% sum.atLeast1_atMost_eq
tff(fact_3839_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Nb: nat,S: A] :
          ( sums(A,aa(nat,fun(nat,A),aTP_Lamp_bm(fun(nat,A),fun(nat,fun(nat,A)),F2),Nb),S)
        <=> sums(A,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,Nb)))) ) ) ).

% sums_iff_shift
tff(fact_3840_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Nb: nat,S: A] :
          ( sums(A,aa(nat,fun(nat,A),aTP_Lamp_bm(fun(nat,A),fun(nat,fun(nat,A)),F2),Nb),aa(A,A,aa(A,fun(A,A),minus_minus(A),S),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,Nb))))
        <=> sums(A,F2,S) ) ) ).

% sums_iff_shift'
tff(fact_3841_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A,Nb: nat] :
          ( sums(A,F2,S)
         => sums(A,aa(nat,fun(nat,A),aTP_Lamp_bm(fun(nat,A),fun(nat,fun(nat,A)),F2),Nb),aa(A,A,aa(A,fun(A,A),minus_minus(A),S),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,Nb)))) ) ) ).

% sums_split_initial_segment
tff(fact_3842_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_aw(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb)) ) ) ).

% prod.atLeast1_atMost_eq
tff(fact_3843_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),Mm: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_co(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,Mm)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ) ).

% sum_bounds_lt_plus1
tff(fact_3844_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)),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),Xb),one_one(A))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xb)),set_ord_lessThan(nat,Nb))) ) ) ).

% power_diff_1_eq
tff(fact_3845_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: 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),Xb),Nb)) = 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)),Xb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xb)),set_ord_lessThan(nat,Nb))) ) ) ).

% one_diff_power_eq
tff(fact_3846_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Nb: nat] :
          ( ( Xb != one_one(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xb)),set_ord_lessThan(nat,Nb)) = 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),Xb),Nb)),one_one(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_3847_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_bm(fun(nat,A),fun(nat,fun(nat,A)),F2),K))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,K))) ) ) ) ).

% suminf_split_initial_segment
tff(fact_3848_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_bm(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,K))) ) ) ) ).

% suminf_minus_initial_segment
tff(fact_3849_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Nb: nat] :
          ( summable(A,F2)
         => ( ! [M: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,M)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,Nb))),suminf(A,F2)) ) ) ) ).

% sum_less_suminf
tff(fact_3850_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xb)),set_ord_lessThan(nat,Nb)) = $ite(Xb = one_one(A),aa(nat,A,semiring_1_of_nat(A),Nb),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),Xb),Nb)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))) ) ) ).

% sum_gp_strict
tff(fact_3851_lemma__termdiff1,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Z: A,H: A,Ma: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_eb(A,fun(A,fun(nat,fun(nat,A))),Z),H),Ma)),set_ord_lessThan(nat,Ma)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ec(A,fun(A,fun(nat,fun(nat,A))),Z),H),Ma)),set_ord_lessThan(nat,Ma)) ) ) ).

% lemma_termdiff1
tff(fact_3852_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: 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),Xb),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_ed(A,fun(nat,fun(A,fun(nat,A))),Xb),Nb),Y)),set_ord_lessThan(nat,Nb))) ) ) ).

% power_diff_sumr2
tff(fact_3853_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: 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),Xb),aa(nat,nat,suc,Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_ee(A,fun(nat,fun(A,fun(nat,A))),Xb),Nb),Y)),set_ord_lessThan(nat,aa(nat,nat,suc,Nb)))) ) ) ).

% diff_power_eq_sum
tff(fact_3854_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,F2: fun(nat,A),K6: A,K: nat] :
          ( ! [P4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P4),Nb)
             => aa(A,$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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K6)) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_3855_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Nb: nat,I: nat] :
          ( summable(A,F2)
         => ( ! [M: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,M)) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),I)
             => ( aa(A,$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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,Nb))),suminf(A,F2)) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_3856_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: 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),Xb),Nb)) = 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)),Xb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ef(A,fun(nat,fun(nat,A)),Xb),Nb)),set_ord_lessThan(nat,Nb))) ) ) ).

% one_diff_power_eq'
tff(fact_3857_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xb: real,Nb: nat,Diff: fun(nat,fun(A,real))] :
          ( ( Xb = zero_zero(real) )
         => ( ( Nb != zero_zero(nat) )
           => ( aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_eg(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Xb),Diff)),set_ord_lessThan(nat,Nb)) = aa(A,real,aa(nat,fun(A,real),Diff,zero_zero(nat)),zero_zero(A)) ) ) ) ) ).

% Maclaurin_zero
tff(fact_3858_sum__split__even__odd,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real),Nb: nat] : ( aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,real),fun(nat,real),aTP_Lamp_eh(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))),Nb))) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ei(fun(nat,real),fun(nat,real),F2)),set_ord_lessThan(nat,Nb))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ej(fun(nat,real),fun(nat,real),G)),set_ord_lessThan(nat,Nb))) ) ).

% sum_split_even_odd
tff(fact_3859_Maclaurin__sin__bound,axiom,
    ! [Xb: real,Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),sin(real,Xb)),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_dt(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Xb)),Nb))) ).

% Maclaurin_sin_bound
tff(fact_3860_sum__pos__lt__pair,axiom,
    ! [F2: fun(nat,real),K: nat] :
      ( summable(real,F2)
     => ( ! [D5: 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)))),D5)))),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)))),D5)),one_one(nat))))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),F2),set_ord_lessThan(nat,K))),suminf(real,F2)) ) ) ).

% sum_pos_lt_pair
tff(fact_3861_Maclaurin__exp__lt,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != zero_zero(real) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ? [T2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T2))
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),Xb))
            & ( exp(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ek(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,exp(real,T2),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_3862_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z: A,Nb: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(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)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb)),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),times_times(A),H),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_em(A,fun(A,fun(nat,fun(nat,A))),H),Z),Nb)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ) ).

% lemma_termdiff2
tff(fact_3863_Maclaurin__sin__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T2: real] : ( sin(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_dt(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ).

% Maclaurin_sin_expansion
tff(fact_3864_Maclaurin__sin__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),Xb))
      & ( sin(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_dt(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_3865_Maclaurin__cos__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),Xb))
      & ( aa(real,real,cos(real),Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ds(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,cos(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_3866_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C2: fun(nat,A),Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_en(fun(nat,A),fun(A,fun(nat,A)),C2),Xb))
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_eo(fun(nat,A),fun(A,fun(nat,A)),C2),Xb),suminf(A,aa(A,fun(nat,A),aTP_Lamp_en(fun(nat,A),fun(A,fun(nat,A)),C2),Xb))) ) ) ).

% diffs_equiv
tff(fact_3867_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => bij_betw(nat,complex,aTP_Lamp_ep(nat,fun(nat,complex),Nb),set_ord_lessThan(nat,Nb),collect(complex,aTP_Lamp_ak(nat,fun(complex,$o),Nb))) ) ).

% bij_betw_roots_unity
tff(fact_3868_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Ma: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dj(A,fun(nat,A),A2)),set_ord_atMost(nat,Ma)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Ma)),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),Ma),one_one(nat)))) ) ) ).

% gbinomial_partial_row_sum
tff(fact_3869_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_eq(nat,fun(nat,A),Nb)),set_ord_atMost(nat,Nb))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) ) ) ) ).

% choose_even_sum
tff(fact_3870_card__atMost,axiom,
    ! [U: nat] : ( aa(set(nat),nat,finite_card(nat),set_ord_atMost(nat,U)) = aa(nat,nat,suc,U) ) ).

% card_atMost
tff(fact_3871_sum_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_atMost(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_atMost(nat,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ) ).

% sum.atMost_Suc
tff(fact_3872_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_atMost(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_atMost(nat,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ) ).

% prod.atMost_Suc
tff(fact_3873_atMost__0,axiom,
    set_ord_atMost(nat,zero_zero(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_3874_atMost__atLeast0,axiom,
    ! [Nb: nat] : ( set_ord_atMost(nat,Nb) = set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb) ) ).

% atMost_atLeast0
tff(fact_3875_lessThan__Suc__atMost,axiom,
    ! [K: nat] : ( set_ord_lessThan(nat,aa(nat,nat,suc,K)) = set_ord_atMost(nat,K) ) ).

% lessThan_Suc_atMost
tff(fact_3876_diffs__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X3: nat] : ( aa(nat,A,diffs(A,aTP_Lamp_er(fun(nat,A),fun(nat,A),C2)),X3) = aa(A,A,uminus_uminus(A),aa(nat,A,diffs(A,C2),X3)) ) ) ).

% diffs_minus
tff(fact_3877_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)),insert2(nat),aa(nat,nat,suc,K)),set_ord_atMost(nat,K)) ) ).

% atMost_Suc
tff(fact_3878_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_3879_exp__fdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X3: nat] : ( aa(nat,A,diffs(A,aTP_Lamp_es(nat,A)),X3) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,X3)) ) ) ).

% exp_fdiffs
tff(fact_3880_sum__choose__upper,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_et(nat,fun(nat,nat),Ma)),set_ord_atMost(nat,Nb)) = aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,Ma)) ) ).

% sum_choose_upper
tff(fact_3881_diffs__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X3: nat] : ( aa(nat,A,diffs(A,C2),X3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,X3))),aa(nat,A,C2,aa(nat,nat,suc,X3))) ) ) ).

% diffs_def
tff(fact_3882_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_atMost(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_co(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Nb))) ) ) ).

% sum.atMost_Suc_shift
tff(fact_3883_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),I: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ea(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_3884_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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A2),set_ord_atMost(nat,N))),B3)
           => summable(A,A2) ) ) ) ).

% bounded_imp_summable
tff(fact_3885_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_atMost(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_aw(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Nb))) ) ) ).

% prod.atMost_Suc_shift
tff(fact_3886_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_eu(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_ord_atMost(nat,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ew(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_ord_lessThan(nat,Nb)) ) ) ).

% sum.nested_swap'
tff(fact_3887_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ex(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_ord_atMost(nat,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ez(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_ord_lessThan(nat,Nb)) ) ) ).

% prod.nested_swap'
tff(fact_3888_sum__choose__lower,axiom,
    ! [R2: nat,Nb: nat] : ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_fa(nat,fun(nat,nat),R2)),set_ord_atMost(nat,Nb)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R2),Nb))),Nb) ) ).

% sum_choose_lower
tff(fact_3889_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,Ma: nat] : ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_fb(nat,fun(nat,nat),Nb)),set_ord_atMost(nat,Ma)) = 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),Nb),Ma)),one_one(nat))),Ma) ) ).

% choose_rising_sum(2)
tff(fact_3890_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,Ma: nat] : ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_fb(nat,fun(nat,nat),Nb)),set_ord_atMost(nat,Ma)) = 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),Nb),Ma)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))) ) ).

% choose_rising_sum(1)
tff(fact_3891_diffs__cos__coeff,axiom,
    ! [X3: nat] : ( aa(nat,real,diffs(real,cos_coeff),X3) = aa(real,real,uminus_uminus(real),sin_coeff(X3)) ) ).

% diffs_cos_coeff
tff(fact_3892_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C2: fun(nat,A),Nb: nat,K: nat] :
          ( ! [W2: A] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fc(fun(nat,A),fun(A,fun(nat,A)),C2),W2)),set_ord_atMost(nat,Nb)) = zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => ( aa(nat,A,C2,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_3893_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat] :
          ( ! [X4: A] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(A,fun(nat,A)),C2),X4)),set_ord_atMost(nat,Nb)) = zero_zero(A) )
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
             => ( aa(nat,A,C2,I3) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_3894_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_atMost(nat,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_co(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb))) ) ) ).

% sum.atMost_shift
tff(fact_3895_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_atMost(nat,Ma))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)))) ) ) ).

% sum_up_index_split
tff(fact_3896_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_atMost(nat,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_aw(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb))) ) ) ).

% prod.atMost_shift
tff(fact_3897_atLeast1__atMost__eq__remove0,axiom,
    ! [Nb: nat] : ( set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_atMost(nat,Nb)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat)))) ) ).

% atLeast1_atMost_eq_remove0
tff(fact_3898_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fe(A,fun(nat,A),A2)),set_ord_atMost(nat,Nb)) = 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),Nb))),one_one(A))),Nb) ) ) ).

% gbinomial_parallel_sum
tff(fact_3899_sum__choose__diagonal,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_ff(nat,fun(nat,fun(nat,nat)),Ma),Nb)),set_ord_atMost(nat,Ma)) = aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),Ma) ) ) ).

% sum_choose_diagonal
tff(fact_3900_vandermonde,axiom,
    ! [Ma: nat,Nb: nat,R2: nat] : ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_fg(nat,fun(nat,fun(nat,fun(nat,nat))),Ma),Nb),R2)),set_ord_atMost(nat,R2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)),R2) ) ).

% vandermonde
tff(fact_3901_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: 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)),Xb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xb)),set_ord_atMost(nat,Nb))) = 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),Xb),aa(nat,nat,suc,Nb))) ) ) ).

% sum_gp_basic
tff(fact_3902_polyfun__roots__card,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Nb: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),collect(A,aa(nat,fun(A,$o),aTP_Lamp_fh(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))),Nb) ) ) ) ).

% polyfun_roots_card
tff(fact_3903_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),A2: A,Nb: nat] :
          ( ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fi(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),set_ord_atMost(nat,Nb)) = zero_zero(A) )
         => ~ ! [B4: fun(nat,A)] :
                ~ ! [Z4: A] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fi(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),set_ord_atMost(nat,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z4),A2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fi(fun(nat,A),fun(A,fun(nat,A)),B4),Z4)),set_ord_lessThan(nat,Nb))) ) ) ) ).

% polyfun_linear_factor_root
tff(fact_3904_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),Nb: nat,A2: A] :
        ? [B4: fun(nat,A)] :
        ! [Z4: A] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fi(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),set_ord_atMost(nat,Nb)) = 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),Z4),A2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fi(fun(nat,A),fun(A,fun(nat,A)),B4),Z4)),set_ord_lessThan(nat,Nb)))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fi(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),set_ord_atMost(nat,Nb))) ) ) ).

% polyfun_linear_factor
tff(fact_3905_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Ma: nat,Nb: nat,Xb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xb)),set_or1337092689740270186AtMost(nat,Ma,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Ma)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xb)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma)))) ) ) ) ).

% sum_power_shift
tff(fact_3906_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_fj(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_fj(fun(nat,A),fun(nat,real),B2))
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fl(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ).

% summable_Cauchy_product
tff(fact_3907_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_fj(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_fj(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_fl(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ) ).

% Cauchy_product
tff(fact_3908_binomial,axiom,
    ! [A2: nat,B2: nat,Nb: 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)),Nb) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_fm(nat,fun(nat,fun(nat,fun(nat,nat))),A2),B2),Nb)),set_ord_atMost(nat,Nb)) ) ).

% binomial
tff(fact_3909_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(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))),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_db(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Nb)) ) ) ).

% sum.in_pairs_0
tff(fact_3910_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Ma: nat,A2: fun(nat,A),Nb: nat,B2: fun(nat,A),Xb: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),I2)
             => ( aa(nat,A,A2,I2) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),J2)
               => ( aa(nat,A,B2,J2) = zero_zero(A) ) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fi(fun(nat,A),fun(A,fun(nat,A)),A2),Xb)),set_ord_atMost(nat,Ma))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fi(fun(nat,A),fun(A,fun(nat,A)),B2),Xb)),set_ord_atMost(nat,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_fo(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A2),B2),Xb)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb))) ) ) ) ) ).

% polynomial_product
tff(fact_3911_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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))),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_bd(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Nb)) ) ) ).

% prod.in_pairs_0
tff(fact_3912_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat,K: A] :
          ( ! [X4: A] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(A,fun(nat,A)),C2),X4)),set_ord_atMost(nat,Nb)) = K )
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X4: nat] :
                ( member(nat,X4,set_or1337092689740270186AtMost(nat,one_one(nat),Nb))
               => ( aa(nat,A,C2,X4) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_3913_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Ma: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fp(A,fun(nat,A),A2)),set_ord_atMost(nat,Ma)) = 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))),Ma)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),Ma)) ) ) ).

% gbinomial_sum_lower_neg
tff(fact_3914_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,Nb: 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)),Nb) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fq(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Nb)),set_ord_atMost(nat,Nb)) ) ) ).

% binomial_ring
tff(fact_3915_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,B2: A,Nb: nat] : ( comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),Nb) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fr(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Nb)),set_ord_atMost(nat,Nb)) ) ) ).

% pochhammer_binomial_sum
tff(fact_3916_polynomial__product__nat,axiom,
    ! [Ma: nat,A2: fun(nat,nat),Nb: nat,B2: fun(nat,nat),Xb: nat] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),I2)
         => ( aa(nat,nat,A2,I2) = zero_zero(nat) ) )
     => ( ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),J2)
           => ( aa(nat,nat,B2,J2) = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_fs(fun(nat,nat),fun(nat,fun(nat,nat)),A2),Xb)),set_ord_atMost(nat,Ma))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_fs(fun(nat,nat),fun(nat,fun(nat,nat)),B2),Xb)),set_ord_atMost(nat,Nb))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_fu(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A2),B2),Xb)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_3917_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_fj(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_fj(fun(nat,A),fun(nat,real),B2))
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fl(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_3918_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P3)
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fv(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,P3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fw(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),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_3919_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P3)
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fx(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,P3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fy(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),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_3920_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Ma: nat,A2: A,Xb: A,Y: A] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_fz(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Ma),A2),Xb),Y)),set_ord_atMost(nat,Ma)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ga(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Ma),A2),Xb),Y)),set_ord_atMost(nat,Ma)) ) ) ).

% gbinomial_partial_sum_poly
tff(fact_3921_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,Z: A,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb) = A2 )
          <=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_gb(nat,fun(A,fun(A,fun(nat,A))),Nb),Z),A2)),set_ord_atMost(nat,Nb)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_3922_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xb)),set_ord_atMost(nat,Nb)) = $ite(Xb = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))),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),Xb),aa(nat,nat,suc,Nb))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))) ) ) ).

% sum_gp0
tff(fact_3923_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( ( Nb != one_one(nat) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gc(nat,fun(nat,A),Nb)),set_ord_atMost(nat,Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_3924_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Ma: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gd(nat,fun(nat,A),Ma)),set_ord_atMost(nat,Ma)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ma) ) ) ).

% gbinomial_sum_nat_pow2
tff(fact_3925_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Ma: nat,A2: A,Xb: A,Y: A] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_fz(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Ma),A2),Xb),Y)),set_ord_atMost(nat,Ma)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ge(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Ma),A2),Xb),Y)),set_ord_atMost(nat,Ma)) ) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_3926_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,A2: fun(nat,A),Xb: A,Y: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fi(fun(nat,A),fun(A,fun(nat,A)),A2),Xb)),set_ord_atMost(nat,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fi(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),set_ord_atMost(nat,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_gg(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A2),Xb),Y)),set_ord_lessThan(nat,Nb))) ) ) ) ).

% polyfun_diff_alt
tff(fact_3927_binomial__r__part__sum,axiom,
    ! [Ma: nat] : ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma)),one_one(nat)))),set_ord_atMost(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),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma)) ) ).

% binomial_r_part_sum
tff(fact_3928_choose__linear__sum,axiom,
    ! [Nb: nat] : ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_gh(nat,fun(nat,nat),Nb)),set_ord_atMost(nat,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),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),Nb),one_one(nat)))) ) ).

% choose_linear_sum
tff(fact_3929_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gi(nat,fun(nat,A),Nb)),set_ord_atMost(nat,Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_3930_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E2: real,C2: fun(nat,A),Nb: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => ? [M8: real] :
            ! [Z4: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),M8),real_V7770717601297561774m_norm(A,Z4))
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_bx(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),set_ord_atMost(nat,Nb)))),aa(real,real,aa(real,fun(real,real),times_times(real),E2),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Z4)),aa(nat,nat,suc,Nb)))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_3931_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,A2: fun(nat,A),Xb: A,Y: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fi(fun(nat,A),fun(A,fun(nat,A)),A2),Xb)),set_ord_atMost(nat,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fi(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),set_ord_atMost(nat,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_gk(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A2),Xb),Y)),set_ord_lessThan(nat,Nb))) ) ) ) ).

% polyfun_diff
tff(fact_3932_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Ma: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),Ma))),one_one(A)))),set_ord_atMost(nat,Ma)) = 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))),Ma)) ) ) ).

% gbinomial_r_part_sum
tff(fact_3933_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gl(nat,fun(nat,A),Nb)),set_ord_atMost(nat,Nb))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) ) ) ) ).

% choose_odd_sum
tff(fact_3934_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_gn(A,fun(A,fun(nat,A)),Xb),Y),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Y))) ) ).

% sin_x_sin_y
tff(fact_3935_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_gp(A,fun(A,fun(nat,A)),Xb),Y),aa(A,A,cos(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))) ) ).

% sums_cos_x_plus_y
tff(fact_3936_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_gr(A,fun(A,fun(nat,A)),Xb),Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,cos(A),Xb)),aa(A,A,cos(A),Y))) ) ).

% cos_x_cos_y
tff(fact_3937_exp__first__two__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : ( exp(A,Xb) = 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)),Xb)),suminf(A,aTP_Lamp_gs(A,fun(nat,A),Xb))) ) ) ).

% exp_first_two_terms
tff(fact_3938_scaleR__zero__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real] : ( real_V8093663219630862766scaleR(A,A2,zero_zero(A)) = zero_zero(A) ) ) ).

% scaleR_zero_right
tff(fact_3939_scaleR__cancel__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xb: A,B2: real] :
          ( ( real_V8093663219630862766scaleR(A,A2,Xb) = real_V8093663219630862766scaleR(A,B2,Xb) )
        <=> ( ( A2 = B2 )
            | ( Xb = zero_zero(A) ) ) ) ) ).

% scaleR_cancel_right
tff(fact_3940_scaleR__minus__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xb: A] : ( real_V8093663219630862766scaleR(A,A2,aa(A,A,uminus_uminus(A),Xb)) = aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,A2,Xb)) ) ) ).

% scaleR_minus_right
tff(fact_3941_scaleR__one,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] : ( real_V8093663219630862766scaleR(A,one_one(real),Xb) = Xb ) ) ).

% scaleR_one
tff(fact_3942_scaleR__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xb: A] :
          ( ( real_V8093663219630862766scaleR(A,A2,Xb) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(real) )
            | ( Xb = zero_zero(A) ) ) ) ) ).

% scaleR_eq_0_iff
tff(fact_3943_scaleR__zero__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] : ( real_V8093663219630862766scaleR(A,zero_zero(real),Xb) = zero_zero(A) ) ) ).

% scaleR_zero_left
tff(fact_3944_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),real_V8093663219630862766scaleR(A,U,A2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),real_V8093663219630862766scaleR(A,U,B2)) )
        <=> ( ( A2 = B2 )
            | ( U = one_one(real) ) ) ) ) ).

% scaleR_eq_iff
tff(fact_3945_scaleR__minus__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xb: A] : ( real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),A2),Xb) = aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,A2,Xb)) ) ) ).

% scaleR_minus_left
tff(fact_3946_scaleR__left_Ominus,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: real,Xaa: A] : ( real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),Xb),Xaa) = aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,Xb,Xaa)) ) ) ).

% scaleR_left.minus
tff(fact_3947_scaleR__minus1__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] : ( real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),one_one(real)),Xb) = aa(A,A,uminus_uminus(A),Xb) ) ) ).

% scaleR_minus1_left
tff(fact_3948_scaleR__collapse,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,A2: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),U),A2)),real_V8093663219630862766scaleR(A,U,A2)) = A2 ) ) ).

% scaleR_collapse
tff(fact_3949_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [V2: num,W: num,A2: A] : ( real_V8093663219630862766scaleR(A,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),V2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = real_V8093663219630862766scaleR(A,divide_divide(real,aa(num,real,numeral_numeral(real),W),aa(num,real,numeral_numeral(real),V2)),A2) ) ) ).

% inverse_scaleR_times
tff(fact_3950_scaleR__half__double,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: A] : ( real_V8093663219630862766scaleR(A,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_3951_scaleR__right__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,A2: real,B2: real] :
          ( ( Xb != zero_zero(A) )
         => ( ( real_V8093663219630862766scaleR(A,A2,Xb) = real_V8093663219630862766scaleR(A,B2,Xb) )
           => ( A2 = B2 ) ) ) ) ).

% scaleR_right_imp_eq
tff(fact_3952_scaleR__right__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xb: A,Y: A] : ( real_V8093663219630862766scaleR(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,A2,Y)) ) ) ).

% scaleR_right_distrib
tff(fact_3953_scaleR__right__diff__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xb: A,Y: A] : ( real_V8093663219630862766scaleR(A,A2,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,A2,Y)) ) ) ).

% scaleR_right_diff_distrib
tff(fact_3954_scaleR__left__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,Xb: A] : ( real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,B2,Xb)) ) ) ).

% scaleR_left_distrib
tff(fact_3955_scaleR__left_Oadd,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: real,Y: real,Xaa: A] : ( real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y),Xaa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,Xb,Xaa)),real_V8093663219630862766scaleR(A,Y,Xaa)) ) ) ).

% scaleR_left.add
tff(fact_3956_of__real__def,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R2: real] : ( real_Vector_of_real(A,R2) = real_V8093663219630862766scaleR(A,R2,one_one(A)) ) ) ).

% of_real_def
tff(fact_3957_scaleR__left__diff__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,Xb: A] : ( real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2),Xb) = aa(A,A,aa(A,fun(A,A),minus_minus(A),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,B2,Xb)) ) ) ).

% scaleR_left_diff_distrib
tff(fact_3958_scaleR__left_Odiff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: real,Y: real,Xaa: A] : ( real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y),Xaa) = aa(A,A,aa(A,fun(A,A),minus_minus(A),real_V8093663219630862766scaleR(A,Xb,Xaa)),real_V8093663219630862766scaleR(A,Y,Xaa)) ) ) ).

% scaleR_left.diff
tff(fact_3959_inverse__scaleR__distrib,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A2: real,Xb: A] : ( aa(A,A,inverse_inverse(A),real_V8093663219630862766scaleR(A,A2,Xb)) = real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),A2),aa(A,A,inverse_inverse(A),Xb)) ) ) ).

% inverse_scaleR_distrib
tff(fact_3960_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,B2,Xb)) ) ) ) ).

% scaleR_right_mono
tff(fact_3961_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: real,A2: real,C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,A2,C2)),real_V8093663219630862766scaleR(A,B2,C2)) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_3962_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,V2: real,A2: A,Xb: A] :
          ( ( real_V8093663219630862766scaleR(A,divide_divide(real,U,V2),A2) = Xb )
        <=> $ite(V2 = zero_zero(real),Xb = zero_zero(A),real_V8093663219630862766scaleR(A,U,A2) = real_V8093663219630862766scaleR(A,V2,Xb)) ) ) ).

% vector_fraction_eq_iff
tff(fact_3963_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,U: real,V2: real,A2: A] :
          ( ( Xb = real_V8093663219630862766scaleR(A,divide_divide(real,U,V2),A2) )
        <=> $ite(V2 = zero_zero(real),Xb = zero_zero(A),real_V8093663219630862766scaleR(A,V2,Xb) = real_V8093663219630862766scaleR(A,U,A2)) ) ) ).

% eq_vector_fraction_iff
tff(fact_3964_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E2: A,C2: A,B2: real,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,A2,E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,B2,E2)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2),E2)),C2)),D3) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_3965_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E2: A,C2: A,B2: real,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,A2,E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,B2,E2)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2),E2)),D3)) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_3966_sgn__div__norm,axiom,
    ! [A: $tType] :
      ( real_V6567297691418259687v_norm(A)
     => ! [Xb: A] : ( aa(A,A,sgn_sgn(A),Xb) = real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,Xb)),Xb) ) ) ).

% sgn_div_norm
tff(fact_3967_complex__sgn__def,axiom,
    ! [Xb: complex] : ( aa(complex,complex,sgn_sgn(complex),Xb) = real_V8093663219630862766scaleR(complex,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(complex,Xb)),Xb) ) ).

% complex_sgn_def
tff(fact_3968_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),real_V8093663219630862766scaleR(A,A2,B2))
        <=> ( ( aa(real,$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_3969_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,A2,B2)),zero_zero(A))
        <=> ( ( aa(real,$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_3970_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,Xb: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,B2,Y)) ) ) ) ) ) ).

% scaleR_mono
tff(fact_3971_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,C2: A,D3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,A2,C2)),real_V8093663219630862766scaleR(A,B2,D3)) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_3972_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xb: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,A2,Xb)),zero_zero(A)) ) ) ).

% split_scaleR_neg_le
tff(fact_3973_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),real_V8093663219630862766scaleR(A,A2,B2)) ) ) ).

% split_scaleR_pos_le
tff(fact_3974_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),real_V8093663219630862766scaleR(A,A2,Xb)) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_3975_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,A2,Xb)),zero_zero(A)) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_3976_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,A2,Xb)),zero_zero(A)) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_3977_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),real_V8093663219630862766scaleR(A,A2,B2)) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_3978_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xb: A,A2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),one_one(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,A2,Xb)),Xb) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_3979_scaleR__2,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] : ( real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),bit0(one2)),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb) ) ) ).

% scaleR_2
tff(fact_3980_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Ma: real,Xb: A,C2: A,Y: A] :
          ( ( Ma != zero_zero(real) )
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,Ma,Xb)),C2) = Y )
          <=> ( Xb = aa(A,A,aa(A,fun(A,A),minus_minus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Ma),Y)),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Ma),C2)) ) ) ) ) ).

% real_vector_affinity_eq
tff(fact_3981_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Ma: real,Y: A,Xb: A,C2: A] :
          ( ( Ma != zero_zero(real) )
         => ( ( Y = aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,Ma,Xb)),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Ma),Y)),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Ma),C2)) = Xb ) ) ) ) ).

% real_vector_eq_affinity
tff(fact_3982_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,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),real_V8093663219630862766scaleR(A,C2,A2)) ) ) ) ).

% pos_divideR_le_eq
tff(fact_3983_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,$o),ord_less_eq(A),A2),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,C2,A2)),B2) ) ) ) ).

% pos_le_divideR_eq
tff(fact_3984_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,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,C2,A2)),B2) ) ) ) ).

% neg_divideR_le_eq
tff(fact_3985_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,$o),ord_less_eq(A),A2),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),real_V8093663219630862766scaleR(A,C2,A2)) ) ) ) ).

% neg_le_divideR_eq
tff(fact_3986_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,$o),ord_less(A),A2),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),real_V8093663219630862766scaleR(A,C2,A2)) ) ) ) ).

% neg_less_divideR_eq
tff(fact_3987_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,$o),ord_less(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),real_V8093663219630862766scaleR(A,C2,A2)),B2) ) ) ) ).

% neg_divideR_less_eq
tff(fact_3988_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,$o),ord_less(A),A2),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),real_V8093663219630862766scaleR(A,C2,A2)),B2) ) ) ) ).

% pos_less_divideR_eq
tff(fact_3989_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,$o),ord_less(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),real_V8093663219630862766scaleR(A,C2,A2)) ) ) ) ).

% pos_divideR_less_eq
tff(fact_3990_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A2: real,Xb: A] :
          ( ( A2 != zero_zero(real) )
         => ( ( Xb != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),real_V8093663219630862766scaleR(A,A2,Xb)) = real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),A2),aa(A,A,inverse_inverse(A),Xb)) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_3991_summable__exp__generic,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(A,aTP_Lamp_gt(A,fun(nat,A),Xb)) ) ).

% summable_exp_generic
tff(fact_3992_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,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,C2,A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_3993_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,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),real_V8093663219630862766scaleR(A,C2,A2)) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_3994_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,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),real_V8093663219630862766scaleR(A,C2,A2)) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_3995_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,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,C2,A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_3996_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,$o),ord_less(A),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),real_V8093663219630862766scaleR(A,C2,A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_3997_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,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),real_V8093663219630862766scaleR(A,C2,A2)) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_3998_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,$o),ord_less(A),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),real_V8093663219630862766scaleR(A,C2,A2)) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_3999_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,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),real_V8093663219630862766scaleR(A,C2,A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_4000_exp__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_gt(A,fun(nat,A),Xb),exp(A,Xb)) ) ).

% exp_converges
tff(fact_4001_exp__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X3: A] : ( exp(A,X3) = suminf(A,aTP_Lamp_gt(A,fun(nat,A),X3)) ) ) ).

% exp_def
tff(fact_4002_summable__norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(real,aTP_Lamp_gu(A,fun(nat,real),Xb)) ) ).

% summable_norm_exp
tff(fact_4003_sin__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_gv(A,fun(nat,A),Xb),sin(A,Xb)) ) ).

% sin_minus_converges
tff(fact_4004_cos__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_gw(A,fun(nat,A),Xb),aa(A,A,cos(A),Xb)) ) ).

% cos_minus_converges
tff(fact_4005_cosh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : ( cosh(A,Xb) = 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),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb)))) ) ) ).

% cosh_def
tff(fact_4006_sinh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : ( sinh(A,Xb) = 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),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb)))) ) ) ).

% sinh_def
tff(fact_4007_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,Y: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb) )
         => ( real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gx(A,fun(A,fun(nat,fun(nat,A))),Xb),Y),Nb)),set_ord_atMost(nat,Nb)) ) ) ) ).

% exp_series_add_commuting
tff(fact_4008_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : ( exp(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),suminf(A,aTP_Lamp_gy(A,fun(nat,A),Xb))) ) ) ).

% exp_first_term
tff(fact_4009_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,K: nat] : ( exp(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gt(A,fun(nat,A),Xb)),set_ord_lessThan(nat,K))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_gz(A,fun(nat,fun(nat,A)),Xb),K))) ) ) ).

% exp_first_terms
tff(fact_4010_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_ha(A,fun(nat,A),Xb),cosh(A,Xb)) ) ).

% cosh_converges
tff(fact_4011_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_hb(A,fun(nat,A),Xb),sinh(A,Xb)) ) ).

% sinh_converges
tff(fact_4012_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,N)),aa(nat,A,X6,aa(nat,nat,suc,N)))
         => topological_monoseq(A,X6) ) ) ).

% mono_SucI1
tff(fact_4013_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N))),aa(nat,A,X6,N))
         => topological_monoseq(A,X6) ) ) ).

% mono_SucI2
tff(fact_4014_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( topological_monoseq(A,X6)
        <=> ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,N4)),aa(nat,A,X6,aa(nat,nat,suc,N4)))
            | ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N4))),aa(nat,A,X6,N4)) ) ) ) ).

% monoseq_Suc
tff(fact_4015_of__nat__code,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] : ( aa(nat,A,semiring_1_of_nat(A),Nb) = semiri8178284476397505188at_aux(A,aTP_Lamp_hc(A,A),Nb,zero_zero(A)) ) ) ).

% of_nat_code
tff(fact_4016_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),Nb: nat,I: A] : ( semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,Nb),I) = semiri8178284476397505188at_aux(A,Inc,Nb,aa(A,A,Inc,I)) ) ) ).

% of_nat_aux.simps(2)
tff(fact_4017_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_4018_monoseq__minus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: fun(nat,A)] :
          ( topological_monoseq(A,A2)
         => topological_monoseq(A,aTP_Lamp_hd(fun(nat,A),fun(nat,A),A2)) ) ) ).

% monoseq_minus
tff(fact_4019_Arg__def,axiom,
    ! [Z: complex] :
      ( arg(Z) = $ite(Z = zero_zero(complex),zero_zero(real),fChoice(real,aTP_Lamp_he(complex,fun(real,$o),Z))) ) ).

% Arg_def
tff(fact_4020_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : ( vEBT_set_vebt(Ta) = collect(nat,vEBT_V8194947554948674370ptions(Ta)) ) ).

% set_vebt_def
tff(fact_4021_vebt__buildup_Opelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( accp(nat,vEBT_v4011308405150292612up_rel,Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ accp(nat,vEBT_v4011308405150292612up_rel,zero_zero(nat)) ) )
         => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                 => ( ( Y = $ite(
                          aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                          $let(
                            half: nat,
                            half:= 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:= 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))) ) ) )
                   => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_4022_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Ma: num,Nb: num] : ( unique8689654367752047608divmod(A,aa(num,num,bit1,Ma),bit0(Nb)) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_hf(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,Ma,Nb)) ) ) ).

% divmod_algorithm_code(6)
tff(fact_4023_some__sym__eq__trivial,axiom,
    ! [A: $tType,Xb: A] : ( fChoice(A,aa(A,fun(A,$o),fequal(A),Xb)) = Xb ) ).

% some_sym_eq_trivial
tff(fact_4024_some__eq__trivial,axiom,
    ! [A: $tType,Xb: A] : ( fChoice(A,aTP_Lamp_hg(A,fun(A,$o),Xb)) = Xb ) ).

% some_eq_trivial
tff(fact_4025_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_4026_someI,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A] :
      ( aa(A,$o,P,Xb)
     => aa(A,$o,P,fChoice(A,P)) ) ).

% someI
tff(fact_4027_Eps__cong,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(A,$o,P,X)
        <=> aa(A,$o,Q,X) )
     => ( fChoice(A,P) = fChoice(A,Q) ) ) ).

% Eps_cong
tff(fact_4028_tfl__some,axiom,
    ! [A: $tType,P5: fun(A,$o),X3: A] :
      ( aa(A,$o,P5,X3)
     => aa(A,$o,P5,fChoice(A,P5)) ) ).

% tfl_some
tff(fact_4029_some__eq__imp,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A,B2: A] :
      ( ( fChoice(A,P) = A2 )
     => ( aa(A,$o,P,B2)
       => aa(A,$o,P,A2) ) ) ).

% some_eq_imp
tff(fact_4030_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),P2: fun(A,$o)] :
      ( ( Xb = fChoice(A,aTP_Lamp_ae(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
          <=> aa(A,$o,P2,X) )
       => ( ! [X_13: A] : aa(A,$o,P2,X_13)
        <=> aa(A,$o,P,Xb) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_4031_verit__sko__forall__indirect,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o)] :
      ( ( Xb = fChoice(A,aTP_Lamp_ae(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X_13: A] : aa(A,$o,P,X_13)
      <=> aa(A,$o,P,Xb) ) ) ).

% verit_sko_forall_indirect
tff(fact_4032_verit__sko__ex__indirect2,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),P2: fun(A,$o)] :
      ( ( Xb = fChoice(A,P) )
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
          <=> aa(A,$o,P2,X) )
       => ( ? [X_13: A] : aa(A,$o,P2,X_13)
        <=> aa(A,$o,P,Xb) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_4033_verit__sko__ex__indirect,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o)] :
      ( ( Xb = fChoice(A,P) )
     => ( ? [X_13: A] : aa(A,$o,P,X_13)
      <=> aa(A,$o,P,Xb) ) ) ).

% verit_sko_ex_indirect
tff(fact_4034_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_4035_verit__sko__forall_H,axiom,
    ! [A: $tType,P: fun(A,$o),A3: $o] :
      ( ( aa(A,$o,P,fChoice(A,aTP_Lamp_ae(fun(A,$o),fun(A,$o),P)))
      <=> (A3) )
     => ( ! [X_13: A] : aa(A,$o,P,X_13)
      <=> (A3) ) ) ).

% verit_sko_forall'
tff(fact_4036_verit__sko__forall,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X_13: A] : aa(A,$o,P,X_13)
    <=> aa(A,$o,P,fChoice(A,aTP_Lamp_ae(fun(A,$o),fun(A,$o),P))) ) ).

% verit_sko_forall
tff(fact_4037_verit__sko__ex_H,axiom,
    ! [A: $tType,P: fun(A,$o),A3: $o] :
      ( ( aa(A,$o,P,fChoice(A,P))
      <=> (A3) )
     => ( ? [X_13: A] : aa(A,$o,P,X_13)
      <=> (A3) ) ) ).

% verit_sko_ex'
tff(fact_4038_some1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X3 ) ) )
     => ( aa(A,$o,P,A2)
       => ( fChoice(A,P) = A2 ) ) ) ).

% some1_equality
tff(fact_4039_some__eq__ex,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
    <=> ? [X_13: A] : aa(A,$o,P,X_13) ) ).

% some_eq_ex
tff(fact_4040_someI2__bex,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,A3)
          & aa(A,$o,P,X3) )
     => ( ! [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,$o),aTP_Lamp_hh(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) ) ) ).

% someI2_bex
tff(fact_4041_someI2__ex,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X_1: A] : aa(A,$o,P,X_1)
     => ( ! [X: A] :
            ( aa(A,$o,P,X)
           => aa(A,$o,Q,X) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2_ex
tff(fact_4042_someI__ex,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X_1: A] : aa(A,$o,P,X_1)
     => aa(A,$o,P,fChoice(A,P)) ) ).

% someI_ex
tff(fact_4043_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_4044_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_4045_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_hi(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ) ).

% divmod_step_nat_def
tff(fact_4046_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_hj(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ) ).

% divmod_step_int_def
tff(fact_4047_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_hk(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ) ).

% divmod_step_def
tff(fact_4048_divmod__nat__if,axiom,
    ! [Ma: nat,Nb: nat] :
      ( divmod_nat(Ma,Nb) = $ite(
          ( ( Nb = zero_zero(nat) )
          | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ),
          aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),Ma),
          aa(product_prod(nat,nat),product_prod(nat,nat),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_hl(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb),Nb)) ) ) ).

% divmod_nat_if
tff(fact_4049_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K: nat,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(set(A),nat,finite_card(A),A3))
     => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(set(A),fun(list(A),$o),aTP_Lamp_hm(nat,fun(set(A),fun(list(A),$o)),K),A3))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_az(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A3))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_4050_arctan__def,axiom,
    ! [Y: real] : ( aa(real,real,arctan,Y) = the(real,aTP_Lamp_hn(real,fun(real,$o),Y)) ) ).

% arctan_def
tff(fact_4051_arcsin__def,axiom,
    ! [Y: real] : ( aa(real,real,arcsin,Y) = the(real,aTP_Lamp_ho(real,fun(real,$o),Y)) ) ).

% arcsin_def
tff(fact_4052_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : ( aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_hp(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hr(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,Nb)) ) ) ).

% sum.triangle_reindex_eq
tff(fact_4053_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : ( aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_hp(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ht(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,Nb)) ) ) ).

% prod.triangle_reindex_eq
tff(fact_4054_distinct__card,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ).

% distinct_card
tff(fact_4055_card__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) )
     => distinct(A,Xs) ) ).

% card_distinct
tff(fact_4056_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xs: list(A),I: nat,J: nat] :
      ( distinct(A,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
         => ( ( aa(nat,A,nth(A,Xs),I) = aa(nat,A,nth(A,Xs),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_4057_distinct__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
    <=> ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
         => ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
             => ( ( I3 != J3 )
               => ( aa(nat,A,nth(A,Xs),I3) != aa(nat,A,nth(A,Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_4058_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : ( aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_hu(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hr(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,Nb)) ) ) ).

% sum.triangle_reindex
tff(fact_4059_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : ( aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_hu(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ht(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,Nb)) ) ) ).

% prod.triangle_reindex
tff(fact_4060_distinct__Ex1,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ? [X: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(list(A),nat,size_size(list(A)),Xs))
            & ( aa(nat,A,nth(A,Xs),X) = Xb )
            & ! [Y4: nat] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y4),aa(list(A),nat,size_size(list(A)),Xs))
                  & ( aa(nat,A,nth(A,Xs),Y4) = Xb ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_4061_bij__betw__nth,axiom,
    ! [A: $tType,Xs: list(A),A3: set(nat),B3: set(A)] :
      ( distinct(A,Xs)
     => ( ( A3 = set_ord_lessThan(nat,aa(list(A),nat,size_size(list(A)),Xs)) )
       => ( ( B3 = aa(list(A),set(A),set2(A),Xs) )
         => bij_betw(nat,A,nth(A,Xs),A3,B3) ) ) ) ).

% bij_betw_nth
tff(fact_4062_of__nat__code__if,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,A,semiring_1_of_nat(A),Nb) = $ite(Nb = zero_zero(nat),zero_zero(A),aa(product_prod(nat,nat),A,product_case_prod(nat,nat,A,aTP_Lamp_hv(nat,fun(nat,A))),divmod_nat(Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% of_nat_code_if
tff(fact_4063_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(set(A),nat,finite_card(A),A3))
       => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_hw(set(A),fun(nat,fun(list(A),$o)),A3),K))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_az(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A3))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_4064_card__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_hx(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A3))),set_ord_atMost(nat,Nb)) ) ) ).

% card_lists_length_le
tff(fact_4065_set__encode__def,axiom,
    nat_set_encode = aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% set_encode_def
tff(fact_4066_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aa(nat,fun(nat,$o)),K))) ).

% finite_Collect_less_nat
tff(fact_4067_finite__interval__int1,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite(int),collect(int,aa(int,fun(int,$o),aTP_Lamp_hy(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int1
tff(fact_4068_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite(int),collect(int,aa(int,fun(int,$o),aTP_Lamp_hz(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int4
tff(fact_4069_List_Ofinite__set,axiom,
    ! [A: $tType,Xs: list(A)] : aa(set(A),$o,finite_finite(A),aa(list(A),set(A),set2(A),Xs)) ).

% List.finite_set
tff(fact_4070_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite(nat),A3)
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ia(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F2)) ) ) ).

% summable_If_finite_set
tff(fact_4071_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite(nat),collect(nat,P))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ib(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F2)) ) ) ).

% summable_If_finite
tff(fact_4072_Eps__case__prod__eq,axiom,
    ! [A: $tType,B: $tType,Xb: A,Y: B] : ( fChoice(product_prod(A,B),product_case_prod(A,B,$o,aa(B,fun(A,fun(B,$o)),aTP_Lamp_ic(A,fun(B,fun(A,fun(B,$o))),Xb),Y))) = aa(B,product_prod(A,B),product_Pair(A,B,Xb),Y) ) ).

% Eps_case_prod_eq
tff(fact_4073_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite(int),collect(int,aa(int,fun(int,$o),aTP_Lamp_id(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int3
tff(fact_4074_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite(int),collect(int,aa(int,fun(int,$o),aTP_Lamp_ie(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int2
tff(fact_4075_set__encode__inverse,axiom,
    ! [A3: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),A3)
     => ( nat_set_decode(aa(set(nat),nat,nat_set_encode,A3)) = A3 ) ) ).

% set_encode_inverse
tff(fact_4076_set__decode__inverse,axiom,
    ! [Nb: nat] : ( aa(set(nat),nat,nat_set_encode,nat_set_decode(Nb)) = Nb ) ).

% set_decode_inverse
tff(fact_4077_sum_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ~ aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = zero_zero(B) ) ) ) ).

% sum.infinite
tff(fact_4078_sum__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [F4: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),F4)
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),F4) = zero_zero(B) )
          <=> ! [X4: A] :
                ( member(A,X4,F4)
               => ( aa(A,B,F2,X4) = zero_zero(B) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4079_prod__zero__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3) = zero_zero(B) )
          <=> ? [X4: A] :
                ( member(A,X4,A3)
                & ( aa(A,B,F2,X4) = zero_zero(B) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_4080_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite(A),set_or1337092689740270186AtMost(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Icc_iff
tff(fact_4081_prod_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ~ aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = one_one(B) ) ) ) ).

% prod.infinite
tff(fact_4082_card_Oinfinite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ aa(set(A),$o,finite_finite(A),A3)
     => ( aa(set(A),nat,finite_card(A),A3) = zero_zero(nat) ) ) ).

% card.infinite
tff(fact_4083_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite(A),set_or5935395276787703475ssThan(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ioo_iff
tff(fact_4084_sum_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_if(A,fun(fun(A,B),fun(A,B)),A2),B2)),S3) = $ite(member(A,A2,S3),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta'
tff(fact_4085_sum_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_ig(A,fun(fun(A,B),fun(A,B)),A2),B2)),S3) = $ite(member(A,A2,S3),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta
tff(fact_4086_prod__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F2),A3) = one_one(nat) )
      <=> ! [X4: A] :
            ( member(A,X4,A3)
           => ( aa(A,nat,F2,X4) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_4087_prod_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_ih(A,fun(fun(A,B),fun(A,B)),A2),B2)),S3) = $ite(member(A,A2,S3),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta
tff(fact_4088_prod_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_ii(A,fun(fun(A,B),fun(A,B)),A2),B2)),S3) = $ite(member(A,A2,S3),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta'
tff(fact_4089_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_4090_finite__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(set(complex),$o,finite_finite(complex),collect(complex,aa(complex,fun(complex,$o),aTP_Lamp_dq(nat,fun(complex,fun(complex,$o)),Nb),C2))) ) ).

% finite_nth_roots
tff(fact_4091_sum_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),Xb: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ~ member(A,Xb,A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)) ) ) ) ) ).

% sum.insert
tff(fact_4092_card__0__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( ( aa(set(A),nat,finite_card(A),A3) = zero_zero(nat) )
      <=> ( A3 = bot_bot(set(A)) ) ) ) ).

% card_0_eq
tff(fact_4093_card__insert__disjoint,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( ~ member(A,Xb,A3)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_4094_prod__pos__nat__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),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_4095_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: fun(nat,A),A3: set(nat)] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ij(fun(nat,A),fun(nat,A),C2)),A3) = $ite(
              ( aa(set(nat),$o,finite_finite(nat),A3)
              & member(nat,zero_zero(nat),A3) ),
              aa(nat,A,C2,zero_zero(nat)),
              zero_zero(A) ) ) ) ).

% sum_zero_power
tff(fact_4096_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => aa(set(list(A)),$o,finite_finite(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_hw(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_distinct_length_eq
tff(fact_4097_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: fun(nat,A),D3: fun(nat,A),A3: set(nat)] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_ik(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D3)),A3) = $ite(
              ( aa(set(nat),$o,finite_finite(nat),A3)
              & member(nat,zero_zero(nat),A3) ),
              divide_divide(A,aa(nat,A,C2,zero_zero(nat)),aa(nat,A,D3,zero_zero(nat))),
              zero_zero(A) ) ) ) ).

% sum_zero_power'
tff(fact_4098_set__encode__insert,axiom,
    ! [A3: set(nat),Nb: nat] :
      ( aa(set(nat),$o,finite_finite(nat),A3)
     => ( ~ member(nat,Nb,A3)
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),Nb),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))),Nb)),aa(set(nat),nat,nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_4099_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,$o,aTP_Lamp_il(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),P))) ) ).

% split_paired_Eps
tff(fact_4100_set__encode__inf,axiom,
    ! [A3: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite(nat),A3)
     => ( aa(set(nat),nat,nat_set_encode,A3) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_4101_set__encode__eq,axiom,
    ! [A3: set(nat),B3: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),A3)
     => ( aa(set(nat),$o,finite_finite(nat),B3)
       => ( ( aa(set(nat),nat,nat_set_encode,A3) = aa(set(nat),nat,nat_set_encode,B3) )
        <=> ( A3 = B3 ) ) ) ) ).

% set_encode_eq
tff(fact_4102_finite__list,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ? [Xs2: list(A)] : ( aa(list(A),set(A),set2(A),Xs2) = A3 ) ) ).

% finite_list
tff(fact_4103_finite__nat__set__iff__bounded,axiom,
    ! [N5: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),N5)
    <=> ? [M3: nat] :
        ! [X4: nat] :
          ( member(nat,X4,N5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),M3) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_4104_bounded__nat__set__is__finite,axiom,
    ! [N5: set(nat),Nb: nat] :
      ( ! [X: nat] :
          ( member(nat,X,N5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Nb) )
     => aa(set(nat),$o,finite_finite(nat),N5) ) ).

% bounded_nat_set_is_finite
tff(fact_4105_finite__set__decode,axiom,
    ! [Nb: nat] : aa(set(nat),$o,finite_finite(nat),nat_set_decode(Nb)) ).

% finite_set_decode
tff(fact_4106_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,$o),I: nat] : aa(set(nat),$o,finite_finite(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_im(fun(nat,$o),fun(nat,fun(nat,$o)),P),I))) ).

% finite_M_bounded_by_nat
tff(fact_4107_finite__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => aa(set(list(A)),$o,finite_finite(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_in(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4108_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ? [X: A] :
                ( member(A,X,S3)
                & ~ ? [Xa: A] :
                      ( member(A,Xa,S3)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4109_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X6: set(A)] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,X6)
               => ? [Xa: A] :
                    ( member(A,Xa,X6)
                    & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Xa) ) )
           => ~ aa(set(A),$o,finite_finite(A),X6) ) ) ) ).

% infinite_growing
tff(fact_4110_prod__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ? [X3: A] :
                ( member(A,X3,A3)
                & ( aa(A,B,F2,X3) = zero_zero(B) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3) = zero_zero(B) ) ) ) ) ).

% prod_zero
tff(fact_4111_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite(A),set_or1337092689740270186AtMost(A,A2,B2)) ) ) ).

% infinite_Icc
tff(fact_4112_finite__maxlen,axiom,
    ! [A: $tType,M9: set(list(A))] :
      ( aa(set(list(A)),$o,finite_finite(list(A)),M9)
     => ? [N: nat] :
        ! [X3: list(A)] :
          ( member(list(A),X3,M9)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),X3)),N) ) ) ).

% finite_maxlen
tff(fact_4113_finite__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => aa(set(list(A)),$o,finite_finite(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_hx(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_length_le
tff(fact_4114_summable__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N5: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite(nat),N5)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N5)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => summable(A,F2) ) ) ) ).

% summable_finite
tff(fact_4115_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite(A),set_or5935395276787703475ssThan(A,A2,B2)) ) ) ).

% infinite_Ioo
tff(fact_4116_finite__distinct__list,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ? [Xs2: list(A)] :
          ( ( aa(list(A),set(A),set2(A),Xs2) = A3 )
          & distinct(A,Xs2) ) ) ).

% finite_distinct_list
tff(fact_4117_sum_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),Xb: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_io(set(A),fun(fun(A,B),fun(A,$o)),I5),Xb)))
         => ( aa(set(A),$o,finite_finite(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_io(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => aa(set(A),$o,finite_finite(A),collect(A,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_ip(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xb),Y))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_4118_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),Xb: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_iq(set(A),fun(fun(A,B),fun(A,$o)),I5),Xb)))
         => ( aa(set(A),$o,finite_finite(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_iq(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => aa(set(A),$o,finite_finite(A),collect(A,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_ir(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xb),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_4119_sum_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_hh(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_is(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A3) ) ) ) ).

% sum.inter_filter
tff(fact_4120_prod_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_hh(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_it(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A3) ) ) ) ).

% prod.inter_filter
tff(fact_4121_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : aa(set(A),$o,finite_finite(A),collect(A,aa(A,fun(A,$o),aTP_Lamp_iu(A,fun(A,fun(A,$o)),A2),B2))) ) ).

% finite_int_segment
tff(fact_4122_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero(int) )
     => aa(set(int),$o,finite_finite(int),collect(int,aTP_Lamp_iv(int,fun(int,$o),I))) ) ).

% finite_divisors_int
tff(fact_4123_even__set__encode__iff,axiom,
    ! [A3: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(nat),nat,nat_set_encode,A3))
      <=> ~ member(nat,zero_zero(nat),A3) ) ) ).

% even_set_encode_iff
tff(fact_4124_sum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ! [X: A] :
                ( member(A,X,A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3) = zero_zero(B) )
            <=> ! [X4: A] :
                  ( member(A,X4,A3)
                 => ( aa(A,B,F2,X4) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4125_sum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [S: set(A),Ta: set(B),G: fun(B,C),I: fun(B,A),F2: fun(A,C)] :
          ( aa(set(A),$o,finite_finite(A),S)
         => ( aa(set(B),$o,finite_finite(B),Ta)
           => ( ! [X: B] :
                  ( member(B,X,Ta)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,G,X)) )
             => ( ! [X: A] :
                    ( member(A,X,S)
                   => ? [Xa: B] :
                        ( member(B,Xa,Ta)
                        & ( aa(B,A,I,Xa) = X )
                        & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,X)),aa(B,C,G,Xa)) ) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),F2),S)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),Ta)) ) ) ) ) ) ).

% sum_le_included
tff(fact_4126_sum__strict__mono__ex1,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(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)) )
           => ( ? [X3: A] :
                  ( member(A,X3,A3)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4127_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R: fun(A,fun(A,$o)),S3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,zero_zero(A)),zero_zero(A))
         => ( ! [X1: A,Y1: A,X23: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X1),X23)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y1),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(A,A,aa(A,fun(A,A),plus_plus(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X23),Y23)) )
           => ( aa(set(B),$o,finite_finite(B),S3)
             => ( ! [X: B] :
                    ( member(B,X,S3)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H,X)),aa(B,A,G,X)) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S3)) ) ) ) ) ) ).

% sum.related
tff(fact_4128_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),P: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B4: A,A7: set(A)] :
                  ( aa(set(A),$o,finite_finite(A),A7)
                 => ( ! [X3: A] :
                        ( member(A,X3,A7)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),B4) )
                   => ( aa(set(A),$o,P,A7)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B4),A7)) ) ) )
             => aa(set(A),$o,P,A3) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4129_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),P: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B4: A,A7: set(A)] :
                  ( aa(set(A),$o,finite_finite(A),A7)
                 => ( ! [X3: A] :
                        ( member(A,X3,A7)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),X3) )
                   => ( aa(set(A),$o,P,A7)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B4),A7)) ) ) )
             => aa(set(A),$o,P,A3) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4130_sum__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(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,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)) ) ) ) ) ).

% sum_strict_mono
tff(fact_4131_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R: fun(A,fun(A,$o)),S3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,one_one(A)),one_one(A))
         => ( ! [X1: A,Y1: A,X23: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X1),X23)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y1),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(A,A,aa(A,fun(A,A),times_times(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),times_times(A),X23),Y23)) )
           => ( aa(set(B),$o,finite_finite(B),S3)
             => ( ! [X: B] :
                    ( member(B,X,S3)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H,X)),aa(B,A,G,X)) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S3)) ) ) ) ) ) ).

% prod.related
tff(fact_4132_sum_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = $ite(member(A,Xb,A3),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3))) ) ) ) ).

% sum.insert_if
tff(fact_4133_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S4: set(A),T3: set(B),S3: set(A),I: fun(B,A),J: fun(A,B),T4: set(B),G: fun(A,C),H: fun(B,C)] :
          ( aa(set(A),$o,finite_finite(A),S4)
         => ( aa(set(B),$o,finite_finite(B),T3)
           => ( ! [A4: A] :
                  ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S4))
                 => ( aa(B,A,I,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S4))
                   => member(B,aa(A,B,J,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),T3)) )
               => ( ! [B4: B] :
                      ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),T3))
                     => ( aa(A,B,J,aa(B,A,I,B4)) = B4 ) )
                 => ( ! [B4: B] :
                        ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),T3))
                       => member(A,aa(B,A,I,B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S4)) )
                   => ( ! [A4: A] :
                          ( member(A,A4,S4)
                         => ( aa(A,C,G,A4) = zero_zero(C) ) )
                     => ( ! [B4: B] :
                            ( member(B,B4,T3)
                           => ( aa(B,C,H,B4) = zero_zero(C) ) )
                       => ( ! [A4: A] :
                              ( member(A,A4,S3)
                             => ( aa(B,C,H,aa(A,B,J,A4)) = aa(A,C,G,A4) ) )
                         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),G),S3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),H),T4) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4134_card__eq__0__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = zero_zero(nat) )
    <=> ( ( A3 = bot_bot(set(A)) )
        | ~ aa(set(A),$o,finite_finite(A),A3) ) ) ).

% card_eq_0_iff
tff(fact_4135_card__ge__0__finite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3))
     => aa(set(A),$o,finite_finite(A),A3) ) ).

% card_ge_0_finite
tff(fact_4136_card__insert__if,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = $ite(member(A,Xb,A3),aa(set(A),nat,finite_card(A),A3),aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A3))) ) ) ).

% card_insert_if
tff(fact_4137_card__Suc__eq__finite,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
    <=> ? [B5: A,B9: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B5),B9) )
          & ~ member(A,B5,B9)
          & ( aa(set(A),nat,finite_card(A),B9) = K )
          & aa(set(A),$o,finite_finite(A),B9) ) ) ).

% card_Suc_eq_finite
tff(fact_4138_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S4: set(A),T3: set(B),S3: set(A),I: fun(B,A),J: fun(A,B),T4: set(B),G: fun(A,C),H: fun(B,C)] :
          ( aa(set(A),$o,finite_finite(A),S4)
         => ( aa(set(B),$o,finite_finite(B),T3)
           => ( ! [A4: A] :
                  ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S4))
                 => ( aa(B,A,I,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S4))
                   => member(B,aa(A,B,J,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),T3)) )
               => ( ! [B4: B] :
                      ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),T3))
                     => ( aa(A,B,J,aa(B,A,I,B4)) = B4 ) )
                 => ( ! [B4: B] :
                        ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),T3))
                       => member(A,aa(B,A,I,B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S4)) )
                   => ( ! [A4: A] :
                          ( member(A,A4,S4)
                         => ( aa(A,C,G,A4) = one_one(C) ) )
                     => ( ! [B4: B] :
                            ( member(B,B4,T3)
                           => ( aa(B,C,H,B4) = one_one(C) ) )
                       => ( ! [A4: A] :
                              ( member(A,A4,S3)
                             => ( aa(B,C,H,aa(A,B,J,A4)) = aa(A,C,G,A4) ) )
                         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),G),S3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H),T4) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_4139_card__less__sym__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( aa(set(A),$o,finite_finite(A),B3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3))) ) ) ) ).

% card_less_sym_Diff
tff(fact_4140_psubset__card__mono,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)) ) ) ).

% psubset_card_mono
tff(fact_4141_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X4: A] :
            ( member(A,X4,A3)
            & ( aa(A,nat,F2,X4) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa3: A] :
                ( member(A,Xa3,A3)
               => ( ( X4 != Xa3 )
                 => ( aa(A,nat,F2,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_4142_sum__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3) = one_one(nat) )
      <=> ? [X4: A] :
            ( member(A,X4,A3)
            & ( aa(A,nat,F2,X4) = one_one(nat) )
            & ! [Xa3: A] :
                ( member(A,Xa3,A3)
               => ( ( X4 != Xa3 )
                 => ( aa(A,nat,F2,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_4143_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S: set(A),F2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite(A),S)
         => ( ! [I2: A] :
                ( member(A,I2,S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),S) = zero_zero(B) )
             => ( member(A,I,S)
               => ( aa(A,B,F2,I) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_4144_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S: set(A),F2: fun(A,B),B3: B,I: A] :
          ( aa(set(A),$o,finite_finite(A),S)
         => ( ! [I2: A] :
                ( member(A,I2,S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),S) = B3 )
             => ( member(A,I,S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),B3) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_4145_sum_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(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_iw(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_4146_prod_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),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_ix(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_4147_finite__divisors__nat,axiom,
    ! [Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => aa(set(nat),$o,finite_finite(nat),collect(nat,aTP_Lamp_iy(nat,fun(nat,$o),Ma))) ) ).

% finite_divisors_nat
tff(fact_4148_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite(nat),A3)
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ia(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F2),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),A3)) ) ) ).

% sums_If_finite_set
tff(fact_4149_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite(nat),collect(nat,P))
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ib(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F2),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),collect(nat,P))) ) ) ).

% sums_If_finite
tff(fact_4150_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N5: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite(nat),N5)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N5)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => sums(A,F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),N5)) ) ) ) ).

% sums_finite
tff(fact_4151_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N5: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite(nat),N5)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N5)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => ( suminf(A,F2) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),N5) ) ) ) ) ).

% suminf_finite
tff(fact_4152_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A] : aa(set(A),$o,finite_finite(A),collect(A,aTP_Lamp_iz(A,fun(A,$o),A2))) ) ).

% finite_abs_int_segment
tff(fact_4153_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N5: set(nat),Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N5),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
     => aa(set(nat),$o,finite_finite(nat),N5) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_4154_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),I: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,I))
             => ( ! [I2: A] :
                    ( member(A,I2,I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),I5)) ) ) ) ) ) ).

% sum_pos2
tff(fact_4155_sum__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,I2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),I5)) ) ) ) ) ).

% sum_pos
tff(fact_4156_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I))
             => ( ! [I2: A] :
                    ( member(A,I2,I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,I2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),I5)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_4157_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),I5)) ) ) ) ) ).

% less_1_prod
tff(fact_4158_sum_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T4: set(A),S3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
                 => ( aa(A,B,G,X) = zero_zero(B) ) )
             => ( ! [X: A] :
                    ( member(A,X,S3)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),T4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),S3) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4159_sum_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T4: set(A),S3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
                 => ( aa(A,B,H,X) = zero_zero(B) ) )
             => ( ! [X: A] :
                    ( member(A,X,S3)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),T4) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4160_sum_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T4: set(A),S3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
                 => ( aa(A,B,G,X) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),T4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S3) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4161_sum_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T4: set(A),S3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
                 => ( aa(A,B,G,X) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),T4) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4162_sum_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C6: set(A),A3: set(A),B3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),C6)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C6)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C6)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C6),A3))
                   => ( aa(A,B,G,A4) = zero_zero(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C6),B3))
                     => ( aa(A,B,H,B4) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),C6) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),C6) )
                   => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),B3) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4163_sum_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C6: set(A),A3: set(A),B3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),C6)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C6)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C6)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C6),A3))
                   => ( aa(A,B,G,A4) = zero_zero(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C6),B3))
                     => ( aa(A,B,H,B4) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),B3) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),C6) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),C6) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4164_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),$o),ord_less_eq(set(A)),B3),A3)
         => ( aa(set(A),$o,finite_finite(A),A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B3)) ) ) ) ) ).

% sum.subset_diff
tff(fact_4165_sum__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A3: set(A),B3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(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(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B3)) ) ) ) ) ).

% sum_diff
tff(fact_4166_card__gt__0__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3))
    <=> ( ( A3 != bot_bot(set(A)) )
        & aa(set(A),$o,finite_finite(A),A3) ) ) ).

% card_gt_0_iff
tff(fact_4167_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(nat,nat,suc,zero_zero(nat)))
      <=> ! [X4: A] :
            ( member(A,X4,A3)
           => ! [Xa3: A] :
                ( member(A,Xa3,A3)
               => ( X4 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_4168_card__le__Suc__iff,axiom,
    ! [A: $tType,Nb: nat,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(set(A),nat,finite_card(A),A3))
    <=> ? [A5: A,B9: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A5),B9) )
          & ~ member(A,A5,B9)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(set(A),nat,finite_card(A),B9))
          & aa(set(A),$o,finite_finite(A),B9) ) ) ).

% card_le_Suc_iff
tff(fact_4169_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),S3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
                 => ( aa(A,B,G,X) = one_one(B) ) )
             => ( ! [X: A] :
                    ( member(A,X,S3)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),S3) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_4170_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),S3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
                 => ( aa(A,B,H,X) = one_one(B) ) )
             => ( ! [X: A] :
                    ( member(A,X,S3)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),T4) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_4171_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),S3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
                 => ( aa(A,B,G,X) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S3) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_4172_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),S3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T4)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
                 => ( aa(A,B,G,X) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T4) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_4173_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C6: set(A),A3: set(A),B3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),C6)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C6)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C6)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C6),A3))
                   => ( aa(A,B,G,A4) = one_one(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C6),B3))
                     => ( aa(A,B,H,B4) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),C6) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),C6) )
                   => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),B3) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_4174_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C6: set(A),A3: set(A),B3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),C6)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C6)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C6)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C6),A3))
                   => ( aa(A,B,G,A4) = one_one(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C6),B3))
                     => ( aa(A,B,H,B4) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),B3) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),C6) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),C6) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_4175_card__Diff__subset,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)) ) ) ) ).

% card_Diff_subset
tff(fact_4176_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ~ aa(set(A),$o,finite_finite(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)),insert2(A),A2),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw
tff(fact_4177_card__psubset,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3))
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3) ) ) ) ).

% card_psubset
tff(fact_4178_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),B3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))) ) ).

% diff_card_le_card_Diff
tff(fact_4179_sum__diff__nat,axiom,
    ! [A: $tType,B3: set(A),A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite(A),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(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),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),B3)) ) ) ) ).

% sum_diff_nat
tff(fact_4180_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => aa(set(A),$o,finite_finite(A),collect(A,aTP_Lamp_al(nat,fun(A,$o),Nb))) ) ) ).

% finite_roots_unity
tff(fact_4181_ex__bij__betw__nat__finite__1,axiom,
    ! [A: $tType,M9: set(A)] :
      ( aa(set(A),$o,finite_finite(A),M9)
     => ? [H2: fun(nat,A)] : bij_betw(nat,A,H2,set_or1337092689740270186AtMost(nat,one_one(nat),aa(set(A),nat,finite_card(A),M9)),M9) ) ).

% ex_bij_betw_nat_finite_1
tff(fact_4182_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S4: set(A),T3: set(B),H: fun(A,B),S3: set(A),T4: set(B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite(A),S4)
         => ( aa(set(B),$o,finite_finite(B),T3)
           => ( bij_betw(A,B,H,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),T3))
             => ( ! [A4: A] :
                    ( member(A,A4,S4)
                   => ( aa(B,C,G,aa(A,B,H,A4)) = zero_zero(C) ) )
               => ( ! [B4: B] :
                      ( member(B,B4,T3)
                     => ( aa(B,C,G,B4) = zero_zero(C) ) )
                 => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_ja(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),T4) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_4183_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S3: A,A3: set(nat),S4: A,F2: fun(nat,A)] :
          ( sums(A,G,S3)
         => ( aa(set(nat),$o,finite_finite(nat),A3)
           => ( ( S4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),S3),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_jb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G),F2)),A3)) )
             => sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_jc(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A3),F2),S4) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_4184_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S4: set(A),T3: set(B),H: fun(A,B),S3: set(A),T4: set(B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite(A),S4)
         => ( aa(set(B),$o,finite_finite(B),T3)
           => ( bij_betw(A,B,H,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),T3))
             => ( ! [A4: A] :
                    ( member(A,A4,S4)
                   => ( aa(B,C,G,aa(A,B,H,A4)) = one_one(C) ) )
               => ( ! [B4: B] :
                      ( member(B,B4,T3)
                     => ( aa(B,C,G,B4) = one_one(C) ) )
                 => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_jd(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),T4) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_4185_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ! [I2: A] :
                ( member(A,I2,A3)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2))
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I2)),aa(A,B,G,I2)) ) )
           => ( ( A3 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)) ) ) ) ) ).

% prod_mono_strict
tff(fact_4186_sum__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [B3: set(A),A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
           => ( ! [B4: A] :
                  ( member(A,B4,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_eq(B),zero_zero(B)),aa(A,B,F2,B4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B3)) ) ) ) ) ).

% sum_mono2
tff(fact_4187_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(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)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ) ).

% sum.insert_remove
tff(fact_4188_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),Xb: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( member(A,Xb,A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(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)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ) ) ).

% sum.remove
tff(fact_4189_sum__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A3: set(A),F2: fun(A,B),A2: A] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(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)),insert2(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,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(A,B,F2,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)) ) ) ) ).

% sum_diff1
tff(fact_4190_card_Oremove,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( member(A,Xb,A3)
       => ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_4191_card_Oinsert__remove,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_4192_card__Suc__Diff1,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( member(A,Xb,A3)
       => ( aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_4193_card__Diff1__less__iff,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3))
    <=> ( aa(set(A),$o,finite_finite(A),A3)
        & member(A,Xb,A3) ) ) ).

% card_Diff1_less_iff
tff(fact_4194_card__Diff2__less,axiom,
    ! [A: $tType,A3: set(A),Xb: A,Y: A] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( member(A,Xb,A3)
       => ( member(A,Y,A3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)) ) ) ) ).

% card_Diff2_less
tff(fact_4195_card__Diff1__less,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( member(A,Xb,A3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)) ) ) ).

% card_Diff1_less
tff(fact_4196_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I5: set(nat)] :
          ( summable(A,F2)
         => ( aa(set(nat),$o,finite_finite(nat),I5)
           => ( ! [N: nat] :
                  ( member(nat,N,aa(set(nat),set(nat),uminus_uminus(set(nat)),I5))
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),I5)),suminf(A,F2)) ) ) ) ) ).

% sum_le_suminf
tff(fact_4197_sum_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B),C2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_je(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C2)),S3) = $ite(member(A,A2,S3),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,B2,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(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)),S3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_4198_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6)
     => ( aa(set(A),$o,finite_finite(A),X6)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),count_list(A,Xs)),X6) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% sum_count_set
tff(fact_4199_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B3: set(A),A3: set(A),B2: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(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,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B3)) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4200_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: A,A3: set(A),F2: fun(A,B)] :
          ( member(A,I,A3)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),bot_bot(set(A)))))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X)) )
           => ( aa(set(A),$o,finite_finite(A),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)) ) ) ) ) ).

% member_le_sum
tff(fact_4201_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B3: set(A),A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
           => ( ! [B4: A] :
                  ( member(A,B4,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_eq(B),one_one(B)),aa(A,B,F2,B4)) )
             => ( ! [A4: A] :
                    ( member(A,A4,A3)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,A4)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B3)) ) ) ) ) ) ).

% prod_mono2
tff(fact_4202_sum__bounded__above__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(B)
     => ! [A3: set(A),F2: fun(A,B),K6: B] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),divide_divide(B,K6,aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A3)))) )
         => ( aa(set(A),$o,finite_finite(A),A3)
           => ( ( A3 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),K6) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_4203_prod__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom_divide(B)
     => ! [A3: set(A),F2: fun(A,B),A2: A] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),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)),insert2(A),A2),bot_bot(set(A))))) = $ite(member(A,A2,A3),divide_divide(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3),aa(A,B,F2,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ) ) ).

% prod_diff1
tff(fact_4204_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Nb: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => aa(set(A),$o,finite_finite(A),collect(A,aa(nat,fun(A,$o),aTP_Lamp_fh(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb))) ) ) ) ).

% polyfun_roots_finite
tff(fact_4205_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat] :
          ( aa(set(A),$o,finite_finite(A),collect(A,aa(nat,fun(A,$o),aTP_Lamp_fh(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))
        <=> ? [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
              & ( aa(nat,A,C2,I3) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_4206_prod__gen__delta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B),C2: B] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_jf(A,fun(fun(A,B),fun(B,fun(A,B))),A2),B2),C2)),S3) = $ite(member(A,A2,S3),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A2)),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),S3)),one_one(nat)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),aa(set(A),nat,finite_card(A),S3))) ) ) ) ).

% prod_gen_delta
tff(fact_4207_card__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_in(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A3)),Nb) ) ) ).

% card_lists_length_eq
tff(fact_4208_polyfun__rootbound,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Nb: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => ( aa(set(A),$o,finite_finite(A),collect(A,aa(nat,fun(A,$o),aTP_Lamp_fh(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),collect(A,aa(nat,fun(A,$o),aTP_Lamp_fh(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))),Nb) ) ) ) ) ).

% polyfun_rootbound
tff(fact_4209_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F4: set(A),I5: set(A),F2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite(A),F4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_jg(set(A),fun(fun(A,B),fun(A,$o)),I5),F2))),F4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),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)),insert2(A),I),bot_bot(set(A))))) = $ite(member(A,I,I5),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F2),I5)),aa(A,B,F2,I)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F2),I5)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_4210_Sum__Ico__nat,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_az(nat,nat)),set_or7035219750837199246ssThan(nat,Ma,Nb)) = 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),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% Sum_Ico_nat
tff(fact_4211_VEBT_Osize_I3_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : ( aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(vEBT_VEBT,size_size(vEBT_VEBT),X13)),aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),X14))),aa(nat,nat,suc,zero_zero(nat))) ) ).

% VEBT.size(3)
tff(fact_4212_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : ( aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(vEBT_VEBT,vEBT_size_VEBT,X13)),aa(vEBT_VEBT,nat,vEBT_size_VEBT,X14))),aa(nat,nat,suc,zero_zero(nat))) ) ).

% VEBT.size_gen(1)
tff(fact_4213_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_4214_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_4215_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_4216_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite(A),set_or7035219750837199246ssThan(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ico_iff
tff(fact_4217_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: fun(B,A)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P3),bot_bot(set(B))) = zero_zero(A) ) ) ).

% sum.empty'
tff(fact_4218_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : ( aa(set(nat),nat,finite_card(nat),set_or7035219750837199246ssThan(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),L) ) ).

% card_atLeastLessThan
tff(fact_4219_atLeastLessThan__singleton,axiom,
    ! [Ma: nat] : ( set_or7035219750837199246ssThan(nat,Ma,aa(nat,nat,suc,Ma)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),Ma),bot_bot(set(nat))) ) ).

% atLeastLessThan_singleton
tff(fact_4220_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),P3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_io(set(A),fun(fun(A,B),fun(A,$o)),I5),P3)))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),P3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),I5)) = $ite(member(A,I,I5),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),P3),I5),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,P3,I)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),P3),I5))) ) ) ) ).

% sum.insert'
tff(fact_4221_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Ma,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Ma,Nb))),aa(nat,A,G,Nb))) ) ) ).

% sum.op_ivl_Suc
tff(fact_4222_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Ma,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Ma,Nb))),aa(nat,A,G,Nb))) ) ) ).

% prod.op_ivl_Suc
tff(fact_4223_ex__bij__betw__finite__nat,axiom,
    ! [A: $tType,M9: set(A)] :
      ( aa(set(A),$o,finite_finite(A),M9)
     => ? [H2: fun(A,nat)] : bij_betw(A,nat,H2,M9,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M9))) ) ).

% ex_bij_betw_finite_nat
tff(fact_4224_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
             => ( B2 = D3 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4225_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
             => ( A2 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4226_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
           => ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D3) )
            <=> ( ( A2 = C2 )
                & ( B2 = D3 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4227_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),I5: set(B)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),collect(B,aa(set(B),fun(B,$o),aTP_Lamp_jh(fun(B,A),fun(set(B),fun(B,$o)),G),I5))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),I5) ) ) ).

% sum.non_neutral'
tff(fact_4228_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite(A),set_or7035219750837199246ssThan(A,A2,B2)) ) ) ).

% infinite_Ico
tff(fact_4229_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Nb)
         => aa(nat,$o,P,M3) )
    <=> ! [X4: nat] :
          ( member(nat,X4,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P,X4) ) ) ).

% all_nat_less_eq
tff(fact_4230_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Nb)
          & aa(nat,$o,P,M3) )
    <=> ? [X4: nat] :
          ( member(nat,X4,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P,X4) ) ) ).

% ex_nat_less_eq
tff(fact_4231_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : ( set_or7035219750837199246ssThan(nat,L,aa(nat,nat,suc,U)) = set_or1337092689740270186AtMost(nat,L,U) ) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_4232_lessThan__atLeast0,axiom,
    ! [Nb: nat] : ( set_ord_lessThan(nat,Nb) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb) ) ).

% lessThan_atLeast0
tff(fact_4233_atLeastLessThan0,axiom,
    ! [Ma: nat] : ( set_or7035219750837199246ssThan(nat,Ma,zero_zero(nat)) = bot_bot(set(nat)) ) ).

% atLeastLessThan0
tff(fact_4234_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_co(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_4235_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N5: set(nat),Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N5),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
     => aa(set(nat),$o,finite_finite(nat),N5) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_4236_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Ma: nat,K: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_cp(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_4237_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_aw(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_4238_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Ma: nat,K: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ax(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_4239_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : ( set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,L),U) = set_or5935395276787703475ssThan(nat,L,U) ) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_4240_sum_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),I5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_ji(fun(A,B),fun(fun(A,B),fun(A,B)),G),H)),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),G),I5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),H),I5)) ) ) ) ).

% sum.distrib_triv'
tff(fact_4241_sum_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_add(B) )
     => ! [A2: A,C2: A,B2: A,D3: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C2 )
         => ( ( B2 = D3 )
           => ( ! [X: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),D3)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),set_or7035219750837199246ssThan(A,A2,B2)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),set_or7035219750837199246ssThan(A,C2,D3)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_4242_prod_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_mult(B) )
     => ! [A2: A,C2: A,B2: A,D3: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C2 )
         => ( ( B2 = D3 )
           => ( ! [X: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),D3)
                   => ( aa(A,B,G,X) = aa(A,B,H,X) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),set_or7035219750837199246ssThan(A,A2,B2)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),set_or7035219750837199246ssThan(A,C2,D3)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_4243_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Ma: nat,Nb: nat,P3: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),P3)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Ma,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,P3))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Ma,P3)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_4244_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Ma: nat,Nb: nat,P3: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),P3)
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,Ma,P3))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,Ma,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,Nb,P3)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_4245_size__list__estimation,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: nat,F2: fun(A,nat)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(A,nat,F2,Xb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),size_list(A,F2,Xs)) ) ) ).

% size_list_estimation
tff(fact_4246_size__list__estimation_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: nat,F2: fun(A,nat)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(A,nat,F2,Xb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),size_list(A,F2,Xs)) ) ) ).

% size_list_estimation'
tff(fact_4247_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,nat),G: fun(A,nat)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,X)),aa(A,nat,G,X)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),size_list(A,F2,Xs)),size_list(A,G,Xs)) ) ).

% size_list_pointwise
tff(fact_4248_atLeast0__lessThan__Suc,axiom,
    ! [Nb: nat] : ( set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),Nb),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% atLeast0_lessThan_Suc
tff(fact_4249_sum_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T4)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
               => ( aa(A,B,G,X) = zero_zero(B) ) )
           => ( ! [X: A] :
                  ( member(A,X,S3)
                 => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),G),T4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),H),S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_4250_sum_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T4: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T4)
         => ( ! [I2: A] :
                ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
               => ( aa(A,B,H,I2) = zero_zero(B) ) )
           => ( ! [X: A] :
                  ( member(A,X,S3)
                 => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),H),T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_4251_sum_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T4)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
               => ( aa(A,B,G,X) = zero_zero(B) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),G),T4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),G),S3) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_4252_sum_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T4)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
               => ( aa(A,B,G,X) = zero_zero(B) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),G),T4) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_4253_subset__card__intvl__is__intvl,axiom,
    ! [A3: set(nat),K: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),A3),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A3))))
     => ( A3 = set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A3))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_4254_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D3))
        <=> ( aa(A,$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),D3) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_4255_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D3))
        <=> ( aa(A,$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),D3) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4256_sum__shift__lb__Suc0__0__upt,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_4257_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(nat,A,G,Nb)) ) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_4258_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Ma: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Ma,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Ma)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),Nb))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_4259_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,aa(fun(nat,A),fun(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_4260_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D3))
        <=> ( aa(A,$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),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_4261_sum_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_io(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( aa(set(A),$o,finite_finite(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_io(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_ji(fun(A,B),fun(fun(A,B),fun(A,B)),G),H)),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),G),I5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),H),I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_4262_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(nat,A,G,Nb)) ) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_4263_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Ma: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Ma,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Ma)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),Nb))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_4264_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)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_4265_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: fun(B,A),I5: set(B)] :
          ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P3),I5) = $ite(aa(set(B),$o,finite_finite(B),collect(B,aa(set(B),fun(B,$o),aTP_Lamp_jh(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),P3),collect(B,aa(set(B),fun(B,$o),aTP_Lamp_jh(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),zero_zero(A)) ) ) ).

% sum.G_def
tff(fact_4266_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Ma: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Nb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Ma,Nb))) ) ) ) ).

% sum.last_plus
tff(fact_4267_ex__bij__betw__nat__finite,axiom,
    ! [A: $tType,M9: set(A)] :
      ( aa(set(A),$o,finite_finite(A),M9)
     => ? [H2: fun(nat,A)] : bij_betw(nat,A,H2,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M9)),M9) ) ).

% ex_bij_betw_nat_finite
tff(fact_4268_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Ma: nat,Nb: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cv(fun(nat,A),fun(nat,A),F2)),set_or7035219750837199246ssThan(nat,Ma,Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,Ma)) ) ) ) ).

% sum_Suc_diff'
tff(fact_4269_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,Ma: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,Ma)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jj(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Ma)),set_or7035219750837199246ssThan(nat,Nb,Ma)) ) ) ).

% sum.atLeastLessThan_rev
tff(fact_4270_atLeastLessThanSuc,axiom,
    ! [Ma: nat,Nb: nat] :
      ( set_or7035219750837199246ssThan(nat,Ma,aa(nat,nat,suc,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),Nb),set_or7035219750837199246ssThan(nat,Ma,Nb)),bot_bot(set(nat))) ) ).

% atLeastLessThanSuc
tff(fact_4271_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_jk(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ew(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ) ).

% sum.nested_swap
tff(fact_4272_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,Ma: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,Ma)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jl(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Ma)),set_or7035219750837199246ssThan(nat,Nb,Ma)) ) ) ).

% prod.atLeastLessThan_rev
tff(fact_4273_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_jm(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ez(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ) ).

% prod.nested_swap
tff(fact_4274_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),K: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_ord_lessThan(nat,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))) ) ) ).

% sum.nat_group
tff(fact_4275_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),K: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jo(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_ord_lessThan(nat,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))) ) ) ).

% prod.nat_group
tff(fact_4276_prod__Suc__fact,axiom,
    ! [Nb: nat] : ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) = semiring_char_0_fact(nat,Nb) ) ).

% prod_Suc_fact
tff(fact_4277_prod__Suc__Suc__fact,axiom,
    ! [Nb: nat] : ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = semiring_char_0_fact(nat,Nb) ) ).

% prod_Suc_Suc_fact
tff(fact_4278_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N5: set(nat),Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N5),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N5)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_4279_card__sum__le__nat__sum,axiom,
    ! [S3: set(nat)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_az(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(nat),nat,finite_card(nat),S3)))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_az(nat,nat)),S3)) ).

% card_sum_le_nat_sum
tff(fact_4280_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Ma,Nb))),aa(nat,A,G,Nb))) ) ) ).

% sum.head_if
tff(fact_4281_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] :
          ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Ma,Nb))),aa(nat,A,G,Nb))) ) ) ).

% prod.head_if
tff(fact_4282_fact__prod__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : ( semiring_char_0_fact(A,Nb) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ) ).

% fact_prod_Suc
tff(fact_4283_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,Ma: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,Ma)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ct(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Ma)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Ma)) ) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_4284_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,Ma: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,Ma)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ay(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Ma)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Ma)) ) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_4285_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : ( comm_s3205402744901411588hammer(A,A2,Nb) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_bb(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ) ).

% pochhammer_prod
tff(fact_4286_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : ( semiring_char_0_fact(A,Nb) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aa(nat,fun(nat,nat),minus_minus(nat),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ) ).

% fact_prod_rev
tff(fact_4287_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A,K: nat] :
          ( sums(A,F2,S)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
           => sums(A,aa(nat,fun(nat,A),aTP_Lamp_jp(fun(nat,A),fun(nat,fun(nat,A)),F2),K),S) ) ) ) ).

% sums_group
tff(fact_4288_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,A2: A] : ( bit_se2584673776208193580ke_bit(A,Nb,A2) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_jq(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ) ).

% take_bit_sum
tff(fact_4289_atLeast1__lessThan__eq__remove0,axiom,
    ! [Nb: nat] : ( set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),Nb) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_lessThan(nat,Nb)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat)))) ) ).

% atLeast1_lessThan_eq_remove0
tff(fact_4290_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( semiring_char_0_fact(A,Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K),Nb)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))) ) ) ) ).

% fact_split
tff(fact_4291_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jr(nat,fun(nat,fun(nat,A)),K),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_4292_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : ( aa(nat,A,gbinomial(A,A2),K) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_js(A,fun(nat,fun(nat,A)),A2),K)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ).

% gbinomial_altdef_of_nat
tff(fact_4293_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)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_jt(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ).

% gbinomial_mult_fact'
tff(fact_4294_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)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_jt(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ).

% gbinomial_mult_fact
tff(fact_4295_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : ( aa(nat,A,gbinomial(A,A2),K) = divide_divide(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_be(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)),semiring_char_0_fact(A,K)) ) ) ).

% gbinomial_prod_rev
tff(fact_4296_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_jg(set(A),fun(fun(A,B),fun(A,$o)),I5),F2)))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),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)),insert2(A),I),bot_bot(set(A))))) = $ite(member(A,I,I5),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F2),I5)),aa(A,B,F2,I)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F2),I5)) ) ) ) ).

% sum_diff1'
tff(fact_4297_sum__power2,axiom,
    ! [K: nat] : ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),one_one(nat)) ) ).

% sum_power2
tff(fact_4298_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,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_ju(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_4299_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I2: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,I2)),aa(nat,A,A2,J2)) ) )
         => ( ! [I2: nat,J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,B2,J2)),aa(nat,A,B2,I2)) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_jv(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_4300_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A2: fun(nat,nat),B2: fun(nat,nat)] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,A2,I2)),aa(nat,nat,A2,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,B2,J2)),aa(nat,nat,B2,I2)) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_jw(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_4301_infinite__int__iff__unbounded,axiom,
    ! [S3: set(int)] :
      ( ~ aa(set(int),$o,finite_finite(int),S3)
    <=> ! [M3: int] :
        ? [N4: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),M3),aa(int,int,abs_abs(int),N4))
          & member(int,N4,S3) ) ) ).

% infinite_int_iff_unbounded
tff(fact_4302_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : ( aa(set(int),nat,finite_card(int),set_or7035219750837199246ssThan(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)) ) ).

% card_atLeastLessThan_int
tff(fact_4303_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa(set(int),$o,finite_finite(int),set_or7035219750837199246ssThan(int,zero_zero(int),U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_4304_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_4305_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( aa(set(int),nat,finite_card(int),set_or7035219750837199246ssThan(int,zero_zero(int),U)) = aa(int,nat,nat2,U) ) ).

% card_atLeastZeroLessThan_int
tff(fact_4306_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_4307_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set(nat)] :
      ( ! [M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),M)
         => ? [N3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N3)
              & member(nat,N3,S3) ) )
     => ~ aa(set(nat),$o,finite_finite(nat),S3) ) ).

% unbounded_k_infinite
tff(fact_4308_infinite__nat__iff__unbounded,axiom,
    ! [S3: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite(nat),S3)
    <=> ! [M3: nat] :
        ? [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N4)
          & member(nat,N4,S3) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_4309_int__ge__less__than2__def,axiom,
    ! [D3: int] : ( int_ge_less_than2(D3) = collect(product_prod(int,int),product_case_prod(int,int,$o,aTP_Lamp_jx(int,fun(int,fun(int,$o)),D3))) ) ).

% int_ge_less_than2_def
tff(fact_4310_int__ge__less__than__def,axiom,
    ! [D3: int] : ( int_ge_less_than(D3) = collect(product_prod(int,int),product_case_prod(int,int,$o,aTP_Lamp_jy(int,fun(int,fun(int,$o)),D3))) ) ).

% int_ge_less_than_def
tff(fact_4311_int__of__nat__def,axiom,
    code_T6385005292777649522of_nat = semiring_1_of_nat(int) ).

% int_of_nat_def
tff(fact_4312_set__update__distinct,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Xb: A] :
      ( distinct(A,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),set(A),set2(A),list_update(A,Xs,Nb,Xb)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(nat,A,nth(A,Xs),Nb)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_4313_nth__list__update__eq,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$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,Xb)),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_4314_set__swap,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),set(A),set2(A),list_update(A,list_update(A,Xs,I,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I))) = aa(list(A),set(A),set2(A),Xs) ) ) ) ).

% set_swap
tff(fact_4315_distinct__swap,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => ( distinct(A,list_update(A,list_update(A,Xs,I,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I)))
        <=> distinct(A,Xs) ) ) ) ).

% distinct_swap
tff(fact_4316_set__update__subsetI,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A),Xb: A,I: nat] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A3)
     => ( member(A,Xb,A3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I,Xb))),A3) ) ) ).

% set_update_subsetI
tff(fact_4317_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xb: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I,Xb))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),aa(list(A),set(A),set2(A),Xs))) ).

% set_update_subset_insert
tff(fact_4318_set__update__memI,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => member(A,Xb,aa(list(A),set(A),set2(A),list_update(A,Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_4319_list__update__same__conv,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ( list_update(A,Xs,I,Xb) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_4320_nth__list__update,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xb: A,J: nat] :
      ( aa(nat,$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,Xb)),J) = $ite(I = J,Xb,aa(nat,A,nth(A,Xs),J)) ) ) ).

% nth_list_update
tff(fact_4321_distinct__list__update,axiom,
    ! [A: $tType,Xs: list(A),A2: A,I: nat] :
      ( distinct(A,Xs)
     => ( ~ member(A,A2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(nat,A,nth(A,Xs),I)),bot_bot(set(A)))))
       => distinct(A,list_update(A,Xs,I,A2)) ) ) ).

% distinct_list_update
tff(fact_4322_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_jz(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ) ).

% divmod_step_integer_def
tff(fact_4323_case__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,A),V2: num,Nb: nat] : ( case_nat(A,A2,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),Nb)) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V2)),Nb)) ) ).

% case_nat_add_eq_if
tff(fact_4324_rec__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,fun(A,A)),V2: num,Nb: nat] :
      ( aa(nat,A,rec_nat(A,A2,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),Nb)) = $let(
          pv: nat,
          pv:= pred_numeral(V2),
          aa(A,A,aa(nat,fun(A,A),F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),Nb)),aa(nat,A,rec_nat(A,A2,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),Nb))) ) ) ).

% rec_nat_add_eq_if
tff(fact_4325_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,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)
              | ~ aa(int,$o,aa(int,fun(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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ).

% or_int_rec
tff(fact_4326_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_4327_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_4328_old_Onat_Osimps_I7_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,fun(A,A)),Nat: nat] : ( aa(nat,A,rec_nat(A,F1,F22),aa(nat,nat,suc,Nat)) = aa(A,A,aa(nat,fun(A,A),F22,Nat),aa(nat,A,rec_nat(A,F1,F22),Nat)) ) ).

% old.nat.simps(7)
tff(fact_4329_old_Onat_Osimps_I6_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,fun(A,A))] : ( aa(nat,A,rec_nat(A,F1,F22),zero_zero(nat)) = F1 ) ).

% old.nat.simps(6)
tff(fact_4330_bit_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% bit.disj_one_right
tff(fact_4331_bit_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,uminus_uminus(A),one_one(A))),Xb) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% bit.disj_one_left
tff(fact_4332_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_4333_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_4334_case__nat__numeral,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,A),V2: num] : ( case_nat(A,A2,F2,aa(num,nat,numeral_numeral(nat),V2)) = aa(nat,A,F2,pred_numeral(V2)) ) ).

% case_nat_numeral
tff(fact_4335_rec__nat__numeral,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,fun(A,A)),V2: num] :
      ( aa(nat,A,rec_nat(A,A2,F2),aa(num,nat,numeral_numeral(nat),V2)) = $let(
          pv: nat,
          pv:= pred_numeral(V2),
          aa(A,A,aa(nat,fun(A,A),F2,pv),aa(nat,A,rec_nat(A,A2,F2),pv)) ) ) ).

% rec_nat_numeral
tff(fact_4336_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_4337_or__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb)) ) ) ).

% or_numerals(8)
tff(fact_4338_bit_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),Xb) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% bit.disj_cancel_left
tff(fact_4339_bit_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% bit.disj_cancel_right
tff(fact_4340_or__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(Xb))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb)) ) ) ).

% or_numerals(5)
tff(fact_4341_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_4342_or__minus__numerals_I2_J,axiom,
    ! [Nb: 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,Nb)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Nb))) ) ).

% or_minus_numerals(2)
tff(fact_4343_or__minus__numerals_I6_J,axiom,
    ! [Nb: 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,Nb)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Nb))) ) ).

% or_minus_numerals(6)
tff(fact_4344_or__minus__minus__numerals,axiom,
    ! [Ma: num,Nb: 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),Ma))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = 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),Ma)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Nb)),one_one(int)))) ) ).

% or_minus_minus_numerals
tff(fact_4345_and__minus__minus__numerals,axiom,
    ! [Ma: num,Nb: 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),Ma))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = 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),Ma)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Nb)),one_one(int)))) ) ).

% and_minus_minus_numerals
tff(fact_4346_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(Xb))),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),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ) ).

% or_numerals(4)
tff(fact_4347_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: 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,Xb))),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),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ) ).

% or_numerals(6)
tff(fact_4348_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: 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,Xb))),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),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ) ).

% or_numerals(7)
tff(fact_4349_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),zero_zero(code_integer)),L) = aa(code_integer,code_integer,uminus_uminus(code_integer),L) ) ).

% minus_integer_code(2)
tff(fact_4350_sgn__integer__code,axiom,
    ! [K: code_integer] :
      ( aa(code_integer,code_integer,sgn_sgn(code_integer),K) = $ite(
          K = zero_zero(code_integer),
          zero_zero(code_integer),
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),one_one(code_integer)) ) ) ).

% sgn_integer_code
tff(fact_4351_bit_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),zero_zero(A)) = Xb ) ) ).

% bit.disj_zero_right
tff(fact_4352_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_4353_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_4354_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_ka(fun(B,A),fun(fun(nat,B),fun(nat,A)),H),F22),Nat) ) ).

% nat.case_distrib
tff(fact_4355_of__nat__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,Nb: nat] : ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Ma),Nb)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,semiring_1_of_nat(A),Ma)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ).

% of_nat_or_eq
tff(fact_4356_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A),X2: nat] : ( case_nat(A,F1,F22,aa(nat,nat,suc,X2)) = aa(nat,A,F22,X2) ) ).

% old.nat.simps(5)
tff(fact_4357_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_4358_OR__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( 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),Xb),Y)) ) ) ).

% OR_lower
tff(fact_4359_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_4360_disjunctive__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( ! [N: nat] :
              ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N)
              | ~ aa(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_4361_plus__and__or,axiom,
    ! [Xb: 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),Xb),Y)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xb),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Y) ) ).

% plus_and_or
tff(fact_4362_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> case_nat($o,$true,aTP_Lamp_kb(nat,$o),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_4363_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> case_nat($o,$false,aTP_Lamp_kc(nat,$o),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_4364_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_4365_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : ( bit_se5668285175392031749et_bit(A,Nb,A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),one_one(A))) ) ) ).

% set_bit_eq_or
tff(fact_4366_zero__natural_Orsp,axiom,
    zero_zero(nat) = zero_zero(nat) ).

% zero_natural.rsp
tff(fact_4367_zero__integer_Orsp,axiom,
    zero_zero(int) = zero_zero(int) ).

% zero_integer.rsp
tff(fact_4368_one__integer_Orsp,axiom,
    one_one(int) = one_one(int) ).

% one_integer.rsp
tff(fact_4369_set__bit__int__def,axiom,
    ! [Nb: nat,K: int] : ( bit_se5668285175392031749et_bit(int,Nb,K) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),one_one(int))) ) ).

% set_bit_int_def
tff(fact_4370_one__natural_Orsp,axiom,
    one_one(nat) = one_one(nat) ).

% one_natural.rsp
tff(fact_4371_less__eq__nat_Osimps_I2_J,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Ma)),Nb)
    <=> case_nat($o,$false,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_4372_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),Xb) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Xb) = 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)) )
               => ( Xb = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_4373_or__not__numerals_I4_J,axiom,
    ! [Ma: num] : ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(Ma))),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_4374_or__not__numerals_I2_J,axiom,
    ! [Nb: 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(Nb)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Nb))) ) ).

% or_not_numerals(2)
tff(fact_4375_diff__Suc,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),aa(nat,nat,suc,Nb)) = case_nat(nat,zero_zero(nat),aTP_Lamp_az(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb)) ) ).

% diff_Suc
tff(fact_4376_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),Y) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),Xb) = Y ) ) ) ) ).

% bit.compl_unique
tff(fact_4377_or__not__numerals_I3_J,axiom,
    ! [Nb: 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,Nb)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Nb))) ) ).

% or_not_numerals(3)
tff(fact_4378_or__not__numerals_I7_J,axiom,
    ! [Ma: num] : ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Ma))),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_4379_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,Xb: A,F2: fun(nat,A),Nb: nat] :
      ( case_nat(A,Xb,F2,Nb) = $ite(Nb = zero_zero(nat),Xb,aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ) ).

% Nitpick.case_nat_unfold
tff(fact_4380_mask__Suc__exp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : ( bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,Nb)) = 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))),Nb)),bit_se2239418461657761734s_mask(A,Nb)) ) ) ).

% mask_Suc_exp
tff(fact_4381_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,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ) ).

% one_or_eq
tff(fact_4382_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,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ) ).

% or_one_eq
tff(fact_4383_mask__Suc__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : ( bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,Nb))) ) ) ).

% mask_Suc_double
tff(fact_4384_OR__upper,axiom,
    ! [Xb: int,Nb: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
       => ( 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))),Nb))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xb),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ) ) ).

% OR_upper
tff(fact_4385_or__not__numerals_I5_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(Ma))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Nb)))) = 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),Ma)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb))))) ) ).

% or_not_numerals(5)
tff(fact_4386_or__not__numerals_I9_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Ma))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Nb)))) = 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),Ma)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb))))) ) ).

% or_not_numerals(9)
tff(fact_4387_or__not__numerals_I8_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Ma))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Nb)))) = 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),Ma)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb))))) ) ).

% or_not_numerals(8)
tff(fact_4388_old_Orec__nat__def,axiom,
    ! [A: $tType,X3: A,Xa: fun(nat,fun(A,A)),Xb3: nat] : ( aa(nat,A,rec_nat(A,X3,Xa),Xb3) = the(A,rec_set_nat(A,X3,Xa,Xb3)) ) ).

% old.rec_nat_def
tff(fact_4389_integer__of__int__code,axiom,
    ! [K: int] :
      ( aa(int,code_integer,code_integer_of_int,K) = $ite(
          aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
          aa(code_integer,code_integer,uminus_uminus(code_integer),aa(int,code_integer,code_integer_of_int,aa(int,int,uminus_uminus(int),K))),
          $ite(
            K = zero_zero(int),
            zero_zero(code_integer),
            $let(
              l: code_integer,
              l:= aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),aa(int,code_integer,code_integer_of_int,divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2))))),
              $ite(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_4390_or__minus__numerals_I5_J,axiom,
    ! [Nb: 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(Nb)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(Nb)))) ) ).

% or_minus_numerals(5)
tff(fact_4391_or__minus__numerals_I1_J,axiom,
    ! [Nb: 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(Nb)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(Nb)))) ) ).

% or_minus_numerals(1)
tff(fact_4392_integer__of__int__eq__of__int,axiom,
    code_integer_of_int = ring_1_of_int(code_integer) ).

% integer_of_int_eq_of_int
tff(fact_4393_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_4394_or__nat__numerals_I4_J,axiom,
    ! [Xb: num] : ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb)) ) ).

% or_nat_numerals(4)
tff(fact_4395_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_4396_or__nat__numerals_I3_J,axiom,
    ! [Xb: num] : ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit0(Xb))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb)) ) ).

% or_nat_numerals(3)
tff(fact_4397_or__minus__numerals_I8_J,axiom,
    ! [Nb: num,Ma: 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,Nb)))),aa(num,int,numeral_numeral(int),Ma)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Ma,bit0(Nb)))) ) ).

% or_minus_numerals(8)
tff(fact_4398_or__minus__numerals_I4_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Nb)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Ma,bit0(Nb)))) ) ).

% or_minus_numerals(4)
tff(fact_4399_or__minus__numerals_I7_J,axiom,
    ! [Nb: num,Ma: 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(Nb)))),aa(num,int,numeral_numeral(int),Ma)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Ma,bitM(Nb)))) ) ).

% or_minus_numerals(7)
tff(fact_4400_or__minus__numerals_I3_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Nb)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Ma,bitM(Nb)))) ) ).

% or_minus_numerals(3)
tff(fact_4401_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_4402_uminus__integer__code_I1_J,axiom,
    aa(code_integer,code_integer,uminus_uminus(code_integer),zero_zero(code_integer)) = zero_zero(code_integer) ).

% uminus_integer_code(1)
tff(fact_4403_uminus__integer_Oabs__eq,axiom,
    ! [Xb: int] : ( aa(code_integer,code_integer,uminus_uminus(code_integer),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,uminus_uminus(int),Xb)) ) ).

% uminus_integer.abs_eq
tff(fact_4404_zero__integer__def,axiom,
    zero_zero(code_integer) = aa(int,code_integer,code_integer_of_int,zero_zero(int)) ).

% zero_integer_def
tff(fact_4405_less__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),aa(int,code_integer,code_integer_of_int,Xaa)),aa(int,code_integer,code_integer_of_int,Xb))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xaa),Xb) ) ).

% less_integer.abs_eq
tff(fact_4406_plus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(int,code_integer,code_integer_of_int,Xaa)),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xaa),Xb)) ) ).

% plus_integer.abs_eq
tff(fact_4407_one__integer__def,axiom,
    one_one(code_integer) = aa(int,code_integer,code_integer_of_int,one_one(int)) ).

% one_integer_def
tff(fact_4408_minus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(int,code_integer,code_integer_of_int,Xaa)),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xaa),Xb)) ) ).

% minus_integer.abs_eq
tff(fact_4409_set__bit__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : ( bit_se5668285175392031749et_bit(nat,Ma,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Nb),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Ma),one_one(nat))) ) ).

% set_bit_nat_def
tff(fact_4410_or__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Ma),Nb) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(nat,int,semiring_1_of_nat(int),Ma)),aa(nat,int,semiring_1_of_nat(int),Nb))) ) ).

% or_nat_def
tff(fact_4411_int__numeral__or__not__num__neg,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Ma,Nb))) ) ).

% int_numeral_or_not_num_neg
tff(fact_4412_int__numeral__not__or__num__neg,axiom,
    ! [Ma: num,Nb: 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),Ma))),aa(num,int,numeral_numeral(int),Nb)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Nb,Ma))) ) ).

% int_numeral_not_or_num_neg
tff(fact_4413_numeral__or__not__num__eq,axiom,
    ! [Ma: num,Nb: num] : ( aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Ma,Nb)) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb)))) ) ).

% numeral_or_not_num_eq
tff(fact_4414_floor__real__def,axiom,
    ! [Xb: real] : ( aa(real,int,archim6421214686448440834_floor(real),Xb) = the(int,aTP_Lamp_kd(real,fun(int,$o),Xb)) ) ).

% floor_real_def
tff(fact_4415_Suc__0__or__eq,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ) ).

% Suc_0_or_eq
tff(fact_4416_or__Suc__0__eq,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Nb),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ) ).

% or_Suc_0_eq
tff(fact_4417_or__nat__rec,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Ma),Nb) = aa(nat,nat,
          aa(nat,fun(nat,nat),plus_plus(nat),
            aa($o,nat,zero_neq_one_of_bool(nat),
              ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma)
              | ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ))),
          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),divide_divide(nat,Ma,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% or_nat_rec
tff(fact_4418_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),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),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),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ) ).

% or_int_unfold
tff(fact_4419_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_4420_rec__nat__0__imp,axiom,
    ! [A: $tType,F2: fun(nat,A),F1: A,F22: fun(nat,fun(A,A))] :
      ( ( F2 = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F2,zero_zero(nat)) = F1 ) ) ).

% rec_nat_0_imp
tff(fact_4421_rec__nat__Suc__imp,axiom,
    ! [A: $tType,F2: fun(nat,A),F1: A,F22: fun(nat,fun(A,A)),Nb: nat] :
      ( ( F2 = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F2,aa(nat,nat,suc,Nb)) = aa(A,A,aa(nat,fun(A,A),F22,Nb),aa(nat,A,F2,Nb)) ) ) ).

% rec_nat_Suc_imp
tff(fact_4422_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_4423_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_4424_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_4425_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_4426_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_4427_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_4428_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_4429_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_4430_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z) ) ) ) ).

% max_less_iff_conj
tff(fact_4431_of__bool__or__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o,Q: $o] :
          ( aa($o,A,zero_neq_one_of_bool(A),
              ( (P)
              | (Q) )) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q))) ) ) ).

% of_bool_or_iff
tff(fact_4432_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)),aa(num,A,numeral_numeral(A),V2),aa(num,A,numeral_numeral(A),U)) ) ) ).

% max_number_of(1)
tff(fact_4433_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : ( aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Xb)) = aa(num,A,numeral_numeral(A),Xb) ) ) ).

% max_0_1(3)
tff(fact_4434_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),Xb)),zero_zero(A)) = aa(num,A,numeral_numeral(A),Xb) ) ) ).

% max_0_1(4)
tff(fact_4435_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_4436_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_4437_max__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : ( aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),aa(num,A,numeral_numeral(A),Xb)) = aa(num,A,numeral_numeral(A),Xb) ) ) ).

% max_0_1(5)
tff(fact_4438_max__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),Xb)),one_one(A)) = aa(num,A,numeral_numeral(A),Xb) ) ) ).

% max_0_1(6)
tff(fact_4439_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = $ite(aa(A,$o,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),V2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ) ).

% max_number_of(4)
tff(fact_4440_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = $ite(aa(A,$o,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),V2)),aa(num,A,numeral_numeral(A),V2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ) ).

% max_number_of(3)
tff(fact_4441_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = $ite(aa(A,$o,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),V2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)),aa(num,A,numeral_numeral(A),U)) ) ) ).

% max_number_of(2)
tff(fact_4442_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X3: A,Xa: A] :
          ( aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa),Xa,X3) ) ) ).

% max_def_raw
tff(fact_4443_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xb)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Y) ) ) ) ).

% less_max_iff_disj
tff(fact_4444_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_4445_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_4446_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_4447_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_4448_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_4449_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_4450_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_4451_of__int__max,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int,Y: int] : ( aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_max(int),Xb),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(int,A,ring_1_of_int(A),Xb)),aa(int,A,ring_1_of_int(A),Y)) ) ) ).

% of_int_max
tff(fact_4452_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: nat,Y: nat] : ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Xb),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ) ).

% of_nat_max
tff(fact_4453_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: 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),Xb),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z)) ) ) ).

% max_diff_distrib_left
tff(fact_4454_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: A,Y: A,Z: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),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),Xb),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z)) ) ) ).

% max_add_distrib_right
tff(fact_4455_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: 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),Xb),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ) ).

% max_add_distrib_left
tff(fact_4456_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,D3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),D3)),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ) ).

% max.mono
tff(fact_4457_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_4458_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_4459_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_4460_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_4461_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_4462_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_4463_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_4464_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xb)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y) ) ) ) ).

% le_max_iff_disj
tff(fact_4465_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_4466_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_4467_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_4468_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_4469_floor__rat__def,axiom,
    ! [Xb: rat] : ( aa(rat,int,archim6421214686448440834_floor(rat),Xb) = the(int,aTP_Lamp_ke(rat,fun(int,$o),Xb)) ) ).

% floor_rat_def
tff(fact_4470_or__nat__unfold,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Ma),Nb) = $ite(
          Ma = zero_zero(nat),
          Nb,
          $ite(Nb = zero_zero(nat),Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),modulo_modulo(nat,Ma,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,Nb,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),divide_divide(nat,Ma,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ) ).

% or_nat_unfold
tff(fact_4471_integer__of__num_I3_J,axiom,
    ! [Nb: num] :
      ( code_integer_of_num(aa(num,num,bit1,Nb)) = $let(
          k: code_integer,
          k:= code_integer_of_num(Nb),
          aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),k),k)),one_one(code_integer)) ) ) ).

% integer_of_num(3)
tff(fact_4472_nat_Osplit__sels_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o),F1: A,F22: fun(nat,A),Nat: nat] :
      ( aa(A,$o,P,case_nat(A,F1,F22,Nat))
    <=> ( ( ( Nat = zero_zero(nat) )
         => aa(A,$o,P,F1) )
        & ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
         => aa(A,$o,P,aa(nat,A,F22,pred(Nat))) ) ) ) ).

% nat.split_sels(1)
tff(fact_4473_max__Suc__Suc,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Ma)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Ma),Nb)) ) ).

% max_Suc_Suc
tff(fact_4474_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_4475_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_4476_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_4477_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_4478_max__0L,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),Nb) = Nb ) ).

% max_0L
tff(fact_4479_max__0R,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),zero_zero(nat)) = Nb ) ).

% max_0R
tff(fact_4480_max__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),pred_numeral(K)),Nb)) ) ).

% max_numeral_Suc
tff(fact_4481_max__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),pred_numeral(K))) ) ).

% max_Suc_numeral
tff(fact_4482_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_4483_nat__add__max__right,axiom,
    ! [Ma: nat,Nb: nat,Q4: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),Q4)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Q4)) ) ).

% nat_add_max_right
tff(fact_4484_nat__add__max__left,axiom,
    ! [Ma: nat,Nb: nat,Q4: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Ma),Nb)),Q4) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Q4)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Q4)) ) ).

% nat_add_max_left
tff(fact_4485_nat__mult__max__left,axiom,
    ! [Ma: nat,Nb: nat,Q4: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Ma),Nb)),Q4) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Q4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q4)) ) ).

% nat_mult_max_left
tff(fact_4486_nat__mult__max__right,axiom,
    ! [Ma: nat,Nb: nat,Q4: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),Q4)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Q4)) ) ).

% nat_mult_max_right
tff(fact_4487_sgn__rat__def,axiom,
    ! [A2: rat] :
      ( aa(rat,rat,sgn_sgn(rat),A2) = $ite(
          A2 = zero_zero(rat),
          zero_zero(rat),
          $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),A2),one_one(rat),aa(rat,rat,uminus_uminus(rat),one_one(rat))) ) ) ).

% sgn_rat_def
tff(fact_4488_nat__minus__add__max,axiom,
    ! [Nb: nat,Ma: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma)),Ma) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),Ma) ) ).

% nat_minus_add_max
tff(fact_4489_max__Suc2,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Ma),aa(nat,nat,suc,Nb)) = case_nat(nat,aa(nat,nat,suc,Nb),aTP_Lamp_kf(nat,fun(nat,nat),Nb),Ma) ) ).

% max_Suc2
tff(fact_4490_max__Suc1,axiom,
    ! [Nb: nat,Ma: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Nb)),Ma) = case_nat(nat,aa(nat,nat,suc,Nb),aTP_Lamp_kg(nat,fun(nat,nat),Nb),Ma) ) ).

% max_Suc1
tff(fact_4491_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one2) = one_one(code_integer) ).

% integer_of_num_triv(1)
tff(fact_4492_pred__def,axiom,
    ! [Nat: nat] : ( pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_az(nat,nat),Nat) ) ).

% pred_def
tff(fact_4493_nat_Osplit__sels_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),F1: A,F22: fun(nat,A),Nat: nat] :
      ( aa(A,$o,P,case_nat(A,F1,F22,Nat))
    <=> ~ ( ( ( Nat = zero_zero(nat) )
            & ~ aa(A,$o,P,F1) )
          | ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
            & ~ aa(A,$o,P,aa(nat,A,F22,pred(Nat))) ) ) ) ).

% nat.split_sels(2)
tff(fact_4494_rat__inverse__code,axiom,
    ! [P3: rat] : ( quotient_of(aa(rat,rat,inverse_inverse(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_kh(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ) ).

% rat_inverse_code
tff(fact_4495_normalize__negative,axiom,
    ! [Q4: int,P3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Q4),zero_zero(int))
     => ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),Q4)) = normalize(aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,uminus_uminus(int),P3)),aa(int,int,uminus_uminus(int),Q4))) ) ) ).

% normalize_negative
tff(fact_4496_bezw__0,axiom,
    ! [Xb: nat] : ( bezw(Xb,zero_zero(nat)) = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ) ).

% bezw_0
tff(fact_4497_prod__decode__aux_Osimps,axiom,
    ! [K: nat,Ma: nat] :
      ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K),Ma) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),K),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Ma),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),Ma)),aa(nat,product_prod(nat,nat),nat_prod_decode_aux(aa(nat,nat,suc,K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),aa(nat,nat,suc,K)))) ) ).

% prod_decode_aux.simps
tff(fact_4498_quotient__of__number_I3_J,axiom,
    ! [K: num] : ( quotient_of(aa(num,rat,numeral_numeral(rat),K)) = aa(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_4499_normalize__denom__zero,axiom,
    ! [P3: int] : ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),zero_zero(int))) = aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)) ) ).

% normalize_denom_zero
tff(fact_4500_rat__one__code,axiom,
    quotient_of(one_one(rat)) = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int)) ).

% rat_one_code
tff(fact_4501_rat__zero__code,axiom,
    quotient_of(zero_zero(rat)) = aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)) ).

% rat_zero_code
tff(fact_4502_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),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_4503_quotient__of__number_I4_J,axiom,
    quotient_of(aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) ).

% quotient_of_number(4)
tff(fact_4504_divide__rat__def,axiom,
    ! [Q4: rat,R2: rat] : ( divide_divide(rat,Q4,R2) = aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),Q4),aa(rat,rat,inverse_inverse(rat),R2)) ) ).

% divide_rat_def
tff(fact_4505_diff__rat__def,axiom,
    ! [Q4: rat,R2: rat] : ( aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Q4),R2) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),Q4),aa(rat,rat,uminus_uminus(rat),R2)) ) ).

% diff_rat_def
tff(fact_4506_quotient__of__div,axiom,
    ! [R2: rat,Nb: int,D3: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),product_Pair(int,int,Nb),D3) )
     => ( R2 = divide_divide(rat,aa(int,rat,ring_1_of_int(rat),Nb),aa(int,rat,ring_1_of_int(rat),D3)) ) ) ).

% quotient_of_div
tff(fact_4507_rat__plus__code,axiom,
    ! [P3: rat,Q4: rat] : ( quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P3),Q4)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_kj(rat,fun(int,fun(int,product_prod(int,int))),Q4)),quotient_of(P3)) ) ).

% rat_plus_code
tff(fact_4508_rat__minus__code,axiom,
    ! [P3: rat,Q4: rat] : ( quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P3),Q4)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_kl(rat,fun(int,fun(int,product_prod(int,int))),Q4)),quotient_of(P3)) ) ).

% rat_minus_code
tff(fact_4509_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q4: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),product_Pair(int,int,P3),Q4) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q4) ) ).

% quotient_of_denom_pos
tff(fact_4510_rat__uminus__code,axiom,
    ! [P3: rat] : ( quotient_of(aa(rat,rat,uminus_uminus(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_km(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ) ).

% rat_uminus_code
tff(fact_4511_normalize__denom__pos,axiom,
    ! [R2: product_prod(int,int),P3: int,Q4: int] :
      ( ( normalize(R2) = aa(int,product_prod(int,int),product_Pair(int,int,P3),Q4) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q4) ) ).

% normalize_denom_pos
tff(fact_4512_normalize__crossproduct,axiom,
    ! [Q4: int,S: int,P3: int,R2: int] :
      ( ( Q4 != zero_zero(int) )
     => ( ( S != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),Q4)) = normalize(aa(int,product_prod(int,int),product_Pair(int,int,R2),S)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P3),S) = aa(int,int,aa(int,fun(int,int),times_times(int),R2),Q4) ) ) ) ) ).

% normalize_crossproduct
tff(fact_4513_rat__less__code,axiom,
    ! [P3: rat,Q4: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),P3),Q4)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aTP_Lamp_ko(rat,fun(int,fun(int,$o)),Q4)),quotient_of(P3)) ) ).

% rat_less_code
tff(fact_4514_prod__decode__aux_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod(nat,nat)] :
      ( ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(Xb),Xaa) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xaa),Xb),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xaa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xb),Xaa)),aa(nat,product_prod(nat,nat),nat_prod_decode_aux(aa(nat,nat,suc,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xaa),aa(nat,nat,suc,Xb)))) ) ) ).

% prod_decode_aux.elims
tff(fact_4515_quotient__of__int,axiom,
    ! [A2: int] : ( quotient_of(aa(int,rat,of_int,A2)) = aa(int,product_prod(int,int),product_Pair(int,int,A2),one_one(int)) ) ).

% quotient_of_int
tff(fact_4516_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : ( aa(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,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_4517_finite__enumerate,axiom,
    ! [S3: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),S3)
     => ? [R4: fun(nat,nat)] :
          ( strict_mono_on(nat,nat,R4,set_ord_lessThan(nat,aa(set(nat),nat,finite_card(nat),S3)))
          & ! [N3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N3),aa(set(nat),nat,finite_card(nat),S3))
             => member(nat,aa(nat,nat,R4,N3),S3) ) ) ) ).

% finite_enumerate
tff(fact_4518_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      ( code_bit_cut_integer(K) = $ite(K = zero_zero(code_integer),aa($o,product_prod(code_integer,$o),product_Pair(code_integer,$o,zero_zero(code_integer)),$false),aa(product_prod(code_integer,code_integer),product_prod(code_integer,$o),product_case_prod(code_integer,code_integer,product_prod(code_integer,$o),aTP_Lamp_kp(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ) ).

% bit_cut_integer_code
tff(fact_4519_drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : ( aa(A,A,bit_se4197421643247451524op_bit(A,Nb),zero_zero(A)) = zero_zero(A) ) ) ).

% drop_bit_of_0
tff(fact_4520_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,Nb: nat,A2: A] : ( aa(A,A,bit_se4197421643247451524op_bit(A,Ma),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2)) = aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)),A2) ) ) ).

% drop_bit_drop_bit
tff(fact_4521_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,B2: $o] :
          ( aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa($o,A,zero_neq_one_of_bool(A),(B2))) = aa($o,A,zero_neq_one_of_bool(A),
              ( ( Nb = zero_zero(nat) )
              & (B2) )) ) ) ).

% drop_bit_of_bool
tff(fact_4522_drop__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se4197421643247451524op_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_4523_drop__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se4197421643247451524op_bit(int,Nb),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_4524_drop__bit__minus__one,axiom,
    ! [Nb: nat] : ( aa(int,int,bit_se4197421643247451524op_bit(int,Nb),aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),one_one(int)) ) ).

% drop_bit_minus_one
tff(fact_4525_drop__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : ( aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa(num,A,numeral_numeral(A),K)) ) ) ).

% drop_bit_Suc_bit0
tff(fact_4526_drop__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : ( aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa(num,A,numeral_numeral(A),K)) ) ) ).

% drop_bit_Suc_bit1
tff(fact_4527_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : ( aa(A,A,bit_se4197421643247451524op_bit(A,Nb),one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ) ).

% drop_bit_of_1
tff(fact_4528_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : ( aa(int,int,bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,Nb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,bit_se4197421643247451524op_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ) ).

% drop_bit_Suc_minus_bit0
tff(fact_4529_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : ( aa(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,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_4530_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : ( aa(int,int,bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,Nb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,bit_se4197421643247451524op_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ) ).

% drop_bit_Suc_minus_bit1
tff(fact_4531_of__nat__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Ma: nat,Nb: nat] : ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4197421643247451524op_bit(nat,Ma),Nb)) = aa(A,A,bit_se4197421643247451524op_bit(A,Ma),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ).

% of_nat_drop_bit
tff(fact_4532_drop__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,Ma: nat] : ( aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa(nat,A,semiring_1_of_nat(A),Ma)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4197421643247451524op_bit(nat,Nb),Ma)) ) ) ).

% drop_bit_of_nat
tff(fact_4533_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( ( bit_se2584673776208193580ke_bit(A,Nb,A2) = A2 )
        <=> ( aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_4534_drop__bit__push__bit__int,axiom,
    ! [Ma: nat,Nb: nat,K: int] : ( aa(int,int,bit_se4197421643247451524op_bit(int,Ma),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K)) = aa(int,int,bit_se4197421643247451524op_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb)),aa(int,int,bit_se4730199178511100633sh_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma)),K)) ) ).

% drop_bit_push_bit_int
tff(fact_4535_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,Nb: nat,A2: A] : ( bit_se2584673776208193580ke_bit(A,Ma,aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2)) = aa(A,A,bit_se4197421643247451524op_bit(A,Nb),bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb),A2)) ) ) ).

% take_bit_drop_bit
tff(fact_4536_drop__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,Nb: nat,A2: A] : ( aa(A,A,bit_se4197421643247451524op_bit(A,Ma),bit_se2584673776208193580ke_bit(A,Nb,A2)) = bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma),aa(A,A,bit_se4197421643247451524op_bit(A,Ma),A2)) ) ) ).

% drop_bit_take_bit
tff(fact_4537_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] : ( divide_divide(A,A2,aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),one_one(A))) = aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2) ) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_4538_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_4539_bits__ident,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2))),bit_se2584673776208193580ke_bit(A,Nb,A2)) = A2 ) ) ).

% bits_ident
tff(fact_4540_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : ( aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Nb)),A2) = aa(A,A,bit_se4197421643247451524op_bit(A,Nb),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% drop_bit_Suc
tff(fact_4541_Rat_Oof__int__def,axiom,
    of_int = ring_1_of_int(rat) ).

% Rat.of_int_def
tff(fact_4542_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,Ma: nat,A2: A] : ( aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),bit_se2584673776208193580ke_bit(A,Ma,aa(A,A,bit_se4197421643247451524op_bit(A,Nb),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),bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb)))) ) ) ).

% slice_eq_mask
tff(fact_4543_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2) = $ite(Nb = zero_zero(nat),A2,aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ) ).

% drop_bit_rec
tff(fact_4544_Frct__code__post_I5_J,axiom,
    ! [K: num] : ( frct(aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),aa(num,int,numeral_numeral(int),K))) = divide_divide(rat,one_one(rat),aa(num,rat,numeral_numeral(rat),K)) ) ).

% Frct_code_post(5)
tff(fact_4545_strict__mono__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F2: fun(A,B),A3: set(A),R2: A,S: A] :
          ( strict_mono_on(A,B,F2,A3)
         => ( member(A,R2,A3)
           => ( member(A,S,A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R2),S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R2)),aa(A,B,F2,S)) ) ) ) ) ) ).

% strict_mono_onD
tff(fact_4546_strict__mono__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [R4: A,S2: A] :
              ( member(A,R4,A3)
             => ( member(A,S2,A3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R4),S2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R4)),aa(A,B,F2,S2)) ) ) )
         => strict_mono_on(A,B,F2,A3) ) ) ).

% strict_mono_onI
tff(fact_4547_strict__mono__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( strict_mono_on(A,B,F2,A3)
        <=> ! [R5: A,S5: A] :
              ( ( member(A,R5,A3)
                & member(A,S5,A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),R5),S5) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R5)),aa(A,B,F2,S5)) ) ) ) ).

% strict_mono_on_def
tff(fact_4548_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : ( aa(nat,nat,bit_se4197421643247451524op_bit(nat,Nb),aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),Nb = zero_zero(nat)) ) ).

% drop_bit_of_Suc_0
tff(fact_4549_Frct__code__post_I9_J,axiom,
    ! [Q4: product_prod(int,int)] : ( aa(rat,rat,uminus_uminus(rat),aa(rat,rat,uminus_uminus(rat),frct(Q4))) = frct(Q4) ) ).

% Frct_code_post(9)
tff(fact_4550_drop__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : ( aa(nat,nat,bit_se4197421643247451524op_bit(nat,Nb),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,bit_se4197421643247451524op_bit(int,Nb),K)) ) ).

% drop_bit_nat_eq
tff(fact_4551_Frct__code__post_I2_J,axiom,
    ! [A2: int] : ( frct(aa(int,product_prod(int,int),product_Pair(int,int,A2),zero_zero(int))) = zero_zero(rat) ) ).

% Frct_code_post(2)
tff(fact_4552_Frct__code__post_I1_J,axiom,
    ! [A2: int] : ( frct(aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),A2)) = zero_zero(rat) ) ).

% Frct_code_post(1)
tff(fact_4553_Frct__code__post_I8_J,axiom,
    ! [A2: int,B2: int] : ( frct(aa(int,product_prod(int,int),product_Pair(int,int,A2),aa(int,int,uminus_uminus(int),B2))) = aa(rat,rat,uminus_uminus(rat),frct(aa(int,product_prod(int,int),product_Pair(int,int,A2),B2))) ) ).

% Frct_code_post(8)
tff(fact_4554_Frct__code__post_I7_J,axiom,
    ! [A2: int,B2: int] : ( frct(aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,uminus_uminus(int),A2)),B2)) = aa(rat,rat,uminus_uminus(rat),frct(aa(int,product_prod(int,int),product_Pair(int,int,A2),B2))) ) ).

% Frct_code_post(7)
tff(fact_4555_Frct__code__post_I3_J,axiom,
    frct(aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int))) = one_one(rat) ).

% Frct_code_post(3)
tff(fact_4556_Frct__code__post_I4_J,axiom,
    ! [K: num] : ( frct(aa(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_4557_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      ( code_divmod_integer(K,L) = $ite(
          K = zero_zero(code_integer),
          aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,zero_zero(code_integer)),zero_zero(code_integer)),
          $ite(
            aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),L),
            $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),K),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_kq(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L))),
            $ite(
              L = zero_zero(code_integer),
              aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,zero_zero(code_integer)),K),
              aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_apsnd(code_integer,code_integer,code_integer),uminus_uminus(code_integer)),
                $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_kr(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ) ).

% divmod_integer_code
tff(fact_4558_nth__rotate1,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,rotate1(A,Xs)),Nb) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,suc,Nb),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate1
tff(fact_4559_set__remove1__eq,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( aa(list(A),set(A),set2(A),remove1(A,Xb,Xs)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))) ) ) ).

% set_remove1_eq
tff(fact_4560_nth__enumerate__eq,axiom,
    ! [A: $tType,Ma: nat,Xs: list(A),Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,Nb,Xs)),Ma) = aa(A,product_prod(nat,A),product_Pair(nat,A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Ma)),aa(nat,A,nth(A,Xs),Ma)) ) ) ).

% nth_enumerate_eq
tff(fact_4561_in__set__remove1,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( ( A2 != B2 )
     => ( member(A,A2,aa(list(A),set(A),set2(A),remove1(A,B2,Xs)))
      <=> member(A,A2,aa(list(A),set(A),set2(A),Xs)) ) ) ).

% in_set_remove1
tff(fact_4562_set__rotate1,axiom,
    ! [A: $tType,Xs: list(A)] : ( aa(list(A),set(A),set2(A),rotate1(A,Xs)) = aa(list(A),set(A),set2(A),Xs) ) ).

% set_rotate1
tff(fact_4563_rotate1__length01,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => ( rotate1(A,Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_4564_notin__set__remove1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ~ member(A,Xb,aa(list(A),set(A),set2(A),remove1(A,Y,Xs))) ) ).

% notin_set_remove1
tff(fact_4565_remove1__idem,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( remove1(A,Xb,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_4566_set__remove1__subset,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),remove1(A,Xb,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_remove1_subset
tff(fact_4567_length__remove1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( aa(list(A),nat,size_size(list(A)),remove1(A,Xb,Xs)) = $ite(member(A,Xb,aa(list(A),set(A),set2(A),Xs)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_remove1
tff(fact_4568_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      ( code_divmod_integer(K,L) = $ite(
          K = zero_zero(code_integer),
          aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,zero_zero(code_integer)),zero_zero(code_integer)),
          $ite(
            L = zero_zero(code_integer),
            aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,zero_zero(code_integer)),K),
            aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(fun(code_integer,code_integer),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,aa(fun(code_integer,fun(code_integer,code_integer)),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,product_apsnd(code_integer,code_integer,code_integer)),times_times(code_integer))),sgn_sgn(code_integer)),L),
              $ite(aa(code_integer,code_integer,sgn_sgn(code_integer),K) = aa(code_integer,code_integer,sgn_sgn(code_integer),L),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_ks(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_4569_xor__minus__numerals_I1_J,axiom,
    ! [Nb: num,K: int] : ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))),K) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),neg_numeral_sub(int,Nb,one2)),K)) ) ).

% xor_minus_numerals(1)
tff(fact_4570_xor__minus__numerals_I2_J,axiom,
    ! [K: int,Nb: num] : ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),neg_numeral_sub(int,Nb,one2))) ) ).

% xor_minus_numerals(2)
tff(fact_4571_div__add__self2__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [Xb: A,B2: B,A2: B] :
          ( nO_MATCH(A,B,Xb,B2)
         => ( ( B2 != zero_zero(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),divide_divide(B,A2,B2)),one_one(B)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_4572_sub__num__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_sub(A,one2,one2) = zero_zero(A) ) ) ).

% sub_num_simps(1)
tff(fact_4573_diff__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Ma: num,Nb: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),Ma)),aa(num,A,numeral_numeral(A),Nb)) = neg_numeral_sub(A,Ma,Nb) ) ) ).

% diff_numeral_simps(1)
tff(fact_4574_sub__num__simps_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : ( neg_numeral_sub(A,bit0(K),bit0(L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K,L)) ) ) ).

% sub_num_simps(6)
tff(fact_4575_sub__num__simps_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : ( neg_numeral_sub(A,aa(num,num,bit1,K),aa(num,num,bit1,L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K,L)) ) ) ).

% sub_num_simps(9)
tff(fact_4576_add__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Ma: num,Nb: num] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),aa(num,A,numeral_numeral(A),Nb)) = neg_numeral_sub(A,Nb,Ma) ) ) ).

% add_neg_numeral_simps(2)
tff(fact_4577_add__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Ma: num,Nb: num] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Ma)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = neg_numeral_sub(A,Ma,Nb) ) ) ).

% add_neg_numeral_simps(1)
tff(fact_4578_semiring__norm_I167_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V2: 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),V2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W)),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,W,V2)),Y) ) ) ).

% semiring_norm(167)
tff(fact_4579_semiring__norm_I166_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V2: num,W: num,Y: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),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),neg_numeral_sub(A,V2,W)),Y) ) ) ).

% semiring_norm(166)
tff(fact_4580_diff__numeral__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Ma: num,Nb: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = neg_numeral_sub(A,Nb,Ma) ) ) ).

% diff_numeral_simps(4)
tff(fact_4581_sub__num__simps_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : ( neg_numeral_sub(A,bit0(K),aa(num,num,bit1,L)) = neg_numeral_dbl_dec(A,neg_numeral_sub(A,K,L)) ) ) ).

% sub_num_simps(7)
tff(fact_4582_sub__num__simps_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : ( neg_numeral_sub(A,aa(num,num,bit1,K),bit0(L)) = neg_numeral_dbl_inc(A,neg_numeral_sub(A,K,L)) ) ) ).

% sub_num_simps(8)
tff(fact_4583_diff__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb)) = neg_numeral_sub(A,one2,Nb) ) ) ).

% diff_numeral_special(1)
tff(fact_4584_diff__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Ma: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),Ma)),one_one(A)) = neg_numeral_sub(A,Ma,one2) ) ) ).

% diff_numeral_special(2)
tff(fact_4585_sub__num__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : ( neg_numeral_sub(A,aa(num,num,bit1,K),one2) = aa(num,A,numeral_numeral(A),bit0(K)) ) ) ).

% sub_num_simps(5)
tff(fact_4586_div__add__self1__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [Xb: A,B2: B,A2: B] :
          ( nO_MATCH(A,B,Xb,B2)
         => ( ( B2 != zero_zero(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),divide_divide(B,A2,B2)),one_one(B)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_4587_not__minus__numeral__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : ( aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = neg_numeral_sub(A,Nb,one2) ) ) ).

% not_minus_numeral_eq
tff(fact_4588_sub__num__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : ( neg_numeral_sub(A,bit0(K),one2) = aa(num,A,numeral_numeral(A),bitM(K)) ) ) ).

% sub_num_simps(4)
tff(fact_4589_add__neg__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),Nb)) = neg_numeral_sub(A,Nb,one2) ) ) ).

% add_neg_numeral_special(4)
tff(fact_4590_add__neg__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Ma: num] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Ma)),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,Ma,one2) ) ) ).

% add_neg_numeral_special(3)
tff(fact_4591_add__neg__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Ma: num] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),one_one(A)) = neg_numeral_sub(A,one2,Ma) ) ) ).

% add_neg_numeral_special(2)
tff(fact_4592_add__neg__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Ma: num] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))) = neg_numeral_sub(A,one2,Ma) ) ) ).

% add_neg_numeral_special(1)
tff(fact_4593_diff__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Ma: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma))),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,one2,Ma) ) ) ).

% diff_numeral_special(8)
tff(fact_4594_diff__numeral__special_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = neg_numeral_sub(A,Nb,one2) ) ) ).

% diff_numeral_special(7)
tff(fact_4595_minus__sub__one__diff__one,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Ma: num] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),neg_numeral_sub(A,Ma,one2))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ma)) ) ) ).

% minus_sub_one_diff_one
tff(fact_4596_sub__num__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [L: num] : ( neg_numeral_sub(A,one2,aa(num,num,bit1,L)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(L))) ) ) ).

% sub_num_simps(3)
tff(fact_4597_sub__num__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [L: num] : ( neg_numeral_sub(A,one2,bit0(L)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bitM(L))) ) ) ).

% sub_num_simps(2)
tff(fact_4598_prod__inversef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),A3: set(B)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),comp(A,A,B,inverse_inverse(A)),F2)),A3) = aa(A,A,inverse_inverse(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3)) ) ) ).

% prod_inversef
tff(fact_4599_neg__numeral__class_Osub__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : ( neg_numeral_sub(A,K,L) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) ) ) ).

% neg_numeral_class.sub_def
tff(fact_4600_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,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(C,B),fun(C,A),comp(B,A,C,H),G)),A3) = aa(B,A,H,aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),A3)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_4601_sub__non__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Ma: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),neg_numeral_sub(A,Nb,Ma)),zero_zero(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Nb),Ma) ) ) ).

% sub_non_positive
tff(fact_4602_sub__non__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Ma: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),neg_numeral_sub(A,Nb,Ma))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Ma),Nb) ) ) ).

% sub_non_negative
tff(fact_4603_sub__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Ma: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),neg_numeral_sub(A,Nb,Ma)),zero_zero(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Nb),Ma) ) ) ).

% sub_negative
tff(fact_4604_sub__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Ma: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),neg_numeral_sub(A,Nb,Ma))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Ma),Nb) ) ) ).

% sub_positive
tff(fact_4605_sub__inc__One__eq,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : ( neg_numeral_sub(A,inc(Nb),one2) = aa(num,A,numeral_numeral(A),Nb) ) ) ).

% sub_inc_One_eq
tff(fact_4606_scale__right__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,A2: real] :
          ( nO_MATCH(A,real,divide_divide(A,Xb,Y),A2)
         => ( real_V8093663219630862766scaleR(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,A2,Y)) ) ) ) ).

% scale_right_distrib_NO_MATCH
tff(fact_4607_scale__right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,A2: real] :
          ( nO_MATCH(A,real,divide_divide(A,Xb,Y),A2)
         => ( real_V8093663219630862766scaleR(A,A2,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,A2,Y)) ) ) ) ).

% scale_right_diff_distrib_NO_MATCH
tff(fact_4608_distrib__right__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xb: A,Y: A,C2: B,A2: B,B2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,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_4609_distrib__left__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xb: A,Y: A,A2: B,B2: B,C2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,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_4610_right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xb: A,Y: A,A2: B,B2: B,C2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,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_4611_left__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xb: A,Y: A,C2: B,A2: B,B2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,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_4612_minus__numeral__eq__not__sub__one,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) = aa(A,A,bit_ri4277139882892585799ns_not(A),neg_numeral_sub(A,Nb,one2)) ) ) ).

% minus_numeral_eq_not_sub_one
tff(fact_4613_power__minus_H,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A,Nb: nat] :
          ( nO_MATCH(A,A,one_one(A),Xb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xb)),Nb) = 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))),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)) ) ) ) ).

% power_minus'
tff(fact_4614_sub__BitM__One__eq,axiom,
    ! [Nb: num] : ( neg_numeral_sub(int,bitM(Nb),one2) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),neg_numeral_sub(int,Nb,one2)) ) ).

% sub_BitM_One_eq
tff(fact_4615_power__int__minus__left__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( division_ring(C)
        & one(A)
        & uminus(A) )
     => ! [Xb: B,A2: C,Nb: int] :
          ( nO_MATCH(A,B,aa(A,A,uminus_uminus(A),one_one(A)),Xb)
         => ( power_int(C,aa(C,C,uminus_uminus(C),A2),Nb) = aa(C,C,aa(C,fun(C,C),times_times(C),power_int(C,aa(C,C,uminus_uminus(C),one_one(C)),Nb)),power_int(C,A2,Nb)) ) ) ) ).

% power_int_minus_left_distrib
tff(fact_4616_scale__left__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,C2: B,A2: real,B2: real] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,Y),C2)
         => ( real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,B2,Xb)) ) ) ) ).

% scale_left_distrib_NO_MATCH
tff(fact_4617_scale__left__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,C2: B,A2: real,B2: real] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,Y),C2)
         => ( real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2),Xb) = aa(A,A,aa(A,fun(A,A),minus_minus(A),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,B2,Xb)) ) ) ) ).

% scale_left_diff_distrib_NO_MATCH
tff(fact_4618_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B)] : ( aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_kt(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_funpow
tff(fact_4619_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),zero_zero(code_integer))
     => ( aa(code_integer,nat,code_nat_of_integer,K) = zero_zero(nat) ) ) ).

% nat_of_integer_non_positive
tff(fact_4620_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_ab(nat,fun(nat,$o)),aTP_Lamp_aa(nat,fun(nat,$o))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_4621_Suc__funpow,axiom,
    ! [Nb: nat] : ( aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),compow(fun(nat,nat)),Nb),suc) = aa(nat,fun(nat,nat),plus_plus(nat),Nb) ) ).

% Suc_funpow
tff(fact_4622_funpow__0,axiom,
    ! [A: $tType,F2: fun(A,A),Xb: A] : ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F2),Xb) = Xb ) ).

% funpow_0
tff(fact_4623_card_Ocomp__fun__commute__on,axiom,
    aa(fun(nat,nat),fun(nat,nat),comp(nat,nat,nat,suc),suc) = aa(fun(nat,nat),fun(nat,nat),comp(nat,nat,nat,suc),suc) ).

% card.comp_fun_commute_on
tff(fact_4624_comp__funpow,axiom,
    ! [A: $tType,B: $tType,Nb: nat,F2: fun(B,B)] : ( aa(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B)),aa(nat,fun(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B))),compow(fun(fun(A,B),fun(A,B))),Nb),comp(B,B,A,F2)) = comp(B,B,A,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),Nb),F2)) ) ).

% comp_funpow
tff(fact_4625_bij__betw__funpow,axiom,
    ! [A: $tType,F2: fun(A,A),S3: set(A),Nb: nat] :
      ( bij_betw(A,A,F2,S3,S3)
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),S3,S3) ) ).

% bij_betw_funpow
tff(fact_4626_funpow__swap1,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat,Xb: A] : ( aa(A,A,F2,aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),Xb)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),aa(A,A,F2,Xb)) ) ).

% funpow_swap1
tff(fact_4627_semilattice__neutr__order_Oneutr__eq__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A] :
      ( semila1105856199041335345_order(A,F2,Z,Less_eq,Less)
     => ( ( Z = aa(A,A,aa(A,fun(A,A),F2,A2),B2) )
      <=> ( ( A2 = Z )
          & ( B2 = Z ) ) ) ) ).

% semilattice_neutr_order.neutr_eq_iff
tff(fact_4628_semilattice__neutr__order_Oeq__neutr__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A] :
      ( semila1105856199041335345_order(A,F2,Z,Less_eq,Less)
     => ( ( aa(A,A,aa(A,fun(A,A),F2,A2),B2) = Z )
      <=> ( ( A2 = Z )
          & ( B2 = Z ) ) ) ) ).

% semilattice_neutr_order.eq_neutr_iff
tff(fact_4629_funpow__mult,axiom,
    ! [A: $tType,Nb: nat,Ma: nat,F2: fun(A,A)] : ( aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Ma),F2)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb)),F2) ) ).

% funpow_mult
tff(fact_4630_funpow__Suc__right,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,A)] : ( aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Nb)),F2) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)),F2) ) ).

% funpow_Suc_right
tff(fact_4631_funpow_Osimps_I2_J,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,A)] : ( aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Nb)),F2) = aa(fun(A,A),fun(A,A),comp(A,A,A,F2),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)) ) ).

% funpow.simps(2)
tff(fact_4632_funpow__add,axiom,
    ! [A: $tType,Ma: nat,Nb: nat,F2: fun(A,A)] : ( aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)),F2) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Ma),F2)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)) ) ).

% funpow_add
tff(fact_4633_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_4634_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_4635_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Ma: nat,K: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% sum.atLeastAtMost_shift_bounds
tff(fact_4636_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Ma: nat,K: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% sum.atLeastLessThan_shift_bounds
tff(fact_4637_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_4638_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_4639_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Ma: nat,K: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_4640_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Ma: nat,K: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_4641_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : ( bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2)) = aa(fun(nat,nat),fun(nat,$o),comp(nat,$o,nat,bit_se5641148757651400278ts_bit(A,A2)),aa(nat,fun(nat,nat),plus_plus(nat),Nb)) ) ) ).

% bit_drop_bit_eq
tff(fact_4642_summable__inverse__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),comp(A,A,nat,inverse_inverse(A)),F2))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_ku(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_inverse_divide
tff(fact_4643_nat__of__integer__code__post_I1_J,axiom,
    aa(code_integer,nat,code_nat_of_integer,zero_zero(code_integer)) = zero_zero(nat) ).

% nat_of_integer_code_post(1)
tff(fact_4644_nat__of__integer_Oabs__eq,axiom,
    ! [Xb: int] : ( aa(code_integer,nat,code_nat_of_integer,aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,nat,nat2,Xb) ) ).

% nat_of_integer.abs_eq
tff(fact_4645_of__nat__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] : ( aa(nat,A,semiring_1_of_nat(A),Nb) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ) ).

% of_nat_def
tff(fact_4646_numeral__add__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num,A2: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),K)),A2) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K)),aa(A,fun(A,A),plus_plus(A),one_one(A))),A2) ) ) ).

% numeral_add_unfold_funpow
tff(fact_4647_nat__of__integer__code__post_I2_J,axiom,
    aa(code_integer,nat,code_nat_of_integer,one_one(code_integer)) = one_one(nat) ).

% nat_of_integer_code_post(2)
tff(fact_4648_numeral__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: num] : ( aa(num,A,numeral_numeral(A),K) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K)),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ) ).

% numeral_unfold_funpow
tff(fact_4649_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_4650_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_4651_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_4652_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_4653_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Ma,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),Ma))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma))) ) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_4654_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Ma,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),Ma))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma))) ) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_4655_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_kv(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_4656_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_kv(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_4657_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_kv(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_4658_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_kv(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_4659_sum_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),Ma: nat,Nb: nat] : ( aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),Ma),aa(nat,int,semiring_1_of_nat(int),Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% sum.atLeast_int_atMost_int_shift
tff(fact_4660_prod_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),Ma: nat,Nb: nat] : ( aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7121269368397514597t_prod(int,A),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),Ma),aa(nat,int,semiring_1_of_nat(int),Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% prod.atLeast_int_atMost_int_shift
tff(fact_4661_sum_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),Ma: nat,Nb: nat] : ( aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),Ma),aa(nat,int,semiring_1_of_nat(int),Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% sum.atLeast_int_lessThan_int_shift
tff(fact_4662_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Ma: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),Ma))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_4663_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Ma: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),Ma))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_4664_prod_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),Ma: nat,Nb: nat] : ( aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7121269368397514597t_prod(int,A),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),Ma),aa(nat,int,semiring_1_of_nat(int),Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% prod.atLeast_int_lessThan_int_shift
tff(fact_4665_Code__Numeral_Onegative__def,axiom,
    code_negative = aa(fun(num,code_integer),fun(num,code_integer),comp(code_integer,code_integer,num,uminus_uminus(code_integer)),numeral_numeral(code_integer)) ).

% Code_Numeral.negative_def
tff(fact_4666_Code__Target__Int_Onegative__def,axiom,
    code_Target_negative = aa(fun(num,int),fun(num,int),comp(int,int,num,uminus_uminus(int)),numeral_numeral(int)) ).

% Code_Target_Int.negative_def
tff(fact_4667_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      ( aa(code_integer,nat,code_nat_of_integer,K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),zero_zero(code_integer)),zero_zero(nat),aa(product_prod(code_integer,code_integer),nat,product_case_prod(code_integer,code_integer,nat,aTP_Lamp_kw(code_integer,fun(code_integer,nat))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ) ).

% nat_of_integer_code
tff(fact_4668_relpowp__fun__conv,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$o)),Xb: A,Y: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Xb),Y)
    <=> ? [F5: fun(nat,A)] :
          ( ( aa(nat,A,F5,zero_zero(nat)) = Xb )
          & ( aa(nat,A,F5,Nb) = Y )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
             => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,F5,I3)),aa(nat,A,F5,aa(nat,nat,suc,I3))) ) ) ) ).

% relpowp_fun_conv
tff(fact_4669_relpowp__1,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : ( aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),one_one(nat)),P) = P ) ).

% relpowp_1
tff(fact_4670_Nat_Ofunpow__code__def,axiom,
    ! [A: $tType] : ( funpow(A) = compow(fun(A,A)) ) ).

% Nat.funpow_code_def
tff(fact_4671_relpowp__Suc__I2,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xb: A,Y: A,Nb: nat,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),P,Xb),Y)
     => ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Y),Z)
       => aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),aa(nat,nat,suc,Nb)),P),Xb),Z) ) ) ).

% relpowp_Suc_I2
tff(fact_4672_relpowp__Suc__E2,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$o)),Xb: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),aa(nat,nat,suc,Nb)),P),Xb),Z)
     => ~ ! [Y3: A] :
            ( aa(A,$o,aa(A,fun(A,$o),P,Xb),Y3)
           => ~ aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Y3),Z) ) ) ).

% relpowp_Suc_E2
tff(fact_4673_relpowp__Suc__D2,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$o)),Xb: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),aa(nat,nat,suc,Nb)),P),Xb),Z)
     => ? [Y3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),P,Xb),Y3)
          & aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Y3),Z) ) ) ).

% relpowp_Suc_D2
tff(fact_4674_relpowp__Suc__I,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$o)),Xb: A,Y: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Xb),Y)
     => ( aa(A,$o,aa(A,fun(A,$o),P,Y),Z)
       => aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),aa(nat,nat,suc,Nb)),P),Xb),Z) ) ) ).

% relpowp_Suc_I
tff(fact_4675_relpowp__Suc__E,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$o)),Xb: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),aa(nat,nat,suc,Nb)),P),Xb),Z)
     => ~ ! [Y3: A] :
            ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Xb),Y3)
           => ~ aa(A,$o,aa(A,fun(A,$o),P,Y3),Z) ) ) ).

% relpowp_Suc_E
tff(fact_4676_relpowp__0__I,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xb: A] : aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),zero_zero(nat)),P),Xb),Xb) ).

% relpowp_0_I
tff(fact_4677_relpowp__0__E,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xb: A,Y: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),zero_zero(nat)),P),Xb),Y)
     => ( Xb = Y ) ) ).

% relpowp_0_E
tff(fact_4678_relpowp_Osimps_I1_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] : ( aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),zero_zero(nat)),R) = fequal(A) ) ).

% relpowp.simps(1)
tff(fact_4679_relpowp__E,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$o)),Xb: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Xb),Z)
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y3: A,M: nat] :
              ( ( Nb = aa(nat,nat,suc,M) )
             => ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),M),P),Xb),Y3)
               => ~ aa(A,$o,aa(A,fun(A,$o),P,Y3),Z) ) ) ) ) ).

% relpowp_E
tff(fact_4680_relpowp__E2,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$o)),Xb: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Xb),Z)
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y3: A,M: nat] :
              ( ( Nb = aa(nat,nat,suc,M) )
             => ( aa(A,$o,aa(A,fun(A,$o),P,Xb),Y3)
               => ~ aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),M),P),Y3),Z) ) ) ) ) ).

% relpowp_E2
tff(fact_4681_relpowp__bot,axiom,
    ! [A: $tType,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),bot_bot(fun(A,fun(A,$o)))) = bot_bot(fun(A,fun(A,$o))) ) ) ).

% relpowp_bot
tff(fact_4682_set__removeAll,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : ( aa(list(A),set(A),set2(A),removeAll(A,Xb,Xs)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))) ) ).

% set_removeAll
tff(fact_4683_bit_Oabstract__boolean__algebra__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => boolea2506097494486148201lgebra(A,bit_se5824344872417868541ns_and(A),bit_se1065995026697491101ons_or(A),bit_ri4277139882892585799ns_not(A),zero_zero(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% bit.abstract_boolean_algebra_axioms
tff(fact_4684_int__of__integer__code,axiom,
    ! [K: code_integer] :
      ( aa(code_integer,int,code_int_of_integer,K) = $ite(
          aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),
          aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),K))),
          $ite(K = zero_zero(code_integer),zero_zero(int),aa(product_prod(code_integer,code_integer),int,product_case_prod(code_integer,code_integer,int,aTP_Lamp_kx(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ) ) ).

% int_of_integer_code
tff(fact_4685_int__of__integer__of__nat,axiom,
    ! [Nb: nat] : ( aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,semiring_1_of_nat(code_integer),Nb)) = aa(nat,int,semiring_1_of_nat(int),Nb) ) ).

% int_of_integer_of_nat
tff(fact_4686_removeAll__id,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( removeAll(A,Xb,Xs) = Xs ) ) ).

% removeAll_id
tff(fact_4687_of__int__integer__of,axiom,
    ! [K: code_integer] : ( aa(int,code_integer,ring_1_of_int(code_integer),aa(code_integer,int,code_int_of_integer,K)) = K ) ).

% of_int_integer_of
tff(fact_4688_int__of__integer__of__int,axiom,
    ! [K: int] : ( aa(code_integer,int,code_int_of_integer,aa(int,code_integer,ring_1_of_int(code_integer),K)) = K ) ).

% int_of_integer_of_int
tff(fact_4689_zero__integer_Orep__eq,axiom,
    aa(code_integer,int,code_int_of_integer,zero_zero(code_integer)) = zero_zero(int) ).

% zero_integer.rep_eq
tff(fact_4690_plus__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : ( aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),Xb),Xaa)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(code_integer,int,code_int_of_integer,Xb)),aa(code_integer,int,code_int_of_integer,Xaa)) ) ).

% plus_integer.rep_eq
tff(fact_4691_uminus__integer_Orep__eq,axiom,
    ! [Xb: code_integer] : ( aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Xb)) = aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,Xb)) ) ).

% uminus_integer.rep_eq
tff(fact_4692_one__integer_Orep__eq,axiom,
    aa(code_integer,int,code_int_of_integer,one_one(code_integer)) = one_one(int) ).

% one_integer.rep_eq
tff(fact_4693_minus__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : ( aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Xb),Xaa)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(code_integer,int,code_int_of_integer,Xb)),aa(code_integer,int,code_int_of_integer,Xaa)) ) ).

% minus_integer.rep_eq
tff(fact_4694_less__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Xb),Xaa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(code_integer,int,code_int_of_integer,Xb)),aa(code_integer,int,code_int_of_integer,Xaa)) ) ).

% less_integer.rep_eq
tff(fact_4695_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),L)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L)) ) ).

% integer_less_iff
tff(fact_4696_nat__of__integer_Orep__eq,axiom,
    ! [Xb: code_integer] : ( aa(code_integer,nat,code_nat_of_integer,Xb) = aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,Xb)) ) ).

% nat_of_integer.rep_eq
tff(fact_4697_length__removeAll__less,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),removeAll(A,Xb,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_removeAll_less
tff(fact_4698_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ~ ? [X3: A] :
                  ( member(A,X3,S3)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S3))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_4699_times__int_Oabs__eq,axiom,
    ! [Xaa: product_prod(nat,nat),Xb: product_prod(nat,nat)] : ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(nat,nat),int,abs_Integ,Xaa)),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_kz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ) ).

% times_int.abs_eq
tff(fact_4700_Gcd__remove0__nat,axiom,
    ! [M9: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),M9)
     => ( gcd_Gcd(nat,M9) = gcd_Gcd(nat,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),M9),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat))))) ) ) ).

% Gcd_remove0_nat
tff(fact_4701_Gcd__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_empty
tff(fact_4702_Gcd__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( ( gcd_Gcd(A,A3) = zero_zero(A) )
        <=> aa(set(A),$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)),insert2(A),zero_zero(A)),bot_bot(set(A)))) ) ) ).

% Gcd_0_iff
tff(fact_4703_Gcd__nat__eq__one,axiom,
    ! [N5: set(nat)] :
      ( member(nat,one_one(nat),N5)
     => ( gcd_Gcd(nat,N5) = one_one(nat) ) ) ).

% Gcd_nat_eq_one
tff(fact_4704_Gcd__1,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( member(A,one_one(A),A3)
         => ( gcd_Gcd(A,A3) = one_one(A) ) ) ) ).

% Gcd_1
tff(fact_4705_int_Oabs__induct,axiom,
    ! [P: fun(int,$o),Xb: int] :
      ( ! [Y3: product_prod(nat,nat)] : aa(int,$o,P,aa(product_prod(nat,nat),int,abs_Integ,Y3))
     => aa(int,$o,P,Xb) ) ).

% int.abs_induct
tff(fact_4706_Gcd__greatest__nat,axiom,
    ! [A3: set(nat),A2: nat] :
      ( ! [B4: nat] :
          ( member(nat,B4,A3)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B4) )
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),gcd_Gcd(nat,A3)) ) ).

% Gcd_greatest_nat
tff(fact_4707_Gcd__dvd__nat,axiom,
    ! [A2: nat,A3: set(nat)] :
      ( member(nat,A2,A3)
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),gcd_Gcd(nat,A3)),A2) ) ).

% Gcd_dvd_nat
tff(fact_4708_Gcd__dvd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( member(A,A2,A3)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),gcd_Gcd(A,A3)),A2) ) ) ).

% Gcd_dvd
tff(fact_4709_dvd__GcdD,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Xb: A,A3: set(A),Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),gcd_Gcd(A,A3))
         => ( member(A,Y,A3)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),Y) ) ) ) ).

% dvd_GcdD
tff(fact_4710_dvd__Gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Xb: A,A3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),gcd_Gcd(A,A3))
        <=> ! [X4: A] :
              ( member(A,X4,A3)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),X4) ) ) ) ).

% dvd_Gcd_iff
tff(fact_4711_Gcd__greatest,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A),A2: A] :
          ( ! [B4: A] :
              ( member(A,B4,A3)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B4) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),gcd_Gcd(A,A3)) ) ) ).

% Gcd_greatest
tff(fact_4712_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X: nat,Y3: nat] : ( Z != aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Y3)) ) ).

% eq_Abs_Integ
tff(fact_4713_Gcd__eq__1__I,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( member(A,A2,A3)
           => ( gcd_Gcd(A,A3) = one_one(A) ) ) ) ) ).

% Gcd_eq_1_I
tff(fact_4714_nat_Oabs__eq,axiom,
    ! [Xb: product_prod(nat,nat)] : ( aa(int,nat,nat2,aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),nat,product_case_prod(nat,nat,nat,minus_minus(nat)),Xb) ) ).

% nat.abs_eq
tff(fact_4715_zero__int__def,axiom,
    zero_zero(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),zero_zero(nat))) ).

% zero_int_def
tff(fact_4716_int__def,axiom,
    ! [Nb: nat] : ( aa(nat,int,semiring_1_of_nat(int),Nb) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Nb),zero_zero(nat))) ) ).

% int_def
tff(fact_4717_uminus__int_Oabs__eq,axiom,
    ! [Xb: product_prod(nat,nat)] : ( aa(int,int,uminus_uminus(int),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_la(nat,fun(nat,product_prod(nat,nat)))),Xb)) ) ).

% uminus_int.abs_eq
tff(fact_4718_one__int__def,axiom,
    one_one(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,one_one(nat)),zero_zero(nat))) ).

% one_int_def
tff(fact_4719_of__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: product_prod(nat,nat)] : ( aa(int,A,ring_1_of_int(A),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),A,product_case_prod(nat,nat,A,aTP_Lamp_lb(nat,fun(nat,A))),Xb) ) ) ).

% of_int.abs_eq
tff(fact_4720_less__int_Oabs__eq,axiom,
    ! [Xaa: product_prod(nat,nat),Xb: product_prod(nat,nat)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(product_prod(nat,nat),int,abs_Integ,Xaa)),aa(product_prod(nat,nat),int,abs_Integ,Xb))
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_ld(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xaa),Xb) ) ).

% less_int.abs_eq
tff(fact_4721_less__eq__int_Oabs__eq,axiom,
    ! [Xaa: product_prod(nat,nat),Xb: product_prod(nat,nat)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(product_prod(nat,nat),int,abs_Integ,Xaa)),aa(product_prod(nat,nat),int,abs_Integ,Xb))
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_lf(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xaa),Xb) ) ).

% less_eq_int.abs_eq
tff(fact_4722_plus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod(nat,nat),Xb: product_prod(nat,nat)] : ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(product_prod(nat,nat),int,abs_Integ,Xaa)),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lh(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ) ).

% plus_int.abs_eq
tff(fact_4723_minus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod(nat,nat),Xb: product_prod(nat,nat)] : ( aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(nat,nat),int,abs_Integ,Xaa)),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lj(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ) ).

% minus_int.abs_eq
tff(fact_4724_semiring__char__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: itself(A)] : ( semiri4206861660011772517g_char(A,Uu) = gcd_Gcd(nat,collect(nat,aTP_Lamp_lk(nat,$o))) ) ) ).

% semiring_char_def
tff(fact_4725_csqrt_Ocode,axiom,
    ! [Z: complex] :
      ( csqrt(Z) = complex2(aa(real,real,sqrt,divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z)),aa(num,real,numeral_numeral(real),bit0(one2)))),
          aa(real,real,
            aa(real,fun(real,real),times_times(real),
              $ite(im(Z) = zero_zero(real),one_one(real),aa(real,real,sgn_sgn(real),im(Z)))),
            aa(real,real,sqrt,divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z)),aa(num,real,numeral_numeral(real),bit0(one2)))))) ) ).

% csqrt.code
tff(fact_4726_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),Nb: nat] :
      ( ! [X: list(A)] :
          ( member(list(A),X,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
         => ( aa(list(A),nat,size_size(list(A)),X) = Nb ) )
     => ( 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)),Nb) ) ) ).

% length_mul_elem
tff(fact_4727_abs__Gcd__eq,axiom,
    ! [K6: set(int)] : ( aa(int,int,abs_abs(int),gcd_Gcd(int,K6)) = gcd_Gcd(int,K6) ) ).

% abs_Gcd_eq
tff(fact_4728_complex__Im__of__int,axiom,
    ! [Z: int] : ( im(aa(int,complex,ring_1_of_int(complex),Z)) = zero_zero(real) ) ).

% complex_Im_of_int
tff(fact_4729_complex__Re__of__int,axiom,
    ! [Z: int] : ( re(aa(int,complex,ring_1_of_int(complex),Z)) = aa(int,real,ring_1_of_int(real),Z) ) ).

% complex_Re_of_int
tff(fact_4730_Re__i__times,axiom,
    ! [Z: complex] : ( re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Z)) = aa(real,real,uminus_uminus(real),im(Z)) ) ).

% Re_i_times
tff(fact_4731_csqrt__minus,axiom,
    ! [Xb: complex] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),im(Xb)),zero_zero(real))
        | ( ( im(Xb) = zero_zero(real) )
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(Xb)) ) )
     => ( csqrt(aa(complex,complex,uminus_uminus(complex),Xb)) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),csqrt(Xb)) ) ) ).

% csqrt_minus
tff(fact_4732_Gcd__greatest__int,axiom,
    ! [A3: set(int),A2: int] :
      ( ! [B4: int] :
          ( member(int,B4,A3)
         => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),B4) )
     => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),gcd_Gcd(int,A3)) ) ).

% Gcd_greatest_int
tff(fact_4733_Gcd__dvd__int,axiom,
    ! [A2: int,A3: set(int)] :
      ( member(int,A2,A3)
     => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),gcd_Gcd(int,A3)),A2) ) ).

% Gcd_dvd_int
tff(fact_4734_uminus__complex_Ocode,axiom,
    ! [Xb: complex] : ( aa(complex,complex,uminus_uminus(complex),Xb) = complex2(aa(real,real,uminus_uminus(real),re(Xb)),aa(real,real,uminus_uminus(real),im(Xb))) ) ).

% uminus_complex.code
tff(fact_4735_complex__is__Int__iff,axiom,
    ! [Z: complex] :
      ( member(complex,Z,ring_1_Ints(complex))
    <=> ( ( im(Z) = zero_zero(real) )
        & ? [I3: int] : ( re(Z) = aa(int,real,ring_1_of_int(real),I3) ) ) ) ).

% complex_is_Int_iff
tff(fact_4736_imaginary__unit_Osimps_I2_J,axiom,
    im(imaginary_unit) = one_one(real) ).

% imaginary_unit.simps(2)
tff(fact_4737_one__complex_Osimps_I2_J,axiom,
    im(one_one(complex)) = zero_zero(real) ).

% one_complex.simps(2)
tff(fact_4738_one__complex_Osimps_I1_J,axiom,
    re(one_one(complex)) = one_one(real) ).

% one_complex.simps(1)
tff(fact_4739_uminus__complex_Osimps_I2_J,axiom,
    ! [Xb: complex] : ( im(aa(complex,complex,uminus_uminus(complex),Xb)) = aa(real,real,uminus_uminus(real),im(Xb)) ) ).

% uminus_complex.simps(2)
tff(fact_4740_uminus__complex_Osimps_I1_J,axiom,
    ! [Xb: complex] : ( re(aa(complex,complex,uminus_uminus(complex),Xb)) = aa(real,real,uminus_uminus(real),re(Xb)) ) ).

% uminus_complex.simps(1)
tff(fact_4741_Gcd__int__greater__eq__0,axiom,
    ! [K6: set(int)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),gcd_Gcd(int,K6)) ).

% Gcd_int_greater_eq_0
tff(fact_4742_inverse__complex_Osimps_I1_J,axiom,
    ! [Xb: complex] : ( re(aa(complex,complex,inverse_inverse(complex),Xb)) = divide_divide(real,re(Xb),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% inverse_complex.simps(1)
tff(fact_4743_inverse__complex_Osimps_I2_J,axiom,
    ! [Xb: complex] : ( im(aa(complex,complex,inverse_inverse(complex),Xb)) = divide_divide(real,aa(real,real,uminus_uminus(real),im(Xb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% inverse_complex.simps(2)
tff(fact_4744_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),divide_divide(real,re(Z),real_V7770717601297561774m_norm(complex,Z))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),divide_divide(real,im(Z),real_V7770717601297561774m_norm(complex,Z))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) ) ) ).

% complex_unit_circle
tff(fact_4745_inverse__complex_Ocode,axiom,
    ! [Xb: complex] : ( aa(complex,complex,inverse_inverse(complex),Xb) = complex2(divide_divide(real,re(Xb),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),divide_divide(real,aa(real,real,uminus_uminus(real),im(Xb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% inverse_complex.code
tff(fact_4746_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z))),zero_zero(real))
    <=> ( re(Z) = aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_4747_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      ( im(csqrt(Z)) = aa(real,real,
          aa(real,fun(real,real),times_times(real),
            $ite(im(Z) = zero_zero(real),one_one(real),aa(real,real,sgn_sgn(real),im(Z)))),
          aa(real,real,sqrt,divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z)),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ).

% csqrt.simps(2)
tff(fact_4748_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( member(complex,R2,real_Vector_Reals(complex))
     => ( im(divide_divide(complex,R2,Z)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),re(R2))),im(Z)),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% Im_Reals_divide
tff(fact_4749_set__n__lists,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : ( aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,Nb,Xs)) = collect(list(A),aa(list(A),fun(list(A),$o),aTP_Lamp_ll(nat,fun(list(A),fun(list(A),$o)),Nb),Xs)) ) ).

% set_n_lists
tff(fact_4750_less__eq__int_Orep__eq,axiom,
    ! [Xb: int,Xaa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),Xaa)
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_lf(nat,fun(nat,fun(product_prod(nat,nat),$o)))),aa(int,product_prod(nat,nat),rep_Integ,Xb)),aa(int,product_prod(nat,nat),rep_Integ,Xaa)) ) ).

% less_eq_int.rep_eq
tff(fact_4751_Reals__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A] :
          ( member(A,aa(A,A,uminus_uminus(A),A2),real_Vector_Reals(A))
        <=> member(A,A2,real_Vector_Reals(A)) ) ) ).

% Reals_minus_iff
tff(fact_4752_Reals__inverse__iff,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [Xb: A] :
          ( member(A,aa(A,A,inverse_inverse(A),Xb),real_Vector_Reals(A))
        <=> member(A,Xb,real_Vector_Reals(A)) ) ) ).

% Reals_inverse_iff
tff(fact_4753_Reals__0,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => member(A,zero_zero(A),real_Vector_Reals(A)) ) ).

% Reals_0
tff(fact_4754_Reals__minus,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A] :
          ( member(A,A2,real_Vector_Reals(A))
         => member(A,aa(A,A,uminus_uminus(A),A2),real_Vector_Reals(A)) ) ) ).

% Reals_minus
tff(fact_4755_Reals__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A2: A] :
          ( member(A,A2,real_Vector_Reals(A))
         => member(A,aa(A,A,inverse_inverse(A),A2),real_Vector_Reals(A)) ) ) ).

% Reals_inverse
tff(fact_4756_Reals__1,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => member(A,one_one(A),real_Vector_Reals(A)) ) ).

% Reals_1
tff(fact_4757_Reals__of__int,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Z: int] : member(A,aa(int,A,ring_1_of_int(A),Z),real_Vector_Reals(A)) ) ).

% Reals_of_int
tff(fact_4758_Reals__diff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,real_Vector_Reals(A))
         => ( member(A,B2,real_Vector_Reals(A))
           => member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),real_Vector_Reals(A)) ) ) ) ).

% Reals_diff
tff(fact_4759_Reals__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,real_Vector_Reals(A))
         => ( member(A,B2,real_Vector_Reals(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),real_Vector_Reals(A)) ) ) ) ).

% Reals_add
tff(fact_4760_Reals__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Nb: nat] : member(A,aa(nat,A,semiring_1_of_nat(A),Nb),real_Vector_Reals(A)) ) ).

% Reals_of_nat
tff(fact_4761_nonzero__Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,real_Vector_Reals(A))
         => ( member(A,B2,real_Vector_Reals(A))
           => ( ( B2 != zero_zero(A) )
             => member(A,divide_divide(A,A2,B2),real_Vector_Reals(A)) ) ) ) ) ).

% nonzero_Reals_divide
tff(fact_4762_nonzero__Reals__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A2: A] :
          ( member(A,A2,real_Vector_Reals(A))
         => ( ( A2 != zero_zero(A) )
           => member(A,aa(A,A,inverse_inverse(A),A2),real_Vector_Reals(A)) ) ) ) ).

% nonzero_Reals_inverse
tff(fact_4763_length__n__lists__elem,axiom,
    ! [A: $tType,Ys2: list(A),Nb: nat,Xs: list(A)] :
      ( member(list(A),Ys2,aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,Nb,Xs)))
     => ( aa(list(A),nat,size_size(list(A)),Ys2) = Nb ) ) ).

% length_n_lists_elem
tff(fact_4764_nat_Orep__eq,axiom,
    ! [Xb: int] : ( aa(int,nat,nat2,Xb) = aa(product_prod(nat,nat),nat,product_case_prod(nat,nat,nat,minus_minus(nat)),aa(int,product_prod(nat,nat),rep_Integ,Xb)) ) ).

% nat.rep_eq
tff(fact_4765_of__int_Orep__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: int] : ( aa(int,A,ring_1_of_int(A),Xb) = aa(product_prod(nat,nat),A,product_case_prod(nat,nat,A,aTP_Lamp_lb(nat,fun(nat,A))),aa(int,product_prod(nat,nat),rep_Integ,Xb)) ) ) ).

% of_int.rep_eq
tff(fact_4766_less__int_Orep__eq,axiom,
    ! [Xb: int,Xaa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xb),Xaa)
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_ld(nat,fun(nat,fun(product_prod(nat,nat),$o)))),aa(int,product_prod(nat,nat),rep_Integ,Xb)),aa(int,product_prod(nat,nat),rep_Integ,Xaa)) ) ).

% less_int.rep_eq
tff(fact_4767_uminus__int__def,axiom,
    uminus_uminus(int) = aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_la(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int_def
tff(fact_4768_prod__encode__def,axiom,
    nat_prod_encode = product_case_prod(nat,nat,nat,aTP_Lamp_lm(nat,fun(nat,nat))) ).

% prod_encode_def
tff(fact_4769_num__of__nat_Osimps_I2_J,axiom,
    ! [Nb: nat] :
      ( aa(nat,num,num_of_nat,aa(nat,nat,suc,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb),inc(aa(nat,num,num_of_nat,Nb)),one2) ) ).

% num_of_nat.simps(2)
tff(fact_4770_prod__encode__eq,axiom,
    ! [Xb: product_prod(nat,nat),Y: product_prod(nat,nat)] :
      ( ( aa(product_prod(nat,nat),nat,nat_prod_encode,Xb) = aa(product_prod(nat,nat),nat,nat_prod_encode,Y) )
    <=> ( Xb = Y ) ) ).

% prod_encode_eq
tff(fact_4771_num__of__nat__numeral__eq,axiom,
    ! [Q4: num] : ( aa(nat,num,num_of_nat,aa(num,nat,numeral_numeral(nat),Q4)) = Q4 ) ).

% num_of_nat_numeral_eq
tff(fact_4772_num__of__nat_Osimps_I1_J,axiom,
    aa(nat,num,num_of_nat,zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_4773_le__prod__encode__1,axiom,
    ! [A2: nat,B2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A2),B2))) ).

% le_prod_encode_1
tff(fact_4774_le__prod__encode__2,axiom,
    ! [B2: nat,A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A2),B2))) ).

% le_prod_encode_2
tff(fact_4775_numeral__num__of__nat,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(num,nat,numeral_numeral(nat),aa(nat,num,num_of_nat,Nb)) = Nb ) ) ).

% numeral_num_of_nat
tff(fact_4776_num__of__nat__One,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),one_one(nat))
     => ( aa(nat,num,num_of_nat,Nb) = one2 ) ) ).

% num_of_nat_One
tff(fact_4777_numeral__num__of__nat__unfold,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          ( aa(num,A,numeral_numeral(A),aa(nat,num,num_of_nat,Nb)) = $ite(Nb = zero_zero(nat),one_one(A),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ).

% numeral_num_of_nat_unfold
tff(fact_4778_num__of__nat__double,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Nb)) = bit0(aa(nat,num,num_of_nat,Nb)) ) ) ).

% num_of_nat_double
tff(fact_4779_num__of__nat__plus__distrib,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(nat,num,num_of_nat,Ma)),aa(nat,num,num_of_nat,Nb)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_4780_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,Ma: nat] : ( aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K),Ma)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),Ma) ) ).

% prod_encode_prod_decode_aux
tff(fact_4781_times__int__def,axiom,
    times_times(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_kz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int_def
tff(fact_4782_minus__int__def,axiom,
    minus_minus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lj(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int_def
tff(fact_4783_plus__int__def,axiom,
    plus_plus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lh(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int_def
tff(fact_4784_num__of__integer_Oabs__eq,axiom,
    ! [Xb: int] : ( aa(code_integer,num,code_num_of_integer,aa(int,code_integer,code_integer_of_int,Xb)) = aa(nat,num,num_of_nat,aa(int,nat,nat2,Xb)) ) ).

% num_of_integer.abs_eq
tff(fact_4785_num__of__integer_Orep__eq,axiom,
    ! [Xb: code_integer] : ( aa(code_integer,num,code_num_of_integer,Xb) = aa(nat,num,num_of_nat,aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,Xb))) ) ).

% num_of_integer.rep_eq
tff(fact_4786_eq__numeral__iff__iszero_I7_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb)) = one_one(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),one2))) ) ) ).

% eq_numeral_iff_iszero(7)
tff(fact_4787_iszero__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: num] :
          ( ring_1_iszero(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),W)) ) ) ).

% iszero_neg_numeral
tff(fact_4788_not__iszero__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [W: num] : ~ ring_1_iszero(A,aa(num,A,numeral_numeral(A),W)) ) ).

% not_iszero_numeral
tff(fact_4789_eq__iff__iszero__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb = Y )
        <=> ring_1_iszero(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) ) ) ).

% eq_iff_iszero_diff
tff(fact_4790_not__iszero__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,one_one(A)) ) ).

% not_iszero_1
tff(fact_4791_iszero__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] :
          ( ring_1_iszero(A,Z)
        <=> ( Z = zero_zero(A) ) ) ) ).

% iszero_def
tff(fact_4792_iszero__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ring_1_iszero(A,zero_zero(A)) ) ).

% iszero_0
tff(fact_4793_eq__numeral__iff__iszero_I9_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num] :
          ( ( aa(num,A,numeral_numeral(A),Xb) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Xb)) ) ) ).

% eq_numeral_iff_iszero(9)
tff(fact_4794_eq__numeral__iff__iszero_I10_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( zero_zero(A) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y)) ) ) ).

% eq_numeral_iff_iszero(10)
tff(fact_4795_not__iszero__Numeral1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(num,A,numeral_numeral(A),one2)) ) ).

% not_iszero_Numeral1
tff(fact_4796_not__iszero__neg__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_iszero_neg_1
tff(fact_4797_eq__numeral__iff__iszero_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num,Y: num] :
          ( ( aa(num,A,numeral_numeral(A),Xb) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,Xb,Y)) ) ) ).

% eq_numeral_iff_iszero(1)
tff(fact_4798_eq__numeral__iff__iszero_I12_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y)) ) ) ).

% eq_numeral_iff_iszero(12)
tff(fact_4799_eq__numeral__iff__iszero_I11_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb)) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Xb)) ) ) ).

% eq_numeral_iff_iszero(11)
tff(fact_4800_not__iszero__neg__Numeral1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) ) ).

% not_iszero_neg_Numeral1
tff(fact_4801_eq__numeral__iff__iszero_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num,Y: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb)) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),Y))) ) ) ).

% eq_numeral_iff_iszero(3)
tff(fact_4802_eq__numeral__iff__iszero_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num,Y: num] :
          ( ( aa(num,A,numeral_numeral(A),Xb) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),Y))) ) ) ).

% eq_numeral_iff_iszero(2)
tff(fact_4803_eq__numeral__iff__iszero_I4_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num,Y: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,Y,Xb)) ) ) ).

% eq_numeral_iff_iszero(4)
tff(fact_4804_eq__numeral__iff__iszero_I6_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,one2,Y)) ) ) ).

% eq_numeral_iff_iszero(6)
tff(fact_4805_eq__numeral__iff__iszero_I5_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num] :
          ( ( aa(num,A,numeral_numeral(A),Xb) = one_one(A) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,Xb,one2)) ) ) ).

% eq_numeral_iff_iszero(5)
tff(fact_4806_eq__numeral__iff__iszero_I8_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( one_one(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Y))) ) ) ).

% eq_numeral_iff_iszero(8)
tff(fact_4807_num__of__integer__code,axiom,
    ! [K: code_integer] :
      ( aa(code_integer,num,code_num_of_integer,K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),one_one(code_integer)),one2,aa(product_prod(code_integer,code_integer),num,product_case_prod(code_integer,code_integer,num,aTP_Lamp_ln(code_integer,fun(code_integer,num))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ) ).

% num_of_integer_code
tff(fact_4808_prod_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),P3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_iq(set(A),fun(fun(A,B),fun(A,$o)),I5),P3)))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),P3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),I5)) = $ite(member(A,I,I5),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),P3),I5),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,P3,I)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),P3),I5))) ) ) ) ).

% prod.insert'
tff(fact_4809_pow_Osimps_I3_J,axiom,
    ! [Xb: num,Y: num] : ( pow(Xb,aa(num,num,bit1,Y)) = aa(num,num,aa(num,fun(num,num),times_times(num),sqr(pow(Xb,Y))),Xb) ) ).

% pow.simps(3)
tff(fact_4810_prod_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: fun(B,A)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P3),bot_bot(set(B))) = one_one(A) ) ) ).

% prod.empty'
tff(fact_4811_prod_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),I5: set(B)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),collect(B,aa(set(B),fun(B,$o),aTP_Lamp_lo(fun(B,A),fun(set(B),fun(B,$o)),G),I5))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),I5) ) ) ).

% prod.non_neutral'
tff(fact_4812_sqr_Osimps_I2_J,axiom,
    ! [Nb: num] : ( sqr(bit0(Nb)) = bit0(bit0(sqr(Nb))) ) ).

% sqr.simps(2)
tff(fact_4813_sqr_Osimps_I1_J,axiom,
    sqr(one2) = one2 ).

% sqr.simps(1)
tff(fact_4814_sqr__conv__mult,axiom,
    ! [Xb: num] : ( sqr(Xb) = aa(num,num,aa(num,fun(num,num),times_times(num),Xb),Xb) ) ).

% sqr_conv_mult
tff(fact_4815_prod_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),T4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T4)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
               => ( aa(A,B,G,X) = one_one(B) ) )
           => ( ! [X: A] :
                  ( member(A,X,S3)
                 => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),T4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),H),S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_4816_prod_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),T4: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T4)
         => ( ! [I2: A] :
                ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
               => ( aa(A,B,H,I2) = one_one(B) ) )
           => ( ! [X: A] :
                  ( member(A,X,S3)
                 => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),H),T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_4817_prod_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),T4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T4)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
               => ( aa(A,B,G,X) = one_one(B) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),T4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),S3) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_4818_prod_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),T4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T4)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
               => ( aa(A,B,G,X) = one_one(B) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),T4) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_4819_prod_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_iq(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( aa(set(A),$o,finite_finite(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_iq(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_lp(fun(A,B),fun(fun(A,B),fun(A,B)),G),H)),I5) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),I5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),H),I5)) ) ) ) ) ).

% prod.distrib'
tff(fact_4820_prod_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: fun(B,A),I5: set(B)] :
          ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P3),I5) = $ite(aa(set(B),$o,finite_finite(B),collect(B,aa(set(B),fun(B,$o),aTP_Lamp_lo(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P3),collect(B,aa(set(B),fun(B,$o),aTP_Lamp_lo(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),one_one(A)) ) ) ).

% prod.G_def
tff(fact_4821_pow_Osimps_I2_J,axiom,
    ! [Xb: num,Y: num] : ( pow(Xb,bit0(Y)) = sqr(pow(Xb,Y)) ) ).

% pow.simps(2)
tff(fact_4822_sqr_Osimps_I3_J,axiom,
    ! [Nb: num] : ( sqr(aa(num,num,bit1,Nb)) = aa(num,num,bit1,bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),sqr(Nb)),Nb))) ) ).

% sqr.simps(3)
tff(fact_4823_in__set__product__lists__length,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] :
      ( member(list(A),Xs,aa(list(list(A)),set(list(A)),set2(list(A)),product_lists(A,Xss)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Xss) ) ) ).

% in_set_product_lists_length
tff(fact_4824_sum__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A3: set(A),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),$o,finite_finite(A),A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_lq(fun(A,$o),fun(A,B),P)),A3) = aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,P)))) ) ) ) ) ).

% sum_of_bool_eq
tff(fact_4825_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Xb: nat,Y: nat] :
      ( image(nat,nat,aTP_Lamp_lr(nat,fun(nat,nat),C2),set_or7035219750837199246ssThan(nat,Xb,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),Xb),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),Xb),Y),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_4826_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_4827_inf__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Xb) = Xb ) ) ).

% inf_idem
tff(fact_4828_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_4829_inf__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) ) ) ).

% inf_left_idem
tff(fact_4830_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_4831_inf__right__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: 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),Xb),Y)),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) ) ) ).

% inf_right_idem
tff(fact_4832_inf__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( semilattice_inf(A)
     => ! [F2: fun(B,A),G: fun(B,A),Xb: 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),Xb) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F2,Xb)),aa(B,A,G,Xb)) ) ) ).

% inf_apply
tff(fact_4833_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_4834_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z) ) ) ) ).

% le_inf_iff
tff(fact_4835_inf__bot__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),Xb) = bot_bot(A) ) ) ).

% inf_bot_left
tff(fact_4836_inf__bot__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),bot_bot(A)) = bot_bot(A) ) ) ).

% inf_bot_right
tff(fact_4837_bij__betw__Suc,axiom,
    ! [M9: set(nat),N5: set(nat)] :
      ( bij_betw(nat,nat,suc,M9,N5)
    <=> ( image(nat,nat,suc,M9) = N5 ) ) ).

% bij_betw_Suc
tff(fact_4838_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(A)] : ( image(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),S3) = S3 ) ) ).

% image_add_0
tff(fact_4839_boolean__algebra_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,uminus_uminus(A),Xb)) = bot_bot(A) ) ) ).

% boolean_algebra.conj_cancel_right
tff(fact_4840_boolean__algebra_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xb)),Xb) = bot_bot(A) ) ) ).

% boolean_algebra.conj_cancel_left
tff(fact_4841_inf__compl__bot__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,uminus_uminus(A),Xb))) = bot_bot(A) ) ) ).

% inf_compl_bot_right
tff(fact_4842_inf__compl__bot__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xb)),Y)) = bot_bot(A) ) ) ).

% inf_compl_bot_left2
tff(fact_4843_inf__compl__bot__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)) = bot_bot(A) ) ) ).

% inf_compl_bot_left1
tff(fact_4844_image__add__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: 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_4845_image__diff__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D3: A,A2: A,B2: A] : ( image(A,A,aa(A,fun(A,A),minus_minus(A),D3),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),D3),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),D3),A2)) ) ) ).

% image_diff_atLeastAtMost
tff(fact_4846_image__uminus__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Y: A] : ( image(A,A,uminus_uminus(A),set_or1337092689740270186AtMost(A,Xb,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),Xb)) ) ) ).

% image_uminus_atLeastAtMost
tff(fact_4847_image__add__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: 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_4848_image__add__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [C2: A,A2: 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_4849_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)
        <=> ( image(A,A,aa(A,fun(A,A),plus_plus(A),A2),A3) = B3 ) ) ) ).

% bij_betw_add
tff(fact_4850_image__uminus__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Y: A] : ( image(A,A,uminus_uminus(A),set_or5935395276787703475ssThan(A,Xb,Y)) = set_or5935395276787703475ssThan(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),Xb)) ) ) ).

% image_uminus_greaterThanLessThan
tff(fact_4851_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_4852_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_4853_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_4854_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : ( image(nat,nat,suc,set_or1337092689740270186AtMost(nat,I,J)) = set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ) ).

% image_Suc_atLeastAtMost
tff(fact_4855_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : ( image(nat,nat,suc,set_or7035219750837199246ssThan(nat,I,J)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ) ).

% image_Suc_atLeastLessThan
tff(fact_4856_bij__betw__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N5: set(nat),A3: set(A)] :
          ( bij_betw(nat,A,semiring_1_of_nat(A),N5,A3)
        <=> ( image(nat,A,semiring_1_of_nat(A),N5) = A3 ) ) ) ).

% bij_betw_of_nat
tff(fact_4857_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : ( image(A,A,aTP_Lamp_ls(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_4858_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D3: A,A2: A,B2: A] : ( image(A,A,aTP_Lamp_lt(A,fun(A,A),D3),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)) ) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_4859_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : ( image(A,A,aTP_Lamp_ls(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_4860_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D3)
         => ( image(A,A,aa(A,fun(A,A),times_times(A),D3),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),D3),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_4861_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D3)
         => ( image(A,A,aTP_Lamp_lu(A,fun(A,A),D3),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,divide_divide(A,A2,D3),divide_divide(A,B2,D3)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_4862_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_4863_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_4864_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_4865_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_4866_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_4867_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_4868_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_4869_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)) ) ) ) ).

% inf_greatest
tff(fact_4870_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_4871_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_4872_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = Y ) ) ) ).

% inf_absorb2
tff(fact_4873_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = Xb ) ) ) ).

% inf_absorb1
tff(fact_4874_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_4875_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_4876_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = Xb ) ) ) ).

% le_iff_inf
tff(fact_4877_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F2: fun(A,fun(A,A)),Xb: A,Y: A] :
          ( ! [X: A,Y3: A] : aa(A,$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,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X),Y3)),Y3)
           => ( ! [X: A,Y3: A,Z2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),F2,Y3),Z2)) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),F2,Xb),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_4878_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_4879_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_4880_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,Xb: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xb) ) ) ).

% le_infI2
tff(fact_4881_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,Xb: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xb) ) ) ).

% le_infI1
tff(fact_4882_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
           => aa(A,$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),D3)) ) ) ) ).

% inf_mono
tff(fact_4883_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)) ) ) ) ).

% le_infI
tff(fact_4884_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),B2) ) ) ) ).

% le_infE
tff(fact_4885_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)),Y) ) ).

% inf_le2
tff(fact_4886_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)),Xb) ) ).

% inf_le1
tff(fact_4887_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)),Xb) ) ).

% inf_sup_ord(1)
tff(fact_4888_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)),Y) ) ).

% inf_sup_ord(2)
tff(fact_4889_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_4890_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_4891_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_4892_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_4893_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_4894_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_4895_less__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,Xb: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xb) ) ) ).

% less_infI2
tff(fact_4896_less__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,Xb: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xb) ) ) ).

% less_infI1
tff(fact_4897_inf__sup__aci_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) ) ) ).

% inf_sup_aci(4)
tff(fact_4898_inf__sup__aci_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A,Z: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),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),Xb),Z)) ) ) ).

% inf_sup_aci(3)
tff(fact_4899_inf__sup__aci_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: 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),Xb),Y)),Z) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)) ) ) ).

% inf_sup_aci(2)
tff(fact_4900_inf__sup__aci_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Xb) ) ) ).

% inf_sup_aci(1)
tff(fact_4901_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_4902_inf__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: 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),Xb),Y)),Z) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)) ) ) ).

% inf_assoc
tff(fact_4903_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_4904_inf__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Xb) ) ) ).

% inf_commute
tff(fact_4905_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_4906_inf__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A,Z: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),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),Xb),Z)) ) ) ).

% inf_left_commute
tff(fact_4907_inf__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_inf(B)
     => ! [F2: fun(A,B),G: fun(A,B),X3: 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),X3) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) ) ) ).

% inf_fun_def
tff(fact_4908_translation__subtract__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S: set(A),Ta: set(A)] : ( image(A,A,aTP_Lamp_lv(A,fun(A,A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),Ta)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(A,A,aTP_Lamp_lv(A,fun(A,A),A2),S)),image(A,A,aTP_Lamp_lv(A,fun(A,A),A2),Ta)) ) ) ).

% translation_subtract_Int
tff(fact_4909_translation__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S: set(A),Ta: 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)),S),Ta)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(A,A,aa(A,fun(A,A),plus_plus(A),A2),S)),image(A,A,aa(A,fun(A,A),plus_plus(A),A2),Ta)) ) ) ).

% translation_Int
tff(fact_4910_diff__eq,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% diff_eq
tff(fact_4911_inf__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: 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),Xb),A2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xb)),B2)) = bot_bot(A) ) ) ).

% inf_cancel_left1
tff(fact_4912_inf__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: 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),Xb)),A2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),B2)) = bot_bot(A) ) ) ).

% inf_cancel_left2
tff(fact_4913_zero__notin__Suc__image,axiom,
    ! [A3: set(nat)] : ~ member(nat,zero_zero(nat),image(nat,nat,suc,A3)) ).

% zero_notin_Suc_image
tff(fact_4914_translation__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S: set(A),Ta: 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)),S),Ta)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),image(A,A,aa(A,fun(A,A),plus_plus(A),A2),S)),image(A,A,aa(A,fun(A,A),plus_plus(A),A2),Ta)) ) ) ).

% translation_diff
tff(fact_4915_translation__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,Ta: set(A)] : ( image(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(set(A),set(A),uminus_uminus(set(A)),Ta)) = aa(set(A),set(A),uminus_uminus(set(A)),image(A,A,aa(A,fun(A,A),plus_plus(A),A2),Ta)) ) ) ).

% translation_Compl
tff(fact_4916_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) = 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)),B3)) ) ).

% Diff_eq
tff(fact_4917_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A3: set(A),F2: fun(nat,A),Nb: nat] :
      ( ( A3 = image(nat,A,F2,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aa(nat,fun(nat,$o)),Nb))) )
     => aa(set(A),$o,finite_finite(A),A3) ) ).

% nat_seg_image_imp_finite
tff(fact_4918_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A3)
    <=> ? [N4: nat,F5: fun(nat,A)] : ( A3 = image(nat,A,F5,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aa(nat,fun(nat,$o)),N4))) ) ) ).

% finite_conv_nat_seg_image
tff(fact_4919_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S: set(A),Ta: set(A)] : ( image(A,A,aTP_Lamp_lv(A,fun(A,A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),Ta)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),image(A,A,aTP_Lamp_lv(A,fun(A,A),A2),S)),image(A,A,aTP_Lamp_lv(A,fun(A,A),A2),Ta)) ) ) ).

% translation_subtract_diff
tff(fact_4920_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,Ta: set(A)] : ( image(A,A,aTP_Lamp_lv(A,fun(A,A),A2),aa(set(A),set(A),uminus_uminus(set(A)),Ta)) = aa(set(A),set(A),uminus_uminus(set(A)),image(A,A,aTP_Lamp_lv(A,fun(A,A),A2),Ta)) ) ) ).

% translation_subtract_Compl
tff(fact_4921_Gcd__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_Gcd(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(A,B,F2,X)),aa(A,B,G,X)) )
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),gcd_Gcd(B,image(A,B,F2,A3))),gcd_Gcd(B,image(A,B,G,A3))) ) ) ).

% Gcd_mono
tff(fact_4922_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% inf_shunt
tff(fact_4923_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_4924_sum_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),B3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(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,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(set(A),fun(A,B),aTP_Lamp_lw(fun(A,B),fun(set(A),fun(A,B)),G),B3)),A3) ) ) ) ).

% sum.inter_restrict
tff(fact_4925_prod_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),B3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),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)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(set(A),fun(A,B),aTP_Lamp_lx(fun(A,B),fun(set(A),fun(A,B)),G),B3)),A3) ) ) ) ).

% prod.inter_restrict
tff(fact_4926_sum_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [A3: set(A),H: fun(A,B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ! [X: A,Y3: A] :
                ( member(A,X,A3)
               => ( member(A,Y3,A3)
                 => ( ( X != Y3 )
                   => ( ( aa(A,B,H,X) = aa(A,B,H,Y3) )
                     => ( aa(B,C,G,aa(A,B,H,X)) = zero_zero(C) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),image(A,B,H,A3)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G),H)),A3) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_4927_prod_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A3: set(A),H: fun(A,B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ! [X: A,Y3: A] :
                ( member(A,X,A3)
               => ( member(A,Y3,A3)
                 => ( ( X != Y3 )
                   => ( ( aa(A,B,H,X) = aa(A,B,H,Y3) )
                     => ( aa(B,C,G,aa(A,B,H,X)) = one_one(C) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),image(A,B,H,A3)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G),H)),A3) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_4928_sum_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T4: set(A),S3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T4)
         => ( aa(set(A),$o,finite_finite(A),S3)
           => ( ! [I2: A] :
                  ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
                 => ( aa(A,B,H,I2) = zero_zero(B) ) )
             => ( ! [I2: A] :
                    ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),T4))
                   => ( aa(A,B,G,I2) = zero_zero(B) ) )
               => ( ! [X: A] :
                      ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),T4))
                     => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
                 => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),T4) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_4929_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K: A,Xb: 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)),insert2(A),Xb),bot_bot(set(A)))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),K),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))),bot_bot(set(A))) ) ) ).

% Iio_Int_singleton
tff(fact_4930_sum_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),B3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))),aa(set(A),B,aa(fun(A,B),fun(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_4931_image__Suc__lessThan,axiom,
    ! [Nb: nat] : ( image(nat,nat,suc,set_ord_lessThan(nat,Nb)) = set_or1337092689740270186AtMost(nat,one_one(nat),Nb) ) ).

% image_Suc_lessThan
tff(fact_4932_image__Suc__atMost,axiom,
    ! [Nb: nat] : ( image(nat,nat,suc,set_ord_atMost(nat,Nb)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,Nb)) ) ).

% image_Suc_atMost
tff(fact_4933_prod_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),S3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T4)
         => ( aa(set(A),$o,finite_finite(A),S3)
           => ( ! [I2: A] :
                  ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T4),S3))
                 => ( aa(A,B,H,I2) = one_one(B) ) )
             => ( ! [I2: A] :
                    ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),T4))
                   => ( aa(A,B,G,I2) = one_one(B) ) )
               => ( ! [X: A] :
                      ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),T4))
                     => ( aa(A,B,G,X) = aa(A,B,H,X) ) )
                 => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),T4) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_4934_card__Diff__subset__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ).

% card_Diff_subset_Int
tff(fact_4935_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : ( set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),image(nat,nat,suc,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_4936_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : ( set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),image(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_4937_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : ( set_ord_lessThan(nat,aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),image(nat,nat,suc,set_ord_lessThan(nat,Nb))) ) ).

% lessThan_Suc_eq_insert_0
tff(fact_4938_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : ( set_ord_atMost(nat,aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),image(nat,nat,suc,set_ord_atMost(nat,Nb))) ) ).

% atMost_Suc_eq_insert_0
tff(fact_4939_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)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ly(fun(A,$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,aa(fun(A,B),fun(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,aa(fun(A,B),fun(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_4940_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)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_lz(fun(A,$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),aa(set(A),B,aa(fun(A,B),fun(set(A),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)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),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_4941_distinct__product__lists,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ! [X: list(A)] :
          ( member(list(A),X,aa(list(list(A)),set(list(A)),set2(list(A)),Xss))
         => distinct(A,X) )
     => distinct(list(A),product_lists(A,Xss)) ) ).

% distinct_product_lists
tff(fact_4942_sum__image__le,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),G: fun(C,B),F2: fun(A,C)] :
          ( aa(set(A),$o,finite_finite(A),I5)
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(C,B,G,aa(A,C,F2,I2))) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),image(A,C,F2,I5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,C),fun(A,B),comp(C,B,A,G),F2)),I5)) ) ) ) ).

% sum_image_le
tff(fact_4943_sum__div__partition,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A3: set(A),F2: fun(A,B),B2: B] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( divide_divide(B,aa(set(A),B,aa(fun(A,B),fun(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,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(B,fun(A,B),aTP_Lamp_ma(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,$o),aTP_Lamp_mb(fun(A,B),fun(B,fun(A,$o)),F2),B2))))),divide_divide(B,aa(set(A),B,aa(fun(A,B),fun(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,$o),aTP_Lamp_mc(fun(A,B),fun(B,fun(A,$o)),F2),B2)))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_4944_distinct__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(list(A),Xs)
     => ( ! [Ys3: list(A)] :
            ( member(list(A),Ys3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys3) )
       => ( ! [Ys3: list(A),Zs: list(A)] :
              ( member(list(A),Ys3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
             => ( member(list(A),Zs,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
               => ( ( Ys3 != Zs )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys3)),aa(list(A),set(A),set2(A),Zs)) = bot_bot(set(A)) ) ) ) )
         => distinct(A,concat(A,Xs)) ) ) ) ).

% distinct_concat
tff(fact_4945_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xb: A,Y: A] :
          ( image(A,A,aa(A,fun(A,A),times_times(A),C2),set_or1337092689740270186AtMost(A,Xb,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),Xb),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),Xb),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),Xb)),bot_bot(set(A))) ) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_4946_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xb: A,Y: A] :
          ( image(A,A,aTP_Lamp_md(A,fun(A,A),C2),set_or1337092689740270186AtMost(A,Xb,Y)) = $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),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),Xb),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),Xb),C2))),
              bot_bot(set(A)) ) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_4947_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ma: A,C2: A,A2: A,B2: A] :
          ( image(A,A,aa(A,fun(A,A),aTP_Lamp_me(A,fun(A,fun(A,A)),Ma),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)),Ma),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),Ma),A2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ma),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),Ma),B2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ma),A2)),C2))) ) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_4948_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ma: A,C2: A,A2: A,B2: A] :
          ( image(A,A,aa(A,fun(A,A),aTP_Lamp_mf(A,fun(A,fun(A,A)),Ma),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)),Ma),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),Ma),A2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ma),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),Ma),B2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ma),A2)),C2))) ) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_4949_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ma: A,C2: A,A2: A,B2: A] :
          ( image(A,A,aa(A,fun(A,A),aTP_Lamp_mg(A,fun(A,fun(A,A)),Ma),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)),Ma),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,Ma)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,Ma)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,Ma)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,Ma)),C2))) ) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_4950_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ma: A,C2: A,A2: A,B2: A] :
          ( image(A,A,aa(A,fun(A,A),aTP_Lamp_mh(A,fun(A,fun(A,A)),Ma),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)),Ma),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,Ma)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,B2,Ma)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,B2,Ma)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,Ma)),C2))) ) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_4951_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S3: set(A),R: set(B),G: fun(A,B),F2: fun(B,C)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( aa(set(B),$o,finite_finite(B),R)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,G,S3)),R)
             => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_mi(fun(A,B),fun(fun(B,C),fun(A,C)),G),F2)),S3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_mk(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S3),G),F2)),R) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_4952_card__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),nat,finite_card(list(A)),shuffles(A,Xs,Ys2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys2))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% card_disjoint_shuffles
tff(fact_4953_distinct__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(A,concat(A,Xs))
    <=> ( distinct(list(A),removeAll(list(A),nil(A),Xs))
        & ! [Ys4: list(A)] :
            ( member(list(A),Ys4,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys4) )
        & ! [Ys4: list(A),Zs2: list(A)] :
            ( ( member(list(A),Ys4,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
              & member(list(A),Zs2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
              & ( Ys4 != Zs2 ) )
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys4)),aa(list(A),set(A),set2(A),Zs2)) = bot_bot(set(A)) ) ) ) ) ).

% distinct_concat_iff
tff(fact_4954_rat__floor__lemma,axiom,
    ! [A2: int,B2: int] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),divide_divide(int,A2,B2))),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2))
      & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,A2,B2)),one_one(int)))) ) ).

% rat_floor_lemma
tff(fact_4955_minus__rat__cancel,axiom,
    ! [A2: int,B2: int] : ( aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),A2)),aa(int,int,uminus_uminus(int),B2)) = aa(int,rat,aa(int,fun(int,rat),fract,A2),B2) ) ).

% minus_rat_cancel
tff(fact_4956_inverse__rat,axiom,
    ! [A2: int,B2: int] : ( aa(rat,rat,inverse_inverse(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)) = aa(int,rat,aa(int,fun(int,rat),fract,B2),A2) ) ).

% inverse_rat
tff(fact_4957_Gcd__abs__eq,axiom,
    ! [K6: set(int)] : ( gcd_Gcd(int,image(int,int,abs_abs(int),K6)) = gcd_Gcd(int,K6) ) ).

% Gcd_abs_eq
tff(fact_4958_set__empty,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),set(A),set2(A),Xs) = bot_bot(set(A)) )
    <=> ( Xs = nil(A) ) ) ).

% set_empty
tff(fact_4959_set__empty2,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( bot_bot(set(A)) = aa(list(A),set(A),set2(A),Xs) )
    <=> ( Xs = nil(A) ) ) ).

% set_empty2
tff(fact_4960_length__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = zero_zero(nat) )
    <=> ( Xs = nil(A) ) ) ).

% length_0_conv
tff(fact_4961_replicate__empty,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( replicate(A,Nb,Xb) = nil(A) )
    <=> ( Nb = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_4962_empty__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( nil(A) = replicate(A,Nb,Xb) )
    <=> ( Nb = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_4963_horner__sum__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A] : ( aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),nil(B)) = zero_zero(A) ) ) ).

% horner_sum_simps(1)
tff(fact_4964_minus__rat,axiom,
    ! [A2: int,B2: int] : ( aa(rat,rat,uminus_uminus(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),A2)),B2) ) ).

% minus_rat
tff(fact_4965_concat__eq__Nil__conv,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ( concat(A,Xss) = nil(A) )
    <=> ! [X4: list(A)] :
          ( member(list(A),X4,aa(list(list(A)),set(list(A)),set2(list(A)),Xss))
         => ( X4 = nil(A) ) ) ) ).

% concat_eq_Nil_conv
tff(fact_4966_Nil__eq__concat__conv,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ( nil(A) = concat(A,Xss) )
    <=> ! [X4: list(A)] :
          ( member(list(A),X4,aa(list(list(A)),set(list(A)),set2(list(A)),Xss))
         => ( X4 = nil(A) ) ) ) ).

% Nil_eq_concat_conv
tff(fact_4967_length__greater__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ( Xs != nil(A) ) ) ).

% length_greater_0_conv
tff(fact_4968_Gcd__int__eq,axiom,
    ! [N5: set(nat)] : ( gcd_Gcd(int,image(nat,int,semiring_1_of_nat(int),N5)) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,N5)) ) ).

% Gcd_int_eq
tff(fact_4969_Gcd__nat__abs__eq,axiom,
    ! [K6: set(int)] : ( gcd_Gcd(nat,image(int,nat,aTP_Lamp_ml(int,nat),K6)) = aa(int,nat,nat2,gcd_Gcd(int,K6)) ) ).

% Gcd_nat_abs_eq
tff(fact_4970_less__rat,axiom,
    ! [B2: int,D3: int,A2: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D3))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A2),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3))) ) ) ) ).

% less_rat
tff(fact_4971_add__rat,axiom,
    ! [B2: int,D3: int,A2: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D3)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),A2),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3)) ) ) ) ).

% add_rat
tff(fact_4972_le__rat,axiom,
    ! [B2: int,D3: int,A2: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D3))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A2),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3))) ) ) ) ).

% le_rat
tff(fact_4973_diff__rat,axiom,
    ! [B2: int,D3: int,A2: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D3)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),A2),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3)) ) ) ) ).

% diff_rat
tff(fact_4974_sgn__rat,axiom,
    ! [A2: int,B2: int] : ( aa(rat,rat,sgn_sgn(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)) = aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),A2)),aa(int,int,sgn_sgn(int),B2))) ) ).

% sgn_rat
tff(fact_4975_eq__rat_I3_J,axiom,
    ! [A2: int,C2: int] : ( aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),A2) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),C2) ) ).

% eq_rat(3)
tff(fact_4976_eq__rat_I2_J,axiom,
    ! [A2: int] : ( aa(int,rat,aa(int,fun(int,rat),fract,A2),zero_zero(int)) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ) ).

% eq_rat(2)
tff(fact_4977_Rat__induct__pos,axiom,
    ! [P: fun(rat,$o),Q4: rat] :
      ( ! [A4: int,B4: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B4)
         => aa(rat,$o,P,aa(int,rat,aa(int,fun(int,rat),fract,A4),B4)) )
     => aa(rat,$o,P,Q4) ) ).

% Rat_induct_pos
tff(fact_4978_eq__rat_I1_J,axiom,
    ! [B2: int,D3: int,A2: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( ( aa(int,rat,aa(int,fun(int,rat),fract,A2),B2) = aa(int,rat,aa(int,fun(int,rat),fract,C2),D3) )
        <=> ( aa(int,int,aa(int,fun(int,int),times_times(int),A2),D3) = aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2) ) ) ) ) ).

% eq_rat(1)
tff(fact_4979_mult__rat__cancel,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero(int) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),times_times(int),C2),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2)) = aa(int,rat,aa(int,fun(int,rat),fract,A2),B2) ) ) ).

% mult_rat_cancel
tff(fact_4980_Fract__of__nat__eq,axiom,
    ! [K: nat] : ( aa(int,rat,aa(int,fun(int,rat),fract,aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) = aa(nat,rat,semiring_1_of_nat(rat),K) ) ).

% Fract_of_nat_eq
tff(fact_4981_rat__number__collapse_I1_J,axiom,
    ! [K: int] : ( aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),K) = zero_zero(rat) ) ).

% rat_number_collapse(1)
tff(fact_4982_rat__number__collapse_I6_J,axiom,
    ! [K: int] : ( aa(int,rat,aa(int,fun(int,rat),fract,K),zero_zero(int)) = zero_zero(rat) ) ).

% rat_number_collapse(6)
tff(fact_4983_empty__set,axiom,
    ! [A: $tType] : ( bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ) ).

% empty_set
tff(fact_4984_list_Osize_I3_J,axiom,
    ! [A: $tType] : ( aa(list(A),nat,size_size(list(A)),nil(A)) = zero_zero(nat) ) ).

% list.size(3)
tff(fact_4985_One__rat__def,axiom,
    one_one(rat) = aa(int,rat,aa(int,fun(int,rat),fract,one_one(int)),one_one(int)) ).

% One_rat_def
tff(fact_4986_length__shuffles,axiom,
    ! [A: $tType,Zs3: list(A),Xs: list(A),Ys2: list(A)] :
      ( member(list(A),Zs3,shuffles(A,Xs,Ys2))
     => ( aa(list(A),nat,size_size(list(A)),Zs3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys2)) ) ) ).

% length_shuffles
tff(fact_4987_replicate__0,axiom,
    ! [A: $tType,Xb: A] : ( replicate(A,zero_zero(nat),Xb) = nil(A) ) ).

% replicate_0
tff(fact_4988_Fract__of__int__eq,axiom,
    ! [K: int] : ( aa(int,rat,aa(int,fun(int,rat),fract,K),one_one(int)) = aa(int,rat,ring_1_of_int(rat),K) ) ).

% Fract_of_int_eq
tff(fact_4989_list_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xb: fun(A,nat)] : ( size_list(A,Xb,nil(A)) = zero_zero(nat) ) ).

% list.size_gen(1)
tff(fact_4990_count__list_Osimps_I1_J,axiom,
    ! [A: $tType,Y: A] : ( aa(A,nat,count_list(A,nil(A)),Y) = zero_zero(nat) ) ).

% count_list.simps(1)
tff(fact_4991_Fract__of__int__quotient,axiom,
    ! [K: int,L: int] : ( aa(int,rat,aa(int,fun(int,rat),fract,K),L) = divide_divide(rat,aa(int,rat,ring_1_of_int(rat),K),aa(int,rat,ring_1_of_int(rat),L)) ) ).

% Fract_of_int_quotient
tff(fact_4992_Zero__rat__def,axiom,
    zero_zero(rat) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ).

% Zero_rat_def
tff(fact_4993_image__int__atLeastAtMost,axiom,
    ! [A2: nat,B2: nat] : ( 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_4994_rat__number__expand_I3_J,axiom,
    ! [K: num] : ( aa(num,rat,numeral_numeral(rat),K) = aa(int,rat,aa(int,fun(int,rat),fract,aa(num,int,numeral_numeral(int),K)),one_one(int)) ) ).

% rat_number_expand(3)
tff(fact_4995_rat__number__collapse_I3_J,axiom,
    ! [W: num] : ( aa(int,rat,aa(int,fun(int,rat),fract,aa(num,int,numeral_numeral(int),W)),one_one(int)) = aa(num,rat,numeral_numeral(rat),W) ) ).

% rat_number_collapse(3)
tff(fact_4996_image__int__atLeastLessThan,axiom,
    ! [A2: nat,B2: nat] : ( 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_4997_zero__less__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A2) ) ) ).

% zero_less_Fract_iff
tff(fact_4998_Fract__less__zero__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),zero_zero(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int)) ) ) ).

% Fract_less_zero_iff
tff(fact_4999_Fract__less__one__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),one_one(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),B2) ) ) ).

% Fract_less_one_iff
tff(fact_5000_one__less__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),A2) ) ) ).

% one_less_Fract_iff
tff(fact_5001_rat__number__collapse_I5_J,axiom,
    aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) = aa(rat,rat,uminus_uminus(rat),one_one(rat)) ).

% rat_number_collapse(5)
tff(fact_5002_Fract__add__one,axiom,
    ! [Nb: int,Ma: int] :
      ( ( Nb != zero_zero(int) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ma),Nb)),Nb) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(int,rat,aa(int,fun(int,rat),fract,Ma),Nb)),one_one(rat)) ) ) ).

% Fract_add_one
tff(fact_5003_distinct__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs3: list(A)] :
      ( distinct(A,Xs)
     => ( distinct(A,Ys2)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
         => ( member(list(A),Zs3,shuffles(A,Xs,Ys2))
           => distinct(A,Zs3) ) ) ) ) ).

% distinct_disjoint_shuffles
tff(fact_5004_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S3: set(int)] :
      ( ~ aa(set(int),$o,finite_finite(int),S3)
    <=> ~ aa(set(nat),$o,finite_finite(nat),image(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int)),S3)) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_5005_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : ( image(int,int,aTP_Lamp_mm(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_5006_zero__le__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),zero_zero(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2) ) ) ).

% zero_le_Fract_iff
tff(fact_5007_Fract__le__zero__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),zero_zero(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int)) ) ) ).

% Fract_le_zero_iff
tff(fact_5008_Fract__le__one__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),one_one(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),B2) ) ) ).

% Fract_le_one_iff
tff(fact_5009_one__le__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),A2) ) ) ).

% one_le_Fract_iff
tff(fact_5010_rat__number__collapse_I4_J,axiom,
    ! [W: num] : ( aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),one_one(int)) = aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),W)) ) ).

% rat_number_collapse(4)
tff(fact_5011_rat__number__expand_I5_J,axiom,
    ! [K: num] : ( aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ) ).

% rat_number_expand(5)
tff(fact_5012_Gcd__int__def,axiom,
    ! [K6: set(int)] : ( gcd_Gcd(int,K6) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,image(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int)),K6))) ) ).

% Gcd_int_def
tff(fact_5013_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) = image(nat,int,semiring_1_of_nat(int),set_ord_lessThan(nat,aa(int,nat,nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_5014_ring__1__class_Oof__int__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_of_int(A) = aa(fun(product_prod(nat,nat),A),fun(int,A),map_fun(int,product_prod(nat,nat),A,A,rep_Integ,id(A)),product_case_prod(nat,nat,A,aTP_Lamp_lb(nat,fun(nat,A)))) ) ) ).

% ring_1_class.of_int_def
tff(fact_5015_take__bit__numeral__minus__numeral__int,axiom,
    ! [Ma: num,Nb: num] : ( bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),Ma),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = case_option(int,num,zero_zero(int),aTP_Lamp_mn(num,fun(num,int),Ma),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),Ma),Nb)) ) ).

% take_bit_numeral_minus_numeral_int
tff(fact_5016_positive__rat,axiom,
    ! [A2: int,B2: int] :
      ( aa(rat,$o,positive,aa(int,rat,aa(int,fun(int,rat),fract,A2),B2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2)) ) ).

% positive_rat
tff(fact_5017_of__nat__eq__id,axiom,
    semiring_1_of_nat(nat) = id(nat) ).

% of_nat_eq_id
tff(fact_5018_id__funpow,axiom,
    ! [A: $tType,Nb: nat] : ( aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),id(A)) = id(A) ) ).

% id_funpow
tff(fact_5019_take__bit__num__simps_I1_J,axiom,
    ! [Ma: num] : ( bit_take_bit_num(zero_zero(nat),Ma) = none(num) ) ).

% take_bit_num_simps(1)
tff(fact_5020_boolean__algebra__class_Ominus__comp__minus,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ( aa(fun(A,A),fun(A,A),comp(A,A,A,uminus_uminus(A)),uminus_uminus(A)) = id(A) ) ) ).

% boolean_algebra_class.minus_comp_minus
tff(fact_5021_group__add__class_Ominus__comp__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ( aa(fun(A,A),fun(A,A),comp(A,A,A,uminus_uminus(A)),uminus_uminus(A)) = id(A) ) ) ).

% group_add_class.minus_comp_minus
tff(fact_5022_push__bit__0__id,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se4730199178511100633sh_bit(A,zero_zero(nat)) = id(A) ) ) ).

% push_bit_0_id
tff(fact_5023_drop__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se4197421643247451524op_bit(A,zero_zero(nat)) = id(A) ) ) ).

% drop_bit_0
tff(fact_5024_take__bit__num__simps_I2_J,axiom,
    ! [Nb: nat] : ( bit_take_bit_num(aa(nat,nat,suc,Nb),one2) = aa(num,option(num),some(num),one2) ) ).

% take_bit_num_simps(2)
tff(fact_5025_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: num,Nb: num] : ( bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),Ma),aa(num,A,numeral_numeral(A),Nb)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),Ma),Nb)) ) ) ).

% take_bit_numeral_numeral
tff(fact_5026_funpow__simps__right_I1_J,axiom,
    ! [A: $tType,F2: fun(A,A)] : ( aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F2) = id(A) ) ).

% funpow_simps_right(1)
tff(fact_5027_less__int__def,axiom,
    ord_less(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(int,fun(int,$o)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(int,$o),rep_Integ,map_fun(int,product_prod(nat,nat),$o,$o,rep_Integ,id($o))),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_ld(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_int_def
tff(fact_5028_less__eq__int__def,axiom,
    ord_less_eq(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(int,fun(int,$o)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(int,$o),rep_Integ,map_fun(int,product_prod(nat,nat),$o,$o,rep_Integ,id($o))),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_lf(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_eq_int_def
tff(fact_5029_nat__def,axiom,
    nat2 = aa(fun(product_prod(nat,nat),nat),fun(int,nat),map_fun(int,product_prod(nat,nat),nat,nat,rep_Integ,id(nat)),product_case_prod(nat,nat,nat,minus_minus(nat))) ).

% nat_def
tff(fact_5030_Rat_Opositive__minus,axiom,
    ! [Xb: rat] :
      ( ~ aa(rat,$o,positive,Xb)
     => ( ( Xb != zero_zero(rat) )
       => aa(rat,$o,positive,aa(rat,rat,uminus_uminus(rat),Xb)) ) ) ).

% Rat.positive_minus
tff(fact_5031_take__bit__num__eq__None__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,Nb: num] :
          ( ( bit_take_bit_num(Ma,Nb) = none(num) )
         => ( bit_se2584673776208193580ke_bit(A,Ma,aa(num,A,numeral_numeral(A),Nb)) = zero_zero(A) ) ) ) ).

% take_bit_num_eq_None_imp
tff(fact_5032_take__bit__num__def,axiom,
    ! [Nb: nat,Ma: num] :
      ( bit_take_bit_num(Nb,Ma) = $ite(bit_se2584673776208193580ke_bit(nat,Nb,aa(num,nat,numeral_numeral(nat),Ma)) = zero_zero(nat),none(num),aa(num,option(num),some(num),aa(nat,num,num_of_nat,bit_se2584673776208193580ke_bit(nat,Nb,aa(num,nat,numeral_numeral(nat),Ma))))) ) ).

% take_bit_num_def
tff(fact_5033_and__minus__numerals_I7_J,axiom,
    ! [Nb: num,Ma: 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(Nb)))),aa(num,int,numeral_numeral(int),Ma)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Ma,bitM(Nb))) ) ).

% and_minus_numerals(7)
tff(fact_5034_and__minus__numerals_I3_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Nb)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Ma,bitM(Nb))) ) ).

% and_minus_numerals(3)
tff(fact_5035_and__minus__numerals_I4_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Nb)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Ma,bit0(Nb))) ) ).

% and_minus_numerals(4)
tff(fact_5036_take__bit__num__simps_I4_J,axiom,
    ! [Nb: nat,Ma: num] : ( bit_take_bit_num(aa(nat,nat,suc,Nb),aa(num,num,bit1,Ma)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Nb,Ma))) ) ).

% take_bit_num_simps(4)
tff(fact_5037_take__bit__num__simps_I3_J,axiom,
    ! [Nb: nat,Ma: num] : ( bit_take_bit_num(aa(nat,nat,suc,Nb),bit0(Ma)) = case_option(option(num),num,none(num),aTP_Lamp_mo(num,option(num)),bit_take_bit_num(Nb,Ma)) ) ).

% take_bit_num_simps(3)
tff(fact_5038_and__minus__numerals_I8_J,axiom,
    ! [Nb: num,Ma: 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,Nb)))),aa(num,int,numeral_numeral(int),Ma)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Ma,bit0(Nb))) ) ).

% and_minus_numerals(8)
tff(fact_5039_and__not__num__eq__None__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( bit_and_not_num(Ma,Nb) = none(num) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb))) = zero_zero(int) ) ) ).

% and_not_num_eq_None_iff
tff(fact_5040_int__numeral__not__and__num,axiom,
    ! [Ma: num,Nb: 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),Ma))),aa(num,int,numeral_numeral(int),Nb)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Nb,Ma)) ) ).

% int_numeral_not_and_num
tff(fact_5041_int__numeral__and__not__num,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Ma,Nb)) ) ).

% int_numeral_and_not_num
tff(fact_5042_measure__function__int,axiom,
    fun_is_measure(int,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int))) ).

% measure_function_int
tff(fact_5043_nth__image,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),L),aa(list(A),nat,size_size(list(A)),Xs))
     => ( image(nat,A,nth(A,Xs),set_or7035219750837199246ssThan(nat,zero_zero(nat),L)) = aa(list(A),set(A),set2(A),take(A,L,Xs)) ) ) ).

% nth_image
tff(fact_5044_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Nb: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,I)))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J))),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_5045_take0,axiom,
    ! [A: $tType,X3: list(A)] : ( take(A,zero_zero(nat),X3) = nil(A) ) ).

% take0
tff(fact_5046_take__eq__Nil,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( take(A,Nb,Xs) = nil(A) )
    <=> ( ( Nb = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil
tff(fact_5047_take__eq__Nil2,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( nil(A) = take(A,Nb,Xs) )
    <=> ( ( Nb = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil2
tff(fact_5048_nth__take,axiom,
    ! [A: $tType,I: nat,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Nb)
     => ( aa(nat,A,nth(A,take(A,Nb,Xs)),I) = aa(nat,A,nth(A,Xs),I) ) ) ).

% nth_take
tff(fact_5049_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(list(A),set(A),set2(A),linord4507533701916653071of_set(A,A3)) = A3 ) ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_5050_in__set__takeD,axiom,
    ! [A: $tType,Xb: A,Nb: nat,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),take(A,Nb,Xs)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% in_set_takeD
tff(fact_5051_take__0,axiom,
    ! [A: $tType,Xs: list(A)] : ( take(A,zero_zero(nat),Xs) = nil(A) ) ).

% take_0
tff(fact_5052_set__take__subset,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,Nb,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_take_subset
tff(fact_5053_set__take__subset__set__take,axiom,
    ! [A: $tType,Ma: nat,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,Ma,Xs))),aa(list(A),set(A),set2(A),take(A,Nb,Xs))) ) ).

% set_take_subset_set_take
tff(fact_5054_nth__take__lemma,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Ys2: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Ys2))
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),K)
             => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys2),I2) ) )
         => ( take(A,K,Xs) = take(A,K,Ys2) ) ) ) ) ).

% nth_take_lemma
tff(fact_5055_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Nb: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J))),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_5056_lex__take__index,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2),lex(A,R2))
     => ~ ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Ys2))
             => ( ( take(A,I2,Xs) = take(A,I2,Ys2) )
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Ys2),I2)),R2) ) ) ) ) ).

% lex_take_index
tff(fact_5057_mask__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat,Ma: nat] : ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ma)) = 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),Ma),Nb))),one_one(A)) ) ) ).

% mask_mod_exp
tff(fact_5058_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_5059_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_5060_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_5061_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_5062_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_5063_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_5064_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_5065_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_5066_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Y) ) ) ) ).

% min_less_iff_conj
tff(fact_5067_min__Suc__Suc,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,Ma)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Ma),Nb)) ) ).

% min_Suc_Suc
tff(fact_5068_min__0L,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),Nb) = zero_zero(nat) ) ).

% min_0L
tff(fact_5069_min__0R,axiom,
    ! [Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Nb),zero_zero(nat)) = zero_zero(nat) ) ).

% min_0R
tff(fact_5070_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V2: 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),V2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)),aa(num,A,numeral_numeral(A),U),aa(num,A,numeral_numeral(A),V2)) ) ) ).

% min_number_of(1)
tff(fact_5071_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),Xb)),zero_zero(A)) = zero_zero(A) ) ) ).

% min_0_1(4)
tff(fact_5072_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : ( aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Xb)) = zero_zero(A) ) ) ).

% min_0_1(3)
tff(fact_5073_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_5074_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_5075_min__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),Xb)),one_one(A)) = one_one(A) ) ) ).

% min_0_1(6)
tff(fact_5076_min__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : ( aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),aa(num,A,numeral_numeral(A),Xb)) = one_one(A) ) ) ).

% min_0_1(5)
tff(fact_5077_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or3652927894154168847AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),U) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_5078_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_5079_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_5080_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite(A),set_or3652927894154168847AtMost(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ioc_iff
tff(fact_5081_image__add__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [C2: A,A2: A,B2: A] : ( image(A,A,aa(A,fun(A,A),plus_plus(A),C2),set_or3652927894154168847AtMost(A,A2,B2)) = set_or3652927894154168847AtMost(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)) ) ) ).

% image_add_greaterThanAtMost
tff(fact_5082_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : ( aa(set(nat),nat,finite_card(nat),set_or3652927894154168847AtMost(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),L) ) ).

% card_greaterThanAtMost
tff(fact_5083_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = $ite(aa(A,$o,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),V2))),aa(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% min_number_of(2)
tff(fact_5084_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = $ite(aa(A,$o,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),V2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) ) ) ).

% min_number_of(3)
tff(fact_5085_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = $ite(aa(A,$o,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),V2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% min_number_of(4)
tff(fact_5086_image__diff__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A,B2: A] : ( image(A,A,aa(A,fun(A,A),minus_minus(A),C2),set_or7035219750837199246ssThan(A,A2,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)) ) ) ).

% image_diff_atLeastLessThan
tff(fact_5087_image__minus__const__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A,B2: A] : ( image(A,A,aa(A,fun(A,A),minus_minus(A),C2),set_or3652927894154168847AtMost(A,A2,B2)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)) ) ) ).

% image_minus_const_greaterThanAtMost
tff(fact_5088_image__uminus__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Y: A] : ( image(A,A,uminus_uminus(A),set_or3652927894154168847AtMost(A,Xb,Y)) = set_or7035219750837199246ssThan(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),Xb)) ) ) ).

% image_uminus_greaterThanAtMost
tff(fact_5089_image__uminus__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Y: A] : ( image(A,A,uminus_uminus(A),set_or7035219750837199246ssThan(A,Xb,Y)) = set_or3652927894154168847AtMost(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),Xb)) ) ) ).

% image_uminus_atLeastLessThan
tff(fact_5090_min__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),pred_numeral(K)),Nb)) ) ).

% min_numeral_Suc
tff(fact_5091_min__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Nb),pred_numeral(K))) ) ).

% min_Suc_numeral
tff(fact_5092_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: nat,Y: nat] : ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Xb),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ) ).

% of_nat_min
tff(fact_5093_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_5094_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_5095_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_5096_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X3: A,Xa: A] :
          ( aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa),X3,Xa) ) ) ).

% min_def_raw
tff(fact_5097_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z) ) ) ) ).

% min_less_iff_disj
tff(fact_5098_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_5099_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_5100_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_5101_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_5102_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
           => aa(A,$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),D3)) ) ) ) ).

% min.mono
tff(fact_5103_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_5104_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_5105_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_5106_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_5107_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_5108_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_5109_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_5110_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_5111_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_5112_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_5113_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_5114_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% min_le_iff_disj
tff(fact_5115_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_5116_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_5117_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_5118_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_5119_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: 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),Xb),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z)) ) ) ).

% min_diff_distrib_left
tff(fact_5120_min__diff,axiom,
    ! [Ma: nat,I: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),I)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),I)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Ma),Nb)),I) ) ).

% min_diff
tff(fact_5121_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: A,Y: A,Z: A] : ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),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),Xb),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z)) ) ) ).

% min_add_distrib_right
tff(fact_5122_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: 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),Xb),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ) ).

% min_add_distrib_left
tff(fact_5123_nat__mult__min__left,axiom,
    ! [Ma: nat,Nb: nat,Q4: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Ma),Nb)),Q4) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Q4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q4)) ) ).

% nat_mult_min_left
tff(fact_5124_nat__mult__min__right,axiom,
    ! [Ma: nat,Nb: nat,Q4: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Nb),Q4)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Q4)) ) ).

% nat_mult_min_right
tff(fact_5125_of__int__min,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int,Y: int] : ( aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_min(int),Xb),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(int,A,ring_1_of_int(A),Xb)),aa(int,A,ring_1_of_int(A),Y)) ) ) ).

% of_int_min
tff(fact_5126_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_5127_inf__min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( inf_inf(A) = ord_min(A) ) ) ).

% inf_min
tff(fact_5128_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% minus_max_eq_min
tff(fact_5129_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% minus_min_eq_max
tff(fact_5130_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : ( set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,L),U) = set_or3652927894154168847AtMost(nat,L,U) ) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_5131_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite(A),set_or3652927894154168847AtMost(A,A2,B2)) ) ) ).

% infinite_Ioc
tff(fact_5132_concat__bit__assoc__sym,axiom,
    ! [Ma: nat,Nb: nat,K: int,L: int,R2: int] : ( bit_concat_bit(Ma,bit_concat_bit(Nb,K,L),R2) = bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Ma),Nb),K,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb),L,R2)) ) ).

% concat_bit_assoc_sym
tff(fact_5133_take__bit__concat__bit__eq,axiom,
    ! [Ma: nat,Nb: nat,K: int,L: int] : ( bit_se2584673776208193580ke_bit(int,Ma,bit_concat_bit(Nb,K,L)) = bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Ma),Nb),K,bit_se2584673776208193580ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb),L)) ) ).

% take_bit_concat_bit_eq
tff(fact_5134_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A,P3: A] :
          ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3))) ) ) ).

% min_mult_distrib_right
tff(fact_5135_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A,P3: A] :
          ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3))) ) ) ).

% max_mult_distrib_right
tff(fact_5136_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,Xb: A,Y: A] :
          ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y))) ) ) ).

% min_mult_distrib_left
tff(fact_5137_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,Xb: A,Y: A] :
          ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y))) ) ) ).

% max_mult_distrib_left
tff(fact_5138_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,P3: A] :
          ( divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),divide_divide(A,Xb,P3)),divide_divide(A,Y,P3)),aa(A,A,aa(A,fun(A,A),ord_max(A),divide_divide(A,Xb,P3)),divide_divide(A,Y,P3))) ) ) ).

% min_divide_distrib_right
tff(fact_5139_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,P3: A] :
          ( divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),divide_divide(A,Xb,P3)),divide_divide(A,Y,P3)),aa(A,A,aa(A,fun(A,A),ord_min(A),divide_divide(A,Xb,P3)),divide_divide(A,Y,P3))) ) ) ).

% max_divide_distrib_right
tff(fact_5140_min__Suc2,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Ma),aa(nat,nat,suc,Nb)) = case_nat(nat,zero_zero(nat),aTP_Lamp_mp(nat,fun(nat,nat),Nb),Ma) ) ).

% min_Suc2
tff(fact_5141_min__Suc1,axiom,
    ! [Nb: nat,Ma: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,Nb)),Ma) = case_nat(nat,zero_zero(nat),aTP_Lamp_mq(nat,fun(nat,nat),Nb),Ma) ) ).

% min_Suc1
tff(fact_5142_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Ma: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Ma,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Ma)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or3652927894154168847AtMost(nat,Ma,Nb))) ) ) ) ).

% sum.head
tff(fact_5143_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D3))
        <=> ( aa(A,$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),D3) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_5144_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D3))
        <=> ( aa(A,$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(A),B2),D3) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_5145_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D3))
        <=> ( aa(A,$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),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_5146_lexord__take__index__conv,axiom,
    ! [A: $tType,Xb: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xb),Y),lexord(A,R2))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xb)),aa(list(A),nat,size_size(list(A)),Y))
          & ( take(A,aa(list(A),nat,size_size(list(A)),Xb),Y) = Xb ) )
        | ? [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xb)),aa(list(A),nat,size_size(list(A)),Y)))
            & ( take(A,I3,Xb) = take(A,I3,Y) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,Xb),I3)),aa(nat,A,nth(A,Y),I3)),R2) ) ) ) ).

% lexord_take_index_conv
tff(fact_5147_listrel1__iff__update,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2),listrel1(A,R2))
    <=> ? [Y5: A,N4: nat] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,Xs),N4)),Y5),R2)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs))
          & ( Ys2 = list_update(A,Xs,N4,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_5148_lenlex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : ( lenlex(A,R2) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),$o,aTP_Lamp_mr(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ) ).

% lenlex_conv
tff(fact_5149_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : ( aa(set(int),nat,finite_card(int),set_or3652927894154168847AtMost(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)) ) ).

% card_greaterThanAtMost_int
tff(fact_5150_inf__int__def,axiom,
    inf_inf(int) = ord_min(int) ).

% inf_int_def
tff(fact_5151_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : ( set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or3652927894154168847AtMost(int,L,U) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_5152_lexord__partial__trans,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A)),Ys2: list(A),Zs3: list(A)] :
      ( ! [X: A,Y3: A,Z2: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y3),R2)
           => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y3),Z2),R2)
             => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Z2),R2) ) ) )
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2),lexord(A,R2))
       => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Ys2),Zs3),lexord(A,R2))
         => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Zs3),lexord(A,R2)) ) ) ) ).

% lexord_partial_trans
tff(fact_5153_find__Some__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Xb: A] :
      ( ( find(A,P,Xs) = aa(A,option(A),some(A),Xb) )
    <=> ? [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,$o,P,aa(nat,A,nth(A,Xs),I3))
          & ( Xb = aa(nat,A,nth(A,Xs),I3) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I3)
             => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_5154_find__Some__iff2,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),Xs: list(A)] :
      ( ( aa(A,option(A),some(A),Xb) = find(A,P,Xs) )
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(A,$o,P,aa(nat,A,nth(A,Xs),I3))
          & ( Xb = aa(nat,A,nth(A,Xs),I3) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I3)
             => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_5155_nth__rotate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rotate(A,Ma),Xs)),Nb) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate
tff(fact_5156_rotate0,axiom,
    ! [A: $tType] : ( rotate(A,zero_zero(nat)) = id(list(A)) ) ).

% rotate0
tff(fact_5157_set__rotate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),rotate(A,Nb),Xs)) = aa(list(A),set(A),set2(A),Xs) ) ).

% set_rotate
tff(fact_5158_rotate__Suc,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : ( aa(list(A),list(A),rotate(A,aa(nat,nat,suc,Nb)),Xs) = rotate1(A,aa(list(A),list(A),rotate(A,Nb),Xs)) ) ).

% rotate_Suc
tff(fact_5159_rotate__length01,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => ( aa(list(A),list(A),rotate(A,Nb),Xs) = Xs ) ) ).

% rotate_length01
tff(fact_5160_rotate__id,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( modulo_modulo(nat,Nb,aa(list(A),nat,size_size(list(A)),Xs)) = zero_zero(nat) )
     => ( aa(list(A),list(A),rotate(A,Nb),Xs) = Xs ) ) ).

% rotate_id
tff(fact_5161_rotate__add,axiom,
    ! [A: $tType,Ma: nat,Nb: nat] : ( rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),rotate(A,Ma)),rotate(A,Nb)) ) ).

% rotate_add
tff(fact_5162_rotate__rotate,axiom,
    ! [A: $tType,Ma: nat,Nb: nat,Xs: list(A)] : ( aa(list(A),list(A),rotate(A,Ma),aa(list(A),list(A),rotate(A,Nb),Xs)) = aa(list(A),list(A),rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)),Xs) ) ).

% rotate_rotate
tff(fact_5163_find__cong,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( Xs = Ys2 )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Ys2))
           => ( aa(A,$o,P,X)
            <=> aa(A,$o,Q,X) ) )
       => ( find(A,P,Xs) = find(A,Q,Ys2) ) ) ) ).

% find_cong
tff(fact_5164_find__None__iff2,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( none(A) = find(A,P,Xs) )
    <=> ~ ? [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P,X4) ) ) ).

% find_None_iff2
tff(fact_5165_find__None__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( find(A,P,Xs) = none(A) )
    <=> ~ ? [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P,X4) ) ) ).

% find_None_iff
tff(fact_5166_nth__zip,axiom,
    ! [A: $tType,B: $tType,I: nat,Xs: list(A),Ys2: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys2))
       => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys2)),I) = aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_5167_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K: nat,Xs: list(A),Xb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
         => ( aa(list(A),A,groups8242544230860333062m_list(A),list_update(A,Xs,K,Xb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),Xb)),aa(nat,A,nth(A,Xs),K)) ) ) ) ).

% sum_list_update
tff(fact_5168_vanishes__mult__bounded,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( ? [A8: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),A8)
          & ! [N: nat] : aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N))),A8) )
     => ( vanishes(Y6)
       => vanishes(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_ms(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y6)) ) ) ).

% vanishes_mult_bounded
tff(fact_5169_vanishes__const,axiom,
    ! [C2: rat] :
      ( vanishes(aTP_Lamp_mt(rat,fun(nat,rat),C2))
    <=> ( C2 = zero_zero(rat) ) ) ).

% vanishes_const
tff(fact_5170_sum__list_ONil,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ( aa(list(A),A,groups8242544230860333062m_list(A),nil(A)) = zero_zero(A) ) ) ).

% sum_list.Nil
tff(fact_5171_sum__list__eq__0__iff,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Ns: list(A)] :
          ( ( aa(list(A),A,groups8242544230860333062m_list(A),Ns) = zero_zero(A) )
        <=> ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Ns))
             => ( X4 = zero_zero(A) ) ) ) ) ).

% sum_list_eq_0_iff
tff(fact_5172_set__zip__rightD,axiom,
    ! [A: $tType,B: $tType,Xb: A,Y: B,Xs: list(A),Ys2: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Xb),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)))
     => member(B,Y,aa(list(B),set(B),set2(B),Ys2)) ) ).

% set_zip_rightD
tff(fact_5173_set__zip__leftD,axiom,
    ! [B: $tType,A: $tType,Xb: A,Y: B,Xs: list(A),Ys2: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Xb),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% set_zip_leftD
tff(fact_5174_in__set__zipE,axiom,
    ! [A: $tType,B: $tType,Xb: A,Y: B,Xs: list(A),Ys2: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Xb),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)))
     => ~ ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
         => ~ member(B,Y,aa(list(B),set(B),set2(B),Ys2)) ) ) ).

% in_set_zipE
tff(fact_5175_zip__same,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A2),B2),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Xs)))
    <=> ( member(A,A2,aa(list(A),set(A),set2(A),Xs))
        & ( A2 = B2 ) ) ) ).

% zip_same
tff(fact_5176_vanishes__minus,axiom,
    ! [X6: fun(nat,rat)] :
      ( vanishes(X6)
     => vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_mu(fun(nat,rat),fun(nat,rat)),X6)) ) ).

% vanishes_minus
tff(fact_5177_vanishes__add,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( vanishes(X6)
     => ( vanishes(Y6)
       => vanishes(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_mv(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y6)) ) ) ).

% vanishes_add
tff(fact_5178_member__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A,Xs: list(A)] :
          ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(list(A),A,groups8242544230860333062m_list(A),Xs)) ) ) ).

% member_le_sum_list
tff(fact_5179_vanishes__diff,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( vanishes(X6)
     => ( vanishes(Y6)
       => vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_mw(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y6)) ) ) ).

% vanishes_diff
tff(fact_5180_in__set__impl__in__set__zip2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),Y: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( member(B,Y,aa(list(B),set(B),set2(B),Ys2))
       => ~ ! [X: A] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_5181_in__set__impl__in__set__zip1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ~ ! [Y3: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Xb),Y3),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_5182_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(list(A),A,groups8242544230860333062m_list(A),Xs)) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_5183_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X) )
         => ( ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = zero_zero(A) )
          <=> ! [X4: A] :
                ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
               => ( X4 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_5184_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),zero_zero(A)) ) ) ).

% sum_list_nonpos
tff(fact_5185_sum__list__replicate,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat,C2: A] : ( aa(list(A),A,groups8242544230860333062m_list(A),replicate(A,Nb,C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),C2) ) ) ).

% sum_list_replicate
tff(fact_5186_distinct__sum__list__conv__Sum,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] :
          ( distinct(A,Xs)
         => ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_mx(A,A)),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% distinct_sum_list_conv_Sum
tff(fact_5187_elem__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [K: nat,Ns: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Ns))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Ns),K)),aa(list(A),A,groups8242544230860333062m_list(A),Ns)) ) ) ).

% elem_le_sum_list
tff(fact_5188_card__length__sum__list__rec,axiom,
    ! [Ma: nat,N5: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Ma)
     => ( aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),$o),aTP_Lamp_my(nat,fun(nat,fun(list(nat),$o)),Ma),N5))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),$o),aTP_Lamp_mz(nat,fun(nat,fun(list(nat),$o)),Ma),N5)))),aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),$o),aTP_Lamp_na(nat,fun(nat,fun(list(nat),$o)),Ma),N5)))) ) ) ).

% card_length_sum_list_rec
tff(fact_5189_card__length__sum__list,axiom,
    ! [Ma: nat,N5: nat] : ( aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),$o),aTP_Lamp_my(nat,fun(nat,fun(list(nat),$o)),Ma),N5))) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),Ma)),one_one(nat))),N5) ) ).

% card_length_sum_list
tff(fact_5190_vanishes__def,axiom,
    ! [X6: fun(nat,rat)] :
      ( vanishes(X6)
    <=> ! [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
         => ? [K3: nat] :
            ! [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N4)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N4))),R5) ) ) ) ).

% vanishes_def
tff(fact_5191_vanishesI,axiom,
    ! [X6: fun(nat,rat)] :
      ( ! [R4: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R4)
         => ? [K4: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),N)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N))),R4) ) )
     => vanishes(X6) ) ).

% vanishesI
tff(fact_5192_vanishesD,axiom,
    ! [X6: fun(nat,rat),R2: rat] :
      ( vanishes(X6)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R2)
       => ? [K2: nat] :
          ! [N3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N3)
           => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N3))),R2) ) ) ) ).

% vanishesD
tff(fact_5193_sum__list__sum__nth,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] : ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),nth(A,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% sum_list_sum_nth
tff(fact_5194_set__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] : ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)) = collect(product_prod(A,B),aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_nb(list(A),fun(list(B),fun(product_prod(A,B),$o)),Xs),Ys2)) ) ).

% set_zip
tff(fact_5195_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6)
     => ( aa(set(A),$o,finite_finite(A),X6)
       => ( aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,F2),Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_nc(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F2)),X6) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_5196_rcis__inverse,axiom,
    ! [R2: real,A2: real] : ( aa(complex,complex,inverse_inverse(complex),rcis(R2,A2)) = rcis(divide_divide(real,one_one(real),R2),aa(real,real,uminus_uminus(real),A2)) ) ).

% rcis_inverse
tff(fact_5197_map__eq__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),G: fun(B,A)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs) = aa(list(B),list(A),map(B,A,G),Xs) )
    <=> ! [X4: B] :
          ( member(B,X4,aa(list(B),set(B),set2(B),Xs))
         => ( aa(B,A,F2,X4) = aa(B,A,G,X4) ) ) ) ).

% map_eq_conv
tff(fact_5198_list_Oset__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),V2: list(B)] : ( aa(list(A),set(A),set2(A),aa(list(B),list(A),map(B,A,F2),V2)) = image(B,A,F2,aa(list(B),set(B),set2(B),V2)) ) ).

% list.set_map
tff(fact_5199_sum__list__0,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B)] : ( aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aTP_Lamp_nd(B,A)),Xs)) = zero_zero(A) ) ) ).

% sum_list_0
tff(fact_5200_nth__map,axiom,
    ! [B: $tType,A: $tType,Nb: nat,Xs: list(A),F2: fun(A,B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,B,nth(B,aa(list(A),list(B),map(A,B,F2),Xs)),Nb) = aa(A,B,F2,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% nth_map
tff(fact_5201_list_Omap__cong,axiom,
    ! [B: $tType,A: $tType,Xb: list(A),Ya: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( Xb = Ya )
     => ( ! [Z2: A] :
            ( member(A,Z2,aa(list(A),set(A),set2(A),Ya))
           => ( aa(A,B,F2,Z2) = aa(A,B,G,Z2) ) )
       => ( aa(list(A),list(B),map(A,B,F2),Xb) = aa(list(A),list(B),map(A,B,G),Ya) ) ) ) ).

% list.map_cong
tff(fact_5202_list_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,Xb: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ! [Z2: A] :
          ( member(A,Z2,aa(list(A),set(A),set2(A),Xb))
         => ( aa(A,B,F2,Z2) = aa(A,B,G,Z2) ) )
     => ( aa(list(A),list(B),map(A,B,F2),Xb) = aa(list(A),list(B),map(A,B,G),Xb) ) ) ).

% list.map_cong0
tff(fact_5203_list_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,Xb: list(A),Xaa: list(A),F2: fun(A,B),Fa: fun(A,B)] :
      ( ! [Z2: A,Za: A] :
          ( member(A,Z2,aa(list(A),set(A),set2(A),Xb))
         => ( member(A,Za,aa(list(A),set(A),set2(A),Xaa))
           => ( ( aa(A,B,F2,Z2) = aa(A,B,Fa,Za) )
             => ( Z2 = Za ) ) ) )
     => ( ( aa(list(A),list(B),map(A,B,F2),Xb) = aa(list(A),list(B),map(A,B,Fa),Xaa) )
       => ( Xb = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_5204_map__ext,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ( aa(A,B,F2,X) = aa(A,B,G,X) ) )
     => ( aa(list(A),list(B),map(A,B,F2),Xs) = aa(list(A),list(B),map(A,B,G),Xs) ) ) ).

% map_ext
tff(fact_5205_map__idI,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,A)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ( aa(A,A,F2,X) = X ) )
     => ( aa(list(A),list(A),map(A,A,F2),Xs) = Xs ) ) ).

% map_idI
tff(fact_5206_map__cong,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( Xs = Ys2 )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Ys2))
           => ( aa(A,B,F2,X) = aa(A,B,G,X) ) )
       => ( aa(list(A),list(B),map(A,B,F2),Xs) = aa(list(A),list(B),map(A,B,G),Ys2) ) ) ) ).

% map_cong
tff(fact_5207_ex__map__conv,axiom,
    ! [B: $tType,A: $tType,Ys2: list(B),F2: fun(A,B)] :
      ( ? [Xs3: list(A)] : ( Ys2 = aa(list(A),list(B),map(A,B,F2),Xs3) )
    <=> ! [X4: B] :
          ( member(B,X4,aa(list(B),set(B),set2(B),Ys2))
         => ? [Xa3: A] : ( X4 = aa(A,B,F2,Xa3) ) ) ) ).

% ex_map_conv
tff(fact_5208_exE__some,axiom,
    ! [A: $tType,P: fun(A,$o),C2: A] :
      ( ? [X_1: A] : aa(A,$o,P,X_1)
     => ( ( C2 = fChoice(A,P) )
       => aa(A,$o,P,C2) ) ) ).

% exE_some
tff(fact_5209_image__set,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : ( image(B,A,F2,aa(list(B),set(B),set2(B),Xs)) = aa(list(A),set(A),set2(A),aa(list(B),list(A),map(B,A,F2),Xs)) ) ).

% image_set
tff(fact_5210_sum__list__addf,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(B,A),G: fun(B,A),Xs: list(B)] : ( aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_cj(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F2),Xs))),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,G),Xs))) ) ) ).

% sum_list_addf
tff(fact_5211_sum__list__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),G: fun(B,A),Xs: list(B)] : ( aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ck(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),Xs)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F2),Xs))),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,G),Xs))) ) ) ).

% sum_list_subtractf
tff(fact_5212_uminus__sum__list__map,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),Xs: list(B)] : ( aa(A,A,uminus_uminus(A),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F2),Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),comp(A,A,B,uminus_uminus(A)),F2)),Xs)) ) ) ).

% uminus_sum_list_map
tff(fact_5213_size__list__conv__sum__list,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] : ( size_list(A,F2,Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,F2),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% size_list_conv_sum_list
tff(fact_5214_sum__list__Suc,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] : ( aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,aTP_Lamp_dn(fun(A,nat),fun(A,nat),F2)),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,F2),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% sum_list_Suc
tff(fact_5215_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),map(A,B,F2),Xs))),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),map(A,B,G),Xs))) ) ) ).

% sum_list_mono
tff(fact_5216_cis__rcis__eq,axiom,
    ! [A2: real] : ( cis(A2) = rcis(one_one(real),A2) ) ).

% cis_rcis_eq
tff(fact_5217_sum__list__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & strict9044650504122735259up_add(B) )
     => ! [Xs: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( ! [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Xs))
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,G,X)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),map(A,B,F2),Xs))),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),map(A,B,G),Xs))) ) ) ) ).

% sum_list_strict_mono
tff(fact_5218_sum_Odistinct__set__conv__list,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(A),G: fun(A,B)] :
          ( distinct(A,Xs)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(list(A),set(A),set2(A),Xs)) = aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),map(A,B,G),Xs)) ) ) ) ).

% sum.distinct_set_conv_list
tff(fact_5219_sum__list__distinct__conv__sum__set,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(A),F2: fun(A,B)] :
          ( distinct(A,Xs)
         => ( aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),map(A,B,F2),Xs)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% sum_list_distinct_conv_sum_set
tff(fact_5220_sum__list__map__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xb: A,Xs: list(A),F2: fun(A,B)] :
          ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
         => ( aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),map(A,B,F2),Xs)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,Xb)),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),map(A,B,F2),remove1(A,Xb,Xs)))) ) ) ) ).

% sum_list_map_remove1
tff(fact_5221_sum__list__triv,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [R2: A,Xs: list(B)] : ( aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aTP_Lamp_ci(A,fun(B,A),R2)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(list(B),nat,size_size(list(B)),Xs))),R2) ) ) ).

% sum_list_triv
tff(fact_5222_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : ( aa(list(A),set(A),set2(A),Xs) = collect(A,aTP_Lamp_ne(list(A),fun(A,$o),Xs)) ) ).

% set_conv_nth
tff(fact_5223_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] : ( aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,F2),Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(list(A),fun(A,nat),aTP_Lamp_nf(fun(A,nat),fun(list(A),fun(A,nat)),F2),Xs)),aa(list(A),set(A),set2(A),Xs)) ) ).

% sum_list_map_eq_sum_count
tff(fact_5224_length__product__lists,axiom,
    ! [A: $tType,Xss: list(list(A))] : ( aa(list(list(A)),nat,size_size(list(list(A))),product_lists(A,Xss)) = aa(nat,nat,foldr(nat,nat,times_times(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xss)),one_one(nat)) ) ).

% length_product_lists
tff(fact_5225_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xb,Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
         => ( Xaa = one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( ( Deg = Xaa )
                & $let(
                    n: nat,
                    n:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                    $let(
                      m2: nat,
                      m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                      ( ! [X4: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                         => vEBT_VEBT_valid(X4,n) )
                      & vEBT_VEBT_valid(Summary,m2)
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                      & case_option($o,product_prod(nat,nat),
                          ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                          & ! [X4: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X7) ) ),
                          product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ng(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_5226_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xb,Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
         => ( Xaa != one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ~ ( ( Deg = Xaa )
                  & $let(
                      n: nat,
                      n:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                      $let(
                        m2: nat,
                        m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                        ( ! [X4: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                           => vEBT_VEBT_valid(X4,n) )
                        & vEBT_VEBT_valid(Summary,m2)
                        & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                        & case_option($o,product_prod(nat,nat),
                            ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                            & ! [X4: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X7) ) ),
                            product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ng(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_5227_foldr__cong,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: A,L: list(B),K: list(B),F2: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
      ( ( A2 = B2 )
     => ( ( L = K )
       => ( ! [A4: A,X: B] :
              ( member(B,X,aa(list(B),set(B),set2(B),L))
             => ( aa(A,A,aa(B,fun(A,A),F2,X),A4) = aa(A,A,aa(B,fun(A,A),G,X),A4) ) )
         => ( aa(A,A,foldr(B,A,F2,L),A2) = aa(A,A,foldr(B,A,G,K),B2) ) ) ) ) ).

% foldr_cong
tff(fact_5228_sum__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A)] : ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = aa(A,A,foldr(A,A,plus_plus(A),Xs),zero_zero(A)) ) ) ).

% sum_list.eq_foldr
tff(fact_5229_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,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),Xs) = aa(A,A,foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_nh(fun(B,A),fun(A,fun(B,fun(A,A))),F2),A2),Xs),zero_zero(A)) ) ) ).

% horner_sum_foldr
tff(fact_5230_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Deg3: nat] :
      ( vEBT_VEBT_valid(vEBT_Node(Mima2,Dega,TreeLista,Summarya),Deg3)
    <=> ( ( Dega = Deg3 )
        & $let(
            n: nat,
            n:= divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2))),
            $let(
              m2: nat,
              m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Dega),n),
              ( ! [X4: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                 => vEBT_VEBT_valid(X4,n) )
              & vEBT_VEBT_valid(Summarya,m2)
              & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
              & case_option($o,product_prod(nat,nat),
                  ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X7)
                  & ! [X4: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X7) ) ),
                  product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ng(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Dega),TreeLista),Summarya),n),m2)),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_5231_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_valid(Xb,Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
         => ( (Y)
          <=> ( Xaa != one_one(nat) ) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( (Y)
              <=> ~ ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        $let(
                          m2: nat,
                          m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                          ( ! [X4: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => vEBT_VEBT_valid(X4,n) )
                          & vEBT_VEBT_valid(Summary,m2)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                              & ! [X4: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X7) ) ),
                              product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ng(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_5232_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_valid(Xb,Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Y)
                <=> ( Xaa = one_one(nat) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),Xaa)) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( ( (Y)
                  <=> ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $let(
                            m2: nat,
                            m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                            ( ! [X4: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => vEBT_VEBT_valid(X4,n) )
                            & vEBT_VEBT_valid(Summary,m2)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                                & ! [X4: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X7) ) ),
                                product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ng(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Mima,Deg,TreeList,Summary)),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_5233_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),Xaa))
               => ( Xaa != one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Mima,Deg,TreeList,Summary)),Xaa))
                 => ~ ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $let(
                            m2: nat,
                            m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                            ( ! [X4: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => vEBT_VEBT_valid(X4,n) )
                            & vEBT_VEBT_valid(Summary,m2)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                                & ! [X4: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X7) ) ),
                                product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ng(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_5234_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),Xaa))
               => ( Xaa = one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Mima,Deg,TreeList,Summary)),Xaa))
                 => ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        $let(
                          m2: nat,
                          m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                          ( ! [X4: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => vEBT_VEBT_valid(X4,n) )
                          & vEBT_VEBT_valid(Summary,m2)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                              & ! [X4: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X7) ) ),
                              product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ng(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_5235_list__eq__iff__zip__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs = Ys2 )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
        & ! [X4: product_prod(A,A)] :
            ( member(product_prod(A,A),X4,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Ys2)))
           => aa(product_prod(A,A),$o,product_case_prod(A,A,$o,fequal(A)),X4) ) ) ) ).

% list_eq_iff_zip_eq
tff(fact_5236_concat__eq__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys2: list(list(A))] :
      ( ! [X: product_prod(list(A),list(A))] :
          ( member(product_prod(list(A),list(A)),X,aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys2)))
         => aa(product_prod(list(A),list(A)),$o,product_case_prod(list(A),list(A),$o,aTP_Lamp_ni(list(A),fun(list(A),$o))),X) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys2) )
       => ( ( concat(A,Xs) = concat(A,Ys2) )
        <=> ( Xs = Ys2 ) ) ) ) ).

% concat_eq_concat_iff
tff(fact_5237_concat__injective,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys2: list(list(A))] :
      ( ( concat(A,Xs) = concat(A,Ys2) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys2) )
       => ( ! [X: product_prod(list(A),list(A))] :
              ( member(product_prod(list(A),list(A)),X,aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys2)))
             => aa(product_prod(list(A),list(A)),$o,product_case_prod(list(A),list(A),$o,aTP_Lamp_ni(list(A),fun(list(A),$o))),X) )
         => ( Xs = Ys2 ) ) ) ) ).

% concat_injective
tff(fact_5238_product__code,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B)] : ( product_product(A,B,aa(list(A),set(A),set2(A),Xs),aa(list(B),set(B),set2(B),Ys2)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_nj(list(B),fun(A,list(product_prod(A,B))),Ys2)),Xs))) ) ).

% product_code
tff(fact_5239_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] : ( aa(list($o),A,aa(A,fun(list($o),A),aa(fun($o,A),fun(A,fun(list($o),A)),groups4207007520872428315er_sum($o,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),bit0(one2))),aa(list(nat),list($o),map(nat,$o,bit_se5641148757651400278ts_bit(A,A2)),upt(zero_zero(nat),Nb))) = bit_se2584673776208193580ke_bit(A,Nb,A2) ) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_5240_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [F2: fun(nat,A),Ns: list(nat)] :
          ( ! [X: nat,Y3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Y3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,X)),aa(nat,A,F2,Y3)) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(nat),list(A),map(nat,A,F2),Ns))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_5241_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_5242_take__upt,axiom,
    ! [I: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Ma)),Nb)
     => ( take(nat,Ma,upt(I,Nb)) = upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Ma)) ) ) ).

% take_upt
tff(fact_5243_length__upt,axiom,
    ! [I: nat,J: nat] : ( aa(list(nat),nat,size_size(list(nat)),upt(I,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I) ) ).

% length_upt
tff(fact_5244_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_5245_strict__sorted__equal,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs)
         => ( sorted_wrt(A,ord_less(A),Ys2)
           => ( ( aa(list(A),set(A),set2(A),Ys2) = aa(list(A),set(A),set2(A),Xs) )
             => ( Ys2 = Xs ) ) ) ) ) ).

% strict_sorted_equal
tff(fact_5246_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : ( set_or7035219750837199246ssThan(nat,I,J) = aa(list(nat),set(nat),set2(nat),upt(I,J)) ) ).

% atLeastLessThan_upt
tff(fact_5247_sorted__wrt__mono__rel,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,fun(A,$o)),Q: fun(A,fun(A,$o))] :
      ( ! [X: A,Y3: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ( member(A,Y3,aa(list(A),set(A),set2(A),Xs))
           => ( aa(A,$o,aa(A,fun(A,$o),P,X),Y3)
             => aa(A,$o,aa(A,fun(A,$o),Q,X),Y3) ) ) )
     => ( sorted_wrt(A,P,Xs)
       => sorted_wrt(A,Q,Xs) ) ) ).

% sorted_wrt_mono_rel
tff(fact_5248_upt__0,axiom,
    ! [I: nat] : ( upt(I,zero_zero(nat)) = nil(nat) ) ).

% upt_0
tff(fact_5249_strict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less(A),nil(A)) ) ).

% strict_sorted_simps(1)
tff(fact_5250_strict__sorted__imp__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% strict_sorted_imp_sorted
tff(fact_5251_sorted__wrt__upt,axiom,
    ! [Ma: nat,Nb: nat] : sorted_wrt(nat,ord_less(nat),upt(Ma,Nb)) ).

% sorted_wrt_upt
tff(fact_5252_map__Suc__upt,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(list(nat),list(nat),map(nat,nat,suc),upt(Ma,Nb)) = upt(aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb)) ) ).

% map_Suc_upt
tff(fact_5253_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : sorted_wrt(A,ord_less(A),linord4507533701916653071of_set(A,A3)) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_5254_atLeastAtMost__upt,axiom,
    ! [Nb: nat,Ma: nat] : ( set_or1337092689740270186AtMost(nat,Nb,Ma) = aa(list(nat),set(nat),set2(nat),upt(Nb,aa(nat,nat,suc,Ma))) ) ).

% atLeastAtMost_upt
tff(fact_5255_atLeast__upt,axiom,
    ! [Nb: nat] : ( set_ord_lessThan(nat,Nb) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),Nb)) ) ).

% atLeast_upt
tff(fact_5256_interv__sum__list__conv__sum__set__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),Ma: nat,Nb: nat] : ( aa(list(A),A,groups8242544230860333062m_list(A),aa(list(nat),list(A),map(nat,A,F2),upt(Ma,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(list(nat),set(nat),set2(nat),upt(Ma,Nb))) ) ) ).

% interv_sum_list_conv_sum_set_nat
tff(fact_5257_sum__set__upt__conv__sum__list__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),Ma: nat,Nb: nat] : ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(list(nat),set(nat),set2(nat),upt(Ma,Nb))) = aa(list(A),A,groups8242544230860333062m_list(A),aa(list(nat),list(A),map(nat,A,F2),upt(Ma,Nb))) ) ) ).

% sum_set_upt_conv_sum_list_nat
tff(fact_5258_greaterThanAtMost__upt,axiom,
    ! [Nb: nat,Ma: nat] : ( set_or3652927894154168847AtMost(nat,Nb,Ma) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,Nb),aa(nat,nat,suc,Ma))) ) ).

% greaterThanAtMost_upt
tff(fact_5259_map__add__upt,axiom,
    ! [Nb: nat,Ma: nat] : ( aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_nk(nat,fun(nat,nat),Nb)),upt(zero_zero(nat),Ma)) = upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)) ) ).

% map_add_upt
tff(fact_5260_map__replicate__trivial,axiom,
    ! [A: $tType,Xb: A,I: nat] : ( aa(list(nat),list(A),map(nat,A,aTP_Lamp_nl(A,fun(nat,A),Xb)),upt(zero_zero(nat),I)) = replicate(A,I,Xb) ) ).

% map_replicate_trivial
tff(fact_5261_greaterThanLessThan__upt,axiom,
    ! [Nb: nat,Ma: nat] : ( set_or5935395276787703475ssThan(nat,Nb,Ma) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,Nb),Ma)) ) ).

% greaterThanLessThan_upt
tff(fact_5262_strict__sorted__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,ord_less(A),L)
        <=> ( sorted_wrt(A,ord_less_eq(A),L)
            & distinct(A,L) ) ) ) ).

% strict_sorted_iff
tff(fact_5263_sorted__distinct__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( sorted_wrt(A,ord_less_eq(A),Ys2)
             => ( distinct(A,Ys2)
               => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys2) )
                 => ( Xs = Ys2 ) ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_5264_sorted__wrt01,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,fun(A,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => sorted_wrt(A,P,Xs) ) ).

% sorted_wrt01
tff(fact_5265_sorted__wrt__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A),I: nat,J: nat] :
      ( sorted_wrt(A,P,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_5266_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A)] :
      ( sorted_wrt(A,P,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_5267_atMost__upto,axiom,
    ! [Nb: nat] : ( set_ord_atMost(nat,Nb) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,Nb))) ) ).

% atMost_upto
tff(fact_5268_enumerate__eq__zip,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : ( enumerate(A,Nb,Xs) = zip(nat,A,upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))),Xs) ) ).

% enumerate_eq_zip
tff(fact_5269_map__decr__upt,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_kv(nat,nat)),upt(aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = upt(Ma,Nb) ) ).

% map_decr_upt
tff(fact_5270_map__nth,axiom,
    ! [A: $tType,Xs: list(A)] : ( aa(list(nat),list(A),map(nat,A,nth(A,Xs)),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = Xs ) ).

% map_nth
tff(fact_5271_sorted01,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
         => sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted01
tff(fact_5272_sorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_5273_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ? [X: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X) = A3 )
              & sorted_wrt(A,ord_less_eq(A),X)
              & distinct(A,X)
              & ! [Y4: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y4) = A3 )
                    & sorted_wrt(A,ord_less_eq(A),Y4)
                    & distinct(A,Y4) )
                 => ( Y4 = X ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_5274_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( linord4507533701916653071of_set(A,aa(list(A),set(A),set2(A),Xs)) = Xs ) ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_5275_nth__map__upt,axiom,
    ! [A: $tType,I: nat,Nb: nat,Ma: nat,F2: fun(nat,A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma))
     => ( aa(nat,A,nth(A,aa(list(nat),list(A),map(nat,A,F2),upt(Ma,Nb))),I) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),I)) ) ) ).

% nth_map_upt
tff(fact_5276_sorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I3))) ) ) ) ).

% sorted_iff_nth_Suc
tff(fact_5277_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ~ ! [L2: list(A)] :
                ( sorted_wrt(A,ord_less(A),L2)
               => ( ( aa(list(A),set(A),set2(A),L2) = A3 )
                 => ( aa(list(A),nat,size_size(list(A)),L2) != aa(set(A),nat,finite_card(A),A3) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_5278_sorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_5279_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J)) ) ) ) ) ).

% sorted_nth_mono
tff(fact_5280_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),I: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(nat),nat,size_size(list(nat)),Ns))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,nth(nat,Ns),I)) ) ) ).

% sorted_wrt_less_idx
tff(fact_5281_enumerate__replicate__eq,axiom,
    ! [A: $tType,Nb: nat,Ma: nat,A2: A] : ( enumerate(A,Nb,replicate(A,Ma,A2)) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_nm(A,fun(nat,product_prod(nat,A)),A2)),upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Ma))) ) ).

% enumerate_replicate_eq
tff(fact_5282_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Ma: nat,F2: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma) )
     => ( ! [I2: nat] :
            ( aa(nat,$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,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),I2)) ) )
       => ( aa(list(nat),list(A),map(nat,A,F2),upt(Ma,Nb)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_5283_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),L: list(A)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ( sorted_wrt(A,ord_less(A),L)
              & ( aa(list(A),set(A),set2(A),L) = A3 )
              & ( aa(list(A),nat,size_size(list(A)),L) = aa(set(A),nat,finite_card(A),A3) ) )
          <=> ( linord4507533701916653071of_set(A,A3) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_5284_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),Nb: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( Nb = zero_zero(nat) ) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
           => ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I2)) = Nb ) )
       => ( transpose(A,Xs) = aa(list(nat),list(list(A)),map(nat,list(A),aTP_Lamp_no(list(list(A)),fun(nat,list(A)),Xs)),upt(zero_zero(nat),Nb)) ) ) ) ).

% transpose_rectangle
tff(fact_5285_folding__insort__key_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),S3)
       => ( aa(set(B),$o,finite_finite(B),A3)
         => ~ ! [L2: list(B)] :
                ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F2),L2))
               => ( ( aa(list(B),set(B),set2(B),L2) = A3 )
                 => ( aa(list(B),nat,size_size(list(B)),L2) != aa(set(B),nat,finite_card(B),A3) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_5286_listrel__iff__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs),Ys2),listrel(A,B,R2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [X4: product_prod(A,B)] :
            ( member(product_prod(A,B),X4,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)))
           => aa(product_prod(A,B),$o,product_case_prod(A,B,$o,aTP_Lamp_np(set(product_prod(A,B)),fun(A,fun(B,$o)),R2)),X4) ) ) ) ).

% listrel_iff_zip
tff(fact_5287_transpose__empty,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( ( transpose(A,Xs) = nil(list(A)) )
    <=> ! [X4: list(A)] :
          ( member(list(A),X4,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
         => ( X4 = nil(A) ) ) ) ).

% transpose_empty
tff(fact_5288_listrel__iff__nth,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs),Ys2),listrel(A,B,R2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs))
           => member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Xs),N4)),aa(nat,B,nth(B,Ys2),N4)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_5289_length__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = aa(nat,nat,foldr(list(A),nat,aTP_Lamp_nq(list(A),fun(nat,nat)),Xs),zero_zero(nat)) ) ).

% length_transpose
tff(fact_5290_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F2: fun(B,A),A3: set(B),L: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),S3)
       => ( aa(set(B),$o,finite_finite(B),A3)
         => ( ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F2),L))
              & ( aa(list(B),set(B),set2(B),L) = A3 )
              & ( aa(list(B),nat,size_size(list(B)),L) = aa(set(B),nat,finite_card(B),A3) ) )
          <=> ( sorted8670434370408473282of_set(A,B,Less_eq,F2,A3) = L ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_5291_length__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = $ite(Xs = nil(list(A)),zero_zero(nat),aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),zero_zero(nat)))) ) ) ).

% length_transpose_sorted
tff(fact_5292_folding__insort__key_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F2: fun(B,A),Xs: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs)),S3)
       => ( sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F2),Xs))
         => ( distinct(B,Xs)
           => ( sorted8670434370408473282of_set(A,B,Less_eq,F2,aa(list(B),set(B),set2(B),Xs)) = Xs ) ) ) ) ) ).

% folding_insort_key.idem_if_sorted_distinct
tff(fact_5293_set__rev,axiom,
    ! [A: $tType,Xs: list(A)] : ( aa(list(A),set(A),set2(A),rev(A,Xs)) = aa(list(A),set(A),set2(A),Xs) ) ).

% set_rev
tff(fact_5294_rotate__rev,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : ( aa(list(A),list(A),rotate(A,Nb),rev(A,Xs)) = rev(A,aa(list(A),list(A),rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),modulo_modulo(nat,Nb,aa(list(A),nat,size_size(list(A)),Xs)))),Xs)) ) ).

% rotate_rev
tff(fact_5295_rev__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,rev(A,Xs)),Nb) = 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,Nb))) ) ) ).

% rev_nth
tff(fact_5296_rev__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
     => ( rev(A,list_update(A,Xs,K,Y)) = list_update(A,rev(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),K)),one_one(nat)),Y) ) ) ).

% rev_update
tff(fact_5297_sorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I3))),aa(nat,A,nth(A,Xs),I3)) ) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_5298_sorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),J3)),aa(nat,A,nth(A,Xs),I3)) ) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_5299_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),J)),aa(nat,A,nth(A,Xs),I)) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_5300_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),S3)
       => ( aa(set(B),$o,finite_finite(B),A3)
         => ( aa(list(B),set(B),set2(B),sorted8670434370408473282of_set(A,B,Less_eq,F2,A3)) = A3 ) ) ) ) ).

% folding_insort_key.set_sorted_key_list_of_set
tff(fact_5301_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( aa(A,A,foldr(A,A,ord_max(A),Xs),Y) = $ite(Xs = nil(A),Y,aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xs),zero_zero(nat))),Y)) ) ) ) ).

% foldr_max_sorted
tff(fact_5302_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_nr(nat,fun(list(A),$o),I),Xs)))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J)),I) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_5303_transpose__column,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_ns(nat,fun(list(A),A),I)),filter2(list(A),aTP_Lamp_nr(nat,fun(list(A),$o),I),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I) ) ) ) ).

% transpose_column
tff(fact_5304_transpose__column__length,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_nr(nat,fun(list(A),$o),I),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I)) ) ) ) ).

% transpose_column_length
tff(fact_5305_filter__True,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X) )
     => ( filter2(A,P,Xs) = Xs ) ) ).

% filter_True
tff(fact_5306_set__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : ( aa(list(A),set(A),set2(A),filter2(A,P,Xs)) = collect(A,aa(list(A),fun(A,$o),aTP_Lamp_nt(fun(A,$o),fun(list(A),fun(A,$o)),P),Xs)) ) ).

% set_filter
tff(fact_5307_filter__False,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P,X) )
     => ( filter2(A,P,Xs) = nil(A) ) ) ).

% filter_False
tff(fact_5308_filter__id__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( filter2(A,P,Xs) = Xs )
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X4) ) ) ).

% filter_id_conv
tff(fact_5309_filter__cong,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( Xs = Ys2 )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Ys2))
           => ( aa(A,$o,P,X)
            <=> aa(A,$o,Q,X) ) )
       => ( filter2(A,P,Xs) = filter2(A,Q,Ys2) ) ) ) ).

% filter_cong
tff(fact_5310_empty__filter__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( nil(A) = filter2(A,P,Xs) )
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P,X4) ) ) ).

% empty_filter_conv
tff(fact_5311_filter__empty__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( filter2(A,P,Xs) = nil(A) )
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P,X4) ) ) ).

% filter_empty_conv
tff(fact_5312_filter__is__subset,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),filter2(A,P,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% filter_is_subset
tff(fact_5313_sum__length__filter__compl,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),filter2(A,P,Xs))),aa(list(A),nat,size_size(list(A)),filter2(A,aTP_Lamp_ae(fun(A,$o),fun(A,$o),P),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ) ).

% sum_length_filter_compl
tff(fact_5314_inter__set__filter,axiom,
    ! [A: $tType,A3: set(A),Xs: list(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),filter2(A,aTP_Lamp_a(set(A),fun(A,$o),A3),Xs)) ) ).

% inter_set_filter
tff(fact_5315_length__filter__less,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P,Xb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),filter2(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% length_filter_less
tff(fact_5316_sum__list__map__filter_H,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [F2: fun(B,A),P: fun(B,$o),Xs: list(B)] : ( aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F2),filter2(B,P,Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_nu(fun(B,A),fun(fun(B,$o),fun(B,A)),F2),P)),Xs)) ) ) ).

% sum_list_map_filter'
tff(fact_5317_sum__list__map__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(B)
     => ! [Xs: list(A),P: fun(A,$o),F2: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xs))
             => ( ~ aa(A,$o,P,X)
               => ( aa(A,B,F2,X) = zero_zero(B) ) ) )
         => ( aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),map(A,B,F2),filter2(A,P,Xs))) = aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),map(A,B,F2),Xs)) ) ) ) ).

% sum_list_map_filter
tff(fact_5318_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y: A] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),filter2(A,aTP_Lamp_nv(A,fun(A,$o),Y),Xs)) ) ).

% set_minus_filter_out
tff(fact_5319_filter__shuffles__disjoint1_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs3: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
     => ( member(list(A),Zs3,shuffles(A,Xs,Ys2))
       => ( filter2(A,aTP_Lamp_nw(list(A),fun(A,$o),Xs),Zs3) = Ys2 ) ) ) ).

% filter_shuffles_disjoint1(2)
tff(fact_5320_filter__shuffles__disjoint1_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs3: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
     => ( member(list(A),Zs3,shuffles(A,Xs,Ys2))
       => ( filter2(A,aTP_Lamp_nx(list(A),fun(A,$o),Xs),Zs3) = Xs ) ) ) ).

% filter_shuffles_disjoint1(1)
tff(fact_5321_filter__shuffles__disjoint2_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs3: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
     => ( member(list(A),Zs3,shuffles(A,Xs,Ys2))
       => ( filter2(A,aTP_Lamp_nw(list(A),fun(A,$o),Ys2),Zs3) = Xs ) ) ) ).

% filter_shuffles_disjoint2(2)
tff(fact_5322_filter__shuffles__disjoint2_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs3: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
     => ( member(list(A),Zs3,shuffles(A,Xs,Ys2))
       => ( filter2(A,aTP_Lamp_nx(list(A),fun(A,$o),Ys2),Zs3) = Ys2 ) ) ) ).

% filter_shuffles_disjoint2(1)
tff(fact_5323_length__filter__conv__card,axiom,
    ! [A: $tType,P3: fun(A,$o),Xs: list(A)] : ( aa(list(A),nat,size_size(list(A)),filter2(A,P3,Xs)) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(list(A),fun(nat,$o),aTP_Lamp_ny(fun(A,$o),fun(list(A),fun(nat,$o)),P3),Xs))) ) ).

% length_filter_conv_card
tff(fact_5324_distinct__length__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( distinct(A,Xs)
     => ( aa(list(A),nat,size_size(list(A)),filter2(A,P,Xs)) = aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),collect(A,P)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% distinct_length_filter
tff(fact_5325_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xss: list(list(B))] : ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_nz(list(B),fun(nat,nat)),Xss),zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_oa(list(B),fun(nat,nat)),filter2(list(B),aTP_Lamp_ob(list(B),$o),Xss)),zero_zero(nat)))) ) ).

% transpose_aux_max
tff(fact_5326_nth__transpose,axiom,
    ! [A: $tType,I: nat,Xs: list(list(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I) = aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_ns(nat,fun(list(A),A),I)),filter2(list(A),aTP_Lamp_nr(nat,fun(list(A),$o),I),Xs)) ) ) ).

% nth_transpose
tff(fact_5327_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : ( aa(nat,nat,foldr(list(A),nat,aTP_Lamp_nq(list(A),fun(nat,nat)),transpose(A,Xs)),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_oc(list(A),$o),Xs)) ) ).

% transpose_max_length
tff(fact_5328_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),Ta: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,aa(list(B),list(A),map(B,A,F2),Xs)))
         => ( filter2(B,aa(A,fun(B,$o),aTP_Lamp_od(fun(B,A),fun(A,fun(B,$o)),F2),Ta),Xs) = takeWhile(B,aa(A,fun(B,$o),aTP_Lamp_od(fun(B,A),fun(A,fun(B,$o)),F2),Ta),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_5329_set__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : ( aa(list(A),set(A),set2(A),nths(A,Xs,I5)) = collect(A,aa(set(nat),fun(A,$o),aTP_Lamp_oe(list(A),fun(set(nat),fun(A,$o)),Xs),I5)) ) ).

% set_nths
tff(fact_5330_Rats__eq__int__div__nat,axiom,
    field_char_0_Rats(real) = collect(real,aTP_Lamp_of(real,$o)) ).

% Rats_eq_int_div_nat
tff(fact_5331_Rats__minus__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A] :
          ( member(A,aa(A,A,uminus_uminus(A),A2),field_char_0_Rats(A))
        <=> member(A,A2,field_char_0_Rats(A)) ) ) ).

% Rats_minus_iff
tff(fact_5332_takeWhile__eq__all__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( takeWhile(A,P,Xs) = Xs )
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X4) ) ) ).

% takeWhile_eq_all_conv
tff(fact_5333_Rats__abs__iff,axiom,
    ! [Xb: real] :
      ( member(real,aa(real,real,abs_abs(real),Xb),field_char_0_Rats(real))
    <=> member(real,Xb,field_char_0_Rats(real)) ) ).

% Rats_abs_iff
tff(fact_5334_in__set__nthsD,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I5: set(nat)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),nths(A,Xs,I5)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% in_set_nthsD
tff(fact_5335_notin__set__nthsI,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I5: set(nat)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ~ member(A,Xb,aa(list(A),set(A),set2(A),nths(A,Xs,I5))) ) ).

% notin_set_nthsI
tff(fact_5336_set__takeWhileD,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),takeWhile(A,P,Xs)))
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
        & aa(A,$o,P,Xb) ) ) ).

% set_takeWhileD
tff(fact_5337_takeWhile__cong,axiom,
    ! [A: $tType,L: list(A),K: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( L = K )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),L))
           => ( aa(A,$o,P,X)
            <=> aa(A,$o,Q,X) ) )
       => ( takeWhile(A,P,L) = takeWhile(A,Q,K) ) ) ) ).

% takeWhile_cong
tff(fact_5338_Rats__of__int,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: int] : member(A,aa(int,A,ring_1_of_int(A),Z),field_char_0_Rats(A)) ) ).

% Rats_of_int
tff(fact_5339_Rats__diff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,field_char_0_Rats(A))
         => ( member(A,B2,field_char_0_Rats(A))
           => member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),field_char_0_Rats(A)) ) ) ) ).

% Rats_diff
tff(fact_5340_Rats__add,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,field_char_0_Rats(A))
         => ( member(A,B2,field_char_0_Rats(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),field_char_0_Rats(A)) ) ) ) ).

% Rats_add
tff(fact_5341_Rats__no__top__le,axiom,
    ! [Xb: real] :
    ? [X: real] :
      ( member(real,X,field_char_0_Rats(real))
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),X) ) ).

% Rats_no_top_le
tff(fact_5342_Rats__dense__in__real,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
     => ? [X: real] :
          ( member(real,X,field_char_0_Rats(real))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),X)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ) ).

% Rats_dense_in_real
tff(fact_5343_Rats__no__bot__less,axiom,
    ! [Xb: real] :
    ? [X: real] :
      ( member(real,X,field_char_0_Rats(real))
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Xb) ) ).

% Rats_no_bot_less
tff(fact_5344_Rats__0,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => member(A,zero_zero(A),field_char_0_Rats(A)) ) ).

% Rats_0
tff(fact_5345_Rats__inverse,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A] :
          ( member(A,A2,field_char_0_Rats(A))
         => member(A,aa(A,A,inverse_inverse(A),A2),field_char_0_Rats(A)) ) ) ).

% Rats_inverse
tff(fact_5346_Rats__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => member(A,one_one(A),field_char_0_Rats(A)) ) ).

% Rats_1
tff(fact_5347_Rats__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Nb: nat] : member(A,aa(nat,A,semiring_1_of_nat(A),Nb),field_char_0_Rats(A)) ) ).

% Rats_of_nat
tff(fact_5348_set__nths__subset,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),nths(A,Xs,I5))),aa(list(A),set(A),set2(A),Xs)) ).

% set_nths_subset
tff(fact_5349_nths__all,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
         => member(nat,I2,I5) )
     => ( nths(A,Xs,I5) = Xs ) ) ).

% nths_all
tff(fact_5350_takeWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))
     => ( aa(nat,A,nth(A,takeWhile(A,P,Xs)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% takeWhile_nth
tff(fact_5351_nth__length__takeWhile,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs))
     => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ).

% nth_length_takeWhile
tff(fact_5352_length__takeWhile__less__P__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xs: list(A)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_5353_Rats__eq__int__div__int,axiom,
    field_char_0_Rats(real) = collect(real,aTP_Lamp_og(real,$o)) ).

% Rats_eq_int_div_int
tff(fact_5354_takeWhile__eq__take__P__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),P: fun(A,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) ) )
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
         => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),Nb)) )
       => ( takeWhile(A,P,Xs) = take(A,Nb,Xs) ) ) ) ).

% takeWhile_eq_take_P_nth
tff(fact_5355_filter__in__nths,axiom,
    ! [A: $tType,Xs: list(A),S: set(nat)] :
      ( distinct(A,Xs)
     => ( filter2(A,aa(set(nat),fun(A,$o),aTP_Lamp_oh(list(A),fun(set(nat),fun(A,$o)),Xs),S),Xs) = nths(A,Xs,S) ) ) ).

% filter_in_nths
tff(fact_5356_length__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : ( aa(list(A),nat,size_size(list(A)),nths(A,Xs,I5)) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(set(nat),fun(nat,$o),aTP_Lamp_oi(list(A),fun(set(nat),fun(nat,$o)),Xs),I5))) ) ).

% length_nths
tff(fact_5357_filter__eq__nths,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : ( filter2(A,P,Xs) = nths(A,Xs,collect(nat,aa(list(A),fun(nat,$o),aTP_Lamp_ny(fun(A,$o),fun(list(A),fun(nat,$o)),P),Xs))) ) ).

% filter_eq_nths
tff(fact_5358_Nats__altdef1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( semiring_1_Nats(A) = collect(A,aTP_Lamp_oj(A,$o)) ) ) ).

% Nats_altdef1
tff(fact_5359_upt__rec__numeral,axiom,
    ! [Ma: num,Nb: num] :
      ( upt(aa(num,nat,numeral_numeral(nat),Ma),aa(num,nat,numeral_numeral(nat),Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),Ma)),aa(num,nat,numeral_numeral(nat),Nb)),cons(nat,aa(num,nat,numeral_numeral(nat),Ma),upt(aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),Ma)),aa(num,nat,numeral_numeral(nat),Nb))),nil(nat)) ) ).

% upt_rec_numeral
tff(fact_5360_Id__on__set,axiom,
    ! [A: $tType,Xs: list(A)] : ( id_on(A,aa(list(A),set(A),set2(A),Xs)) = aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_ok(A,product_prod(A,A))),Xs)) ) ).

% Id_on_set
tff(fact_5361_list_Osimps_I15_J,axiom,
    ! [A: $tType,X21: A,X22: list(A)] : ( aa(list(A),set(A),set2(A),cons(A,X21,X22)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X21),aa(list(A),set(A),set2(A),X22)) ) ).

% list.simps(15)
tff(fact_5362_nth__Cons__Suc,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] : ( aa(nat,A,nth(A,cons(A,Xb,Xs)),aa(nat,nat,suc,Nb)) = aa(nat,A,nth(A,Xs),Nb) ) ).

% nth_Cons_Suc
tff(fact_5363_nth__Cons__0,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : ( aa(nat,A,nth(A,cons(A,Xb,Xs)),zero_zero(nat)) = Xb ) ).

% nth_Cons_0
tff(fact_5364_take__Suc__Cons,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : ( take(A,aa(nat,nat,suc,Nb),cons(A,Xb,Xs)) = cons(A,Xb,take(A,Nb,Xs)) ) ).

% take_Suc_Cons
tff(fact_5365_sum__list_OCons,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xb: A,Xs: list(A)] : ( aa(list(A),A,groups8242544230860333062m_list(A),cons(A,Xb,Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(list(A),A,groups8242544230860333062m_list(A),Xs)) ) ) ).

% sum_list.Cons
tff(fact_5366_nths__singleton,axiom,
    ! [A: $tType,Xb: A,A3: set(nat)] :
      ( nths(A,cons(A,Xb,nil(A)),A3) = $ite(member(nat,zero_zero(nat),A3),cons(A,Xb,nil(A)),nil(A)) ) ).

% nths_singleton
tff(fact_5367_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A,Xb: B,Xs: list(B)] : ( aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),cons(B,Xb,Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F2,Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),Xs))) ) ) ).

% horner_sum_simps(2)
tff(fact_5368_enumerate__simps_I2_J,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : ( enumerate(A,Nb,cons(A,Xb,Xs)) = cons(product_prod(nat,A),aa(A,product_prod(nat,A),product_Pair(nat,A,Nb),Xb),enumerate(A,aa(nat,nat,suc,Nb),Xs)) ) ).

% enumerate_simps(2)
tff(fact_5369_nth__Cons__numeral,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),V2: num] : ( aa(nat,A,nth(A,cons(A,Xb,Xs)),aa(num,nat,numeral_numeral(nat),V2)) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat))) ) ).

% nth_Cons_numeral
tff(fact_5370_take__Cons__numeral,axiom,
    ! [A: $tType,V2: num,Xb: A,Xs: list(A)] : ( take(A,aa(num,nat,numeral_numeral(nat),V2),cons(A,Xb,Xs)) = cons(A,Xb,take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat)),Xs)) ) ).

% take_Cons_numeral
tff(fact_5371_nth__Cons__pos,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,A,nth(A,cons(A,Xb,Xs)),Nb) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_5372_of__nat__in__Nats,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] : member(A,aa(nat,A,semiring_1_of_nat(A),Nb),semiring_1_Nats(A)) ) ).

% of_nat_in_Nats
tff(fact_5373_Nats__induct,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xb: A,P: fun(A,$o)] :
          ( member(A,Xb,semiring_1_Nats(A))
         => ( ! [N: nat] : aa(A,$o,P,aa(nat,A,semiring_1_of_nat(A),N))
           => aa(A,$o,P,Xb) ) ) ) ).

% Nats_induct
tff(fact_5374_Nats__cases,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xb: A] :
          ( member(A,Xb,semiring_1_Nats(A))
         => ~ ! [N: nat] : ( Xb != aa(nat,A,semiring_1_of_nat(A),N) ) ) ) ).

% Nats_cases
tff(fact_5375_Nats__1,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => member(A,one_one(A),semiring_1_Nats(A)) ) ).

% Nats_1
tff(fact_5376_Nats__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,semiring_1_Nats(A))
         => ( member(A,B2,semiring_1_Nats(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),semiring_1_Nats(A)) ) ) ) ).

% Nats_add
tff(fact_5377_Nats__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => member(A,zero_zero(A),semiring_1_Nats(A)) ) ).

% Nats_0
tff(fact_5378_Nats__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,semiring_1_Nats(A))
         => ( member(A,B2,semiring_1_Nats(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),semiring_1_Nats(A)) ) ) ) ).

% Nats_mult
tff(fact_5379_Nats__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [W: num] : member(A,aa(num,A,numeral_numeral(A),W),semiring_1_Nats(A)) ) ).

% Nats_numeral
tff(fact_5380_list__encode_Ocases,axiom,
    ! [Xb: list(nat)] :
      ( ( Xb != nil(nat) )
     => ~ ! [X: nat,Xs2: list(nat)] : ( Xb != cons(nat,X,Xs2) ) ) ).

% list_encode.cases
tff(fact_5381_set__ConsD,axiom,
    ! [A: $tType,Y: A,Xb: A,Xs: list(A)] :
      ( member(A,Y,aa(list(A),set(A),set2(A),cons(A,Xb,Xs)))
     => ( ( Y = Xb )
        | member(A,Y,aa(list(A),set(A),set2(A),Xs)) ) ) ).

% set_ConsD
tff(fact_5382_list_Oset__cases,axiom,
    ! [A: $tType,E2: A,A2: list(A)] :
      ( member(A,E2,aa(list(A),set(A),set2(A),A2))
     => ( ! [Z23: list(A)] : ( A2 != cons(A,E2,Z23) )
       => ~ ! [Z12: A,Z23: list(A)] :
              ( ( A2 = cons(A,Z12,Z23) )
             => ~ member(A,E2,aa(list(A),set(A),set2(A),Z23)) ) ) ) ).

% list.set_cases
tff(fact_5383_list_Oset__intros_I1_J,axiom,
    ! [A: $tType,X21: A,X22: list(A)] : member(A,X21,aa(list(A),set(A),set2(A),cons(A,X21,X22))) ).

% list.set_intros(1)
tff(fact_5384_list_Oset__intros_I2_J,axiom,
    ! [A: $tType,Y: A,X22: list(A),X21: A] :
      ( member(A,Y,aa(list(A),set(A),set2(A),X22))
     => member(A,Y,aa(list(A),set(A),set2(A),cons(A,X21,X22))) ) ).

% list.set_intros(2)
tff(fact_5385_Suc__length__conv,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( aa(nat,nat,suc,Nb) = aa(list(A),nat,size_size(list(A)),Xs) )
    <=> ? [Y5: A,Ys4: list(A)] :
          ( ( Xs = cons(A,Y5,Ys4) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = Nb ) ) ) ).

% Suc_length_conv
tff(fact_5386_length__Suc__conv,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,Nb) )
    <=> ? [Y5: A,Ys4: list(A)] :
          ( ( Xs = cons(A,Y5,Ys4) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = Nb ) ) ) ).

% length_Suc_conv
tff(fact_5387_length__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : ( aa(list(A),nat,size_size(list(A)),cons(A,Xb,Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_Cons
tff(fact_5388_set__subset__Cons,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),cons(A,Xb,Xs))) ).

% set_subset_Cons
tff(fact_5389_distinct_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( distinct(A,cons(A,Xb,Xs))
    <=> ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
        & distinct(A,Xs) ) ) ).

% distinct.simps(2)
tff(fact_5390_replicate__Suc,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : ( replicate(A,aa(nat,nat,suc,Nb),Xb) = cons(A,Xb,replicate(A,Nb,Xb)) ) ).

% replicate_Suc
tff(fact_5391_upt__conv__Cons__Cons,axiom,
    ! [Ma: nat,Nb: nat,Ns: list(nat),Q4: nat] :
      ( ( cons(nat,Ma,cons(nat,Nb,Ns)) = upt(Ma,Q4) )
    <=> ( cons(nat,Nb,Ns) = upt(aa(nat,nat,suc,Ma),Q4) ) ) ).

% upt_conv_Cons_Cons
tff(fact_5392_list__update__code_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I: nat,Y: A] : ( list_update(A,cons(A,Xb,Xs),aa(nat,nat,suc,I),Y) = cons(A,Xb,list_update(A,Xs,I,Y)) ) ).

% list_update_code(3)
tff(fact_5393_list__update__code_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A] : ( list_update(A,cons(A,Xb,Xs),zero_zero(nat),Y) = cons(A,Y,Xs) ) ).

% list_update_code(2)
tff(fact_5394_Nats__diff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,semiring_1_Nats(A))
         => ( member(A,B2,semiring_1_Nats(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),semiring_1_Nats(A)) ) ) ) ) ).

% Nats_diff
tff(fact_5395_Suc__le__length__iff,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ? [X4: A,Ys4: list(A)] :
          ( ( Xs = cons(A,X4,Ys4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Ys4)) ) ) ).

% Suc_le_length_iff
tff(fact_5396_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Ys2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),cons(A,Xb,Ys2))
        <=> ( ! [X4: A] :
                ( member(A,X4,aa(list(A),set(A),set2(A),Ys2))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X4) )
            & sorted_wrt(A,ord_less_eq(A),Ys2) ) ) ) ).

% sorted_simps(2)
tff(fact_5397_strict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Ys2: list(A)] :
          ( sorted_wrt(A,ord_less(A),cons(A,Xb,Ys2))
        <=> ( ! [X4: A] :
                ( member(A,X4,aa(list(A),set(A),set2(A),Ys2))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X4) )
            & sorted_wrt(A,ord_less(A),Ys2) ) ) ) ).

% strict_sorted_simps(2)
tff(fact_5398_sorted__wrt_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xb: A,Ys2: list(A)] :
      ( sorted_wrt(A,P,cons(A,Xb,Ys2))
    <=> ( ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Ys2))
           => aa(A,$o,aa(A,fun(A,$o),P,Xb),X4) )
        & sorted_wrt(A,P,Ys2) ) ) ).

% sorted_wrt.simps(2)
tff(fact_5399_sorted__wrt_Oelims_I3_J,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,$o)),Xaa: list(A)] :
      ( ~ sorted_wrt(A,Xb,Xaa)
     => ~ ! [X: A,Ys3: list(A)] :
            ( ( Xaa = cons(A,X,Ys3) )
           => ( ! [Xa2: A] :
                  ( member(A,Xa2,aa(list(A),set(A),set2(A),Ys3))
                 => aa(A,$o,aa(A,fun(A,$o),Xb,X),Xa2) )
              & sorted_wrt(A,Xb,Ys3) ) ) ) ).

% sorted_wrt.elims(3)
tff(fact_5400_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( upt(I,J) = cons(nat,I,upt(aa(nat,nat,suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_5401_Nats__subset__Ints,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),semiring_1_Nats(A)),ring_1_Ints(A)) ) ).

% Nats_subset_Ints
tff(fact_5402_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A] :
      ( aa(A,nat,count_list(A,cons(A,Xb,Xs)),Y) = $ite(Xb = Y,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,count_list(A,Xs),Y)),one_one(nat)),aa(A,nat,count_list(A,Xs),Y)) ) ).

% count_list.simps(2)
tff(fact_5403_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X22: list(A)] : ( aa(list(A),nat,size_size(list(A)),cons(A,X21,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X22)),aa(nat,nat,suc,zero_zero(nat))) ) ).

% list.size(4)
tff(fact_5404_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : ( n_lists(A,aa(nat,nat,suc,Nb),Xs) = concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),aTP_Lamp_om(list(A),fun(list(A),list(list(A))),Xs)),n_lists(A,Nb,Xs))) ) ).

% n_lists.simps(2)
tff(fact_5405_sorted__wrt_Oelims_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,$o)),Xaa: list(A)] :
      ( sorted_wrt(A,Xb,Xaa)
     => ( ( Xaa != nil(A) )
       => ~ ! [X: A,Ys3: list(A)] :
              ( ( Xaa = cons(A,X,Ys3) )
             => ~ ( ! [Xa: A] :
                      ( member(A,Xa,aa(list(A),set(A),set2(A),Ys3))
                     => aa(A,$o,aa(A,fun(A,$o),Xb,X),Xa) )
                  & sorted_wrt(A,Xb,Ys3) ) ) ) ) ).

% sorted_wrt.elims(2)
tff(fact_5406_sorted__wrt_Oelims_I1_J,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,$o)),Xaa: list(A),Y: $o] :
      ( ( sorted_wrt(A,Xb,Xaa)
      <=> (Y) )
     => ( ( ( Xaa = nil(A) )
         => ~ (Y) )
       => ~ ! [X: A,Ys3: list(A)] :
              ( ( Xaa = cons(A,X,Ys3) )
             => ( (Y)
              <=> ~ ( ! [Xa3: A] :
                        ( member(A,Xa3,aa(list(A),set(A),set2(A),Ys3))
                       => aa(A,$o,aa(A,fun(A,$o),Xb,X),Xa3) )
                    & sorted_wrt(A,Xb,Ys3) ) ) ) ) ) ).

% sorted_wrt.elims(1)
tff(fact_5407_nth__Cons_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] :
      ( aa(nat,A,nth(A,cons(A,Xb,Xs)),Nb) = $ite(Nb = zero_zero(nat),Xb,aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ) ).

% nth_Cons'
tff(fact_5408_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xb: nat,Xs: list(nat)] :
      ( ( upt(I,J) = cons(nat,Xb,Xs) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
        & ( I = Xb )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_5409_upt__rec,axiom,
    ! [I: nat,J: nat] :
      ( upt(I,J) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J),cons(nat,I,upt(aa(nat,nat,suc,I),J)),nil(nat)) ) ).

% upt_rec
tff(fact_5410_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,nat),X21: A,X22: list(A)] : ( size_list(A,Xb,cons(A,X21,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xb,X21)),size_list(A,Xb,X22))),aa(nat,nat,suc,zero_zero(nat))) ) ).

% list.size_gen(2)
tff(fact_5411_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,I)),J)
     => ( linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J)) = cons(nat,aa(nat,nat,suc,I),linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_5412_map__upt__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),Nb: nat] : ( aa(list(nat),list(A),map(nat,A,F2),upt(zero_zero(nat),aa(nat,nat,suc,Nb))) = cons(A,aa(nat,A,F2,zero_zero(nat)),aa(list(nat),list(A),map(nat,A,aTP_Lamp_on(fun(nat,A),fun(nat,A),F2)),upt(zero_zero(nat),Nb))) ) ).

% map_upt_Suc
tff(fact_5413_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),J)
     => ( linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J)) = cons(nat,aa(nat,nat,suc,I),linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_5414_Nats__altdef2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( semiring_1_Nats(A) = collect(A,aTP_Lamp_oo(A,$o)) ) ) ).

% Nats_altdef2
tff(fact_5415_nth__equal__first__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => ( ( aa(nat,A,nth(A,cons(A,Xb,Xs)),Nb) = Xb )
        <=> ( Nb = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5416_nth__non__equal__first__eq,axiom,
    ! [A: $tType,Xb: A,Y: A,Xs: list(A),Nb: nat] :
      ( ( Xb != Y )
     => ( ( aa(nat,A,nth(A,cons(A,Xb,Xs)),Nb) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) = Y )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_5417_take__Cons_H,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] :
      ( take(A,Nb,cons(A,Xb,Xs)) = $ite(Nb = zero_zero(nat),nil(A),cons(A,Xb,take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Xs))) ) ).

% take_Cons'
tff(fact_5418_Cons__replicate__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat,Y: A] :
      ( ( cons(A,Xb,Xs) = replicate(A,Nb,Y) )
    <=> ( ( Xb = Y )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
        & ( Xs = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Xb) ) ) ) ).

% Cons_replicate_eq
tff(fact_5419_Cons__lenlex__iff,axiom,
    ! [A: $tType,Ma: A,Ms: list(A),Nb: A,Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),cons(A,Ma,Ms)),cons(A,Nb,Ns)),lenlex(A,R2))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))
        | ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Ma),Nb),R2) )
        | ( ( Ma = Nb )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Ms),Ns),lenlex(A,R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_5420_list__encode_Oelims,axiom,
    ! [Xb: list(nat),Y: nat] :
      ( ( aa(list(nat),nat,nat_list_encode,Xb) = Y )
     => ( ( ( Xb = nil(nat) )
         => ( Y != zero_zero(nat) ) )
       => ~ ! [X: nat,Xs2: list(nat)] :
              ( ( Xb = cons(nat,X,Xs2) )
             => ( Y != aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),aa(list(nat),nat,nat_list_encode,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_5421_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys2: list(A)] : ( aa(nat,A,nth(A,append(A,Xs,append(A,cons(A,Xb,nil(A)),Ys2))),aa(list(A),nat,size_size(list(A)),Xs)) = Xb ) ).

% concat_inth
tff(fact_5422_the__elem__set,axiom,
    ! [A: $tType,Xb: A] : ( the_elem(A,aa(list(A),set(A),set2(A),cons(A,Xb,nil(A)))) = Xb ) ).

% the_elem_set
tff(fact_5423_length__append,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : ( aa(list(A),nat,size_size(list(A)),append(A,Xs,Ys2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys2)) ) ).

% length_append
tff(fact_5424_sum__list__append,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A),Ys2: list(A)] : ( aa(list(A),A,groups8242544230860333062m_list(A),append(A,Xs,Ys2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),aa(list(A),A,groups8242544230860333062m_list(A),Ys2)) ) ) ).

% sum_list_append
tff(fact_5425_takeWhile__append2,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Ys2: list(A)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X) )
     => ( takeWhile(A,P,append(A,Xs,Ys2)) = append(A,Xs,takeWhile(A,P,Ys2)) ) ) ).

% takeWhile_append2
tff(fact_5426_takeWhile__append1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o),Ys2: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P,Xb)
       => ( takeWhile(A,P,append(A,Xs,Ys2)) = takeWhile(A,P,Xs) ) ) ) ).

% takeWhile_append1
tff(fact_5427_size__list__append,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A),Ys2: list(A)] : ( size_list(A,F2,append(A,Xs,Ys2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(A,F2,Xs)),size_list(A,F2,Ys2)) ) ).

% size_list_append
tff(fact_5428_nth__append__length__plus,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Nb: nat] : ( aa(nat,A,nth(A,append(A,Xs,Ys2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb)) = aa(nat,A,nth(A,Ys2),Nb) ) ).

% nth_append_length_plus
tff(fact_5429_take__append,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys2: list(A)] : ( take(A,Nb,append(A,Xs,Ys2)) = append(A,take(A,Nb,Xs),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),Ys2)) ) ).

% take_append
tff(fact_5430_n__lists__Nil,axiom,
    ! [A: $tType,Nb: nat] :
      ( n_lists(A,Nb,nil(A)) = $ite(Nb = zero_zero(nat),cons(list(A),nil(A),nil(list(A))),nil(list(A))) ) ).

% n_lists_Nil
tff(fact_5431_distinct__append,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( distinct(A,append(A,Xs,Ys2))
    <=> ( distinct(A,Xs)
        & distinct(A,Ys2)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) ) ) ) ).

% distinct_append
tff(fact_5432_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : ( linord4507533701916653071of_set(nat,set_ord_lessThan(nat,aa(nat,nat,suc,K))) = append(nat,linord4507533701916653071of_set(nat,set_ord_lessThan(nat,K)),cons(nat,K,nil(nat))) ) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_5433_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : ( linord4507533701916653071of_set(nat,set_ord_atMost(nat,aa(nat,nat,suc,K))) = append(nat,linord4507533701916653071of_set(nat,set_ord_atMost(nat,K)),cons(nat,aa(nat,nat,suc,K),nil(nat))) ) ).

% sorted_list_of_set_atMost_Suc
tff(fact_5434_enumerate__append__eq,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys2: list(A)] : ( enumerate(A,Nb,append(A,Xs,Ys2)) = append(product_prod(nat,A),enumerate(A,Nb,Xs),enumerate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),Ys2)) ) ).

% enumerate_append_eq
tff(fact_5435_list__encode__eq,axiom,
    ! [Xb: list(nat),Y: list(nat)] :
      ( ( aa(list(nat),nat,nat_list_encode,Xb) = aa(list(nat),nat,nat_list_encode,Y) )
    <=> ( Xb = Y ) ) ).

% list_encode_eq
tff(fact_5436_split__list,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ? [Ys3: list(A),Zs: list(A)] : ( Xs = append(A,Ys3,cons(A,Xb,Zs)) ) ) ).

% split_list
tff(fact_5437_split__list__last,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ? [Ys3: list(A),Zs: list(A)] :
          ( ( Xs = append(A,Ys3,cons(A,Xb,Zs)) )
          & ~ member(A,Xb,aa(list(A),set(A),set2(A),Zs)) ) ) ).

% split_list_last
tff(fact_5438_split__list__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X3) )
     => ? [Ys3: list(A),X: A] :
          ( ? [Zs: list(A)] : ( Xs = append(A,Ys3,cons(A,X,Zs)) )
          & aa(A,$o,P,X) ) ) ).

% split_list_prop
tff(fact_5439_split__list__first,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ? [Ys3: list(A),Zs: list(A)] :
          ( ( Xs = append(A,Ys3,cons(A,Xb,Zs)) )
          & ~ member(A,Xb,aa(list(A),set(A),set2(A),Ys3)) ) ) ).

% split_list_first
tff(fact_5440_split__list__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X3) )
     => ~ ! [Ys3: list(A),X: A] :
            ( ? [Zs: list(A)] : ( Xs = append(A,Ys3,cons(A,X,Zs)) )
           => ~ aa(A,$o,P,X) ) ) ).

% split_list_propE
tff(fact_5441_append__Cons__eq__iff,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys2: list(A),Xs4: list(A),Ys5: list(A)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Ys2))
       => ( ( append(A,Xs,cons(A,Xb,Ys2)) = append(A,Xs4,cons(A,Xb,Ys5)) )
        <=> ( ( Xs = Xs4 )
            & ( Ys2 = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
tff(fact_5442_in__set__conv__decomp,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
    <=> ? [Ys4: list(A),Zs2: list(A)] : ( Xs = append(A,Ys4,cons(A,Xb,Zs2)) ) ) ).

% in_set_conv_decomp
tff(fact_5443_split__list__last__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X3) )
     => ? [Ys3: list(A),X: A,Zs: list(A)] :
          ( ( Xs = append(A,Ys3,cons(A,X,Zs)) )
          & aa(A,$o,P,X)
          & ! [Xa: A] :
              ( member(A,Xa,aa(list(A),set(A),set2(A),Zs))
             => ~ aa(A,$o,P,Xa) ) ) ) ).

% split_list_last_prop
tff(fact_5444_split__list__first__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X3) )
     => ? [Ys3: list(A),X: A] :
          ( ? [Zs: list(A)] : ( Xs = append(A,Ys3,cons(A,X,Zs)) )
          & aa(A,$o,P,X)
          & ! [Xa: A] :
              ( member(A,Xa,aa(list(A),set(A),set2(A),Ys3))
             => ~ aa(A,$o,P,Xa) ) ) ) ).

% split_list_first_prop
tff(fact_5445_split__list__last__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X3) )
     => ~ ! [Ys3: list(A),X: A,Zs: list(A)] :
            ( ( Xs = append(A,Ys3,cons(A,X,Zs)) )
           => ( aa(A,$o,P,X)
             => ~ ! [Xa: A] :
                    ( member(A,Xa,aa(list(A),set(A),set2(A),Zs))
                   => ~ aa(A,$o,P,Xa) ) ) ) ) ).

% split_list_last_propE
tff(fact_5446_split__list__first__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X3) )
     => ~ ! [Ys3: list(A),X: A] :
            ( ? [Zs: list(A)] : ( Xs = append(A,Ys3,cons(A,X,Zs)) )
           => ( aa(A,$o,P,X)
             => ~ ! [Xa: A] :
                    ( member(A,Xa,aa(list(A),set(A),set2(A),Ys3))
                   => ~ aa(A,$o,P,Xa) ) ) ) ) ).

% split_list_first_propE
tff(fact_5447_in__set__conv__decomp__last,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
    <=> ? [Ys4: list(A),Zs2: list(A)] :
          ( ( Xs = append(A,Ys4,cons(A,Xb,Zs2)) )
          & ~ member(A,Xb,aa(list(A),set(A),set2(A),Zs2)) ) ) ).

% in_set_conv_decomp_last
tff(fact_5448_in__set__conv__decomp__first,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
    <=> ? [Ys4: list(A),Zs2: list(A)] :
          ( ( Xs = append(A,Ys4,cons(A,Xb,Zs2)) )
          & ~ member(A,Xb,aa(list(A),set(A),set2(A),Ys4)) ) ) ).

% in_set_conv_decomp_first
tff(fact_5449_split__list__last__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X4) )
    <=> ? [Ys4: list(A),X4: A,Zs2: list(A)] :
          ( ( Xs = append(A,Ys4,cons(A,X4,Zs2)) )
          & aa(A,$o,P,X4)
          & ! [Xa3: A] :
              ( member(A,Xa3,aa(list(A),set(A),set2(A),Zs2))
             => ~ aa(A,$o,P,Xa3) ) ) ) ).

% split_list_last_prop_iff
tff(fact_5450_split__list__first__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X4) )
    <=> ? [Ys4: list(A),X4: A] :
          ( ? [Zs2: list(A)] : ( Xs = append(A,Ys4,cons(A,X4,Zs2)) )
          & aa(A,$o,P,X4)
          & ! [Xa3: A] :
              ( member(A,Xa3,aa(list(A),set(A),set2(A),Ys4))
             => ~ aa(A,$o,P,Xa3) ) ) ) ).

% split_list_first_prop_iff
tff(fact_5451_sorted__wrt__append,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A),Ys2: list(A)] :
      ( sorted_wrt(A,P,append(A,Xs,Ys2))
    <=> ( sorted_wrt(A,P,Xs)
        & sorted_wrt(A,P,Ys2)
        & ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
           => ! [Xa3: A] :
                ( member(A,Xa3,aa(list(A),set(A),set2(A),Ys2))
               => aa(A,$o,aa(A,fun(A,$o),P,X4),Xa3) ) ) ) ) ).

% sorted_wrt_append
tff(fact_5452_replicate__add,axiom,
    ! [A: $tType,Nb: nat,Ma: nat,Xb: A] : ( replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Ma),Xb) = append(A,replicate(A,Nb,Xb),replicate(A,Ma,Xb)) ) ).

% replicate_add
tff(fact_5453_remove1__append,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys2: list(A)] :
      ( remove1(A,Xb,append(A,Xs,Ys2)) = $ite(member(A,Xb,aa(list(A),set(A),set2(A),Xs)),append(A,remove1(A,Xb,Xs),Ys2),append(A,Xs,remove1(A,Xb,Ys2))) ) ).

% remove1_append
tff(fact_5454_sorted__append,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),append(A,Xs,Ys2))
        <=> ( sorted_wrt(A,ord_less_eq(A),Xs)
            & sorted_wrt(A,ord_less_eq(A),Ys2)
            & ! [X4: A] :
                ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
               => ! [Xa3: A] :
                    ( member(A,Xa3,aa(list(A),set(A),set2(A),Ys2))
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa3) ) ) ) ) ) ).

% sorted_append
tff(fact_5455_not__distinct__conv__prefix,axiom,
    ! [A: $tType,As: list(A)] :
      ( ~ distinct(A,As)
    <=> ? [Xs3: list(A),Y5: A,Ys4: list(A)] :
          ( member(A,Y5,aa(list(A),set(A),set2(A),Xs3))
          & distinct(A,Xs3)
          & ( As = append(A,Xs3,cons(A,Y5,Ys4)) ) ) ) ).

% not_distinct_conv_prefix
tff(fact_5456_Cons__eq__filterD,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o),Ys2: list(A)] :
      ( ( cons(A,Xb,Xs) = filter2(A,P,Ys2) )
     => ? [Us: list(A),Vs: list(A)] :
          ( ( Ys2 = append(A,Us,cons(A,Xb,Vs)) )
          & ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Us))
             => ~ aa(A,$o,P,X3) )
          & aa(A,$o,P,Xb)
          & ( Xs = filter2(A,P,Vs) ) ) ) ).

% Cons_eq_filterD
tff(fact_5457_filter__eq__ConsD,axiom,
    ! [A: $tType,P: fun(A,$o),Ys2: list(A),Xb: A,Xs: list(A)] :
      ( ( filter2(A,P,Ys2) = cons(A,Xb,Xs) )
     => ? [Us: list(A),Vs: list(A)] :
          ( ( Ys2 = append(A,Us,cons(A,Xb,Vs)) )
          & ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Us))
             => ~ aa(A,$o,P,X3) )
          & aa(A,$o,P,Xb)
          & ( Xs = filter2(A,P,Vs) ) ) ) ).

% filter_eq_ConsD
tff(fact_5458_Cons__eq__filter__iff,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o),Ys2: list(A)] :
      ( ( cons(A,Xb,Xs) = filter2(A,P,Ys2) )
    <=> ? [Us2: list(A),Vs2: list(A)] :
          ( ( Ys2 = append(A,Us2,cons(A,Xb,Vs2)) )
          & ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Us2))
             => ~ aa(A,$o,P,X4) )
          & aa(A,$o,P,Xb)
          & ( Xs = filter2(A,P,Vs2) ) ) ) ).

% Cons_eq_filter_iff
tff(fact_5459_filter__eq__Cons__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Ys2: list(A),Xb: A,Xs: list(A)] :
      ( ( filter2(A,P,Ys2) = cons(A,Xb,Xs) )
    <=> ? [Us2: list(A),Vs2: list(A)] :
          ( ( Ys2 = append(A,Us2,cons(A,Xb,Vs2)) )
          & ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Us2))
             => ~ aa(A,$o,P,X4) )
          & aa(A,$o,P,Xb)
          & ( Xs = filter2(A,P,Vs2) ) ) ) ).

% filter_eq_Cons_iff
tff(fact_5460_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = append(nat,upt(I,J),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% upt_add_eq_append
tff(fact_5461_list__update__append1,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Ys2: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,append(A,Xs,Ys2),I,Xb) = append(A,list_update(A,Xs,I,Xb),Ys2) ) ) ).

% list_update_append1
tff(fact_5462_remove1__split,axiom,
    ! [A: $tType,A2: A,Xs: list(A),Ys2: list(A)] :
      ( member(A,A2,aa(list(A),set(A),set2(A),Xs))
     => ( ( remove1(A,A2,Xs) = Ys2 )
      <=> ? [Ls: list(A),Rs: list(A)] :
            ( ( Xs = append(A,Ls,cons(A,A2,Rs)) )
            & ~ member(A,A2,aa(list(A),set(A),set2(A),Ls))
            & ( Ys2 = append(A,Ls,Rs) ) ) ) ) ).

% remove1_split
tff(fact_5463_takeWhile__append,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys2: list(A)] :
      ( takeWhile(A,P,append(A,Xs,Ys2)) = $ite(
          ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
           => aa(A,$o,P,X4) ),
          append(A,Xs,takeWhile(A,P,Ys2)),
          takeWhile(A,P,Xs) ) ) ).

% takeWhile_append
tff(fact_5464_nths__append,axiom,
    ! [A: $tType,L: list(A),L4: list(A),A3: set(nat)] : ( nths(A,append(A,L,L4),A3) = append(A,nths(A,L,A3),nths(A,L4,collect(nat,aa(set(nat),fun(nat,$o),aTP_Lamp_op(list(A),fun(set(nat),fun(nat,$o)),L),A3)))) ) ).

% nths_append
tff(fact_5465_list__encode_Osimps_I1_J,axiom,
    aa(list(nat),nat,nat_list_encode,nil(nat)) = zero_zero(nat) ).

% list_encode.simps(1)
tff(fact_5466_length__Suc__conv__rev,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,Nb) )
    <=> ? [Y5: A,Ys4: list(A)] :
          ( ( Xs = append(A,Ys4,cons(A,Y5,nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = Nb ) ) ) ).

% length_Suc_conv_rev
tff(fact_5467_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] : ( aa(list(A),nat,size_size(list(A)),append(A,Xs,cons(A,Xb,nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_append_singleton
tff(fact_5468_nth__append,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Nb: nat] :
      ( aa(nat,A,nth(A,append(A,Xs,Ys2)),Nb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,A,nth(A,Xs),Nb),aa(nat,A,nth(A,Ys2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)))) ) ).

% nth_append
tff(fact_5469_list__update__append,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Nb: nat,Xb: A] :
      ( list_update(A,append(A,Xs,Ys2),Nb,Xb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),append(A,list_update(A,Xs,Nb,Xb),Ys2),append(A,Xs,list_update(A,Ys2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),Xb))) ) ).

% list_update_append
tff(fact_5470_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys2 != nil(A) )
       => ( ( append(A,Xs,Ys2) = append(A,Ys2,Xs) )
         => ? [N: nat,Zs: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),N)
              & ( concat(A,replicate(list(A),N,Zs)) = append(A,Xs,Ys2) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_5471_lexord__same__pref__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs3: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),append(A,Xs,Ys2)),append(A,Xs,Zs3)),lexord(A,R2))
    <=> ( ? [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
            & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X4),X4),R2) )
        | member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Ys2),Zs3),lexord(A,R2)) ) ) ).

% lexord_same_pref_iff
tff(fact_5472_n__lists_Osimps_I1_J,axiom,
    ! [A: $tType,Xs: list(A)] : ( n_lists(A,zero_zero(nat),Xs) = cons(list(A),nil(A),nil(list(A))) ) ).

% n_lists.simps(1)
tff(fact_5473_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B),Ys2: list(B)] : ( aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),append(B,Xs,Ys2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(list(B),nat,size_size(list(B)),Xs))),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),Ys2))) ) ) ).

% horner_sum_append
tff(fact_5474_nths__Cons,axiom,
    ! [A: $tType,Xb: A,L: list(A),A3: set(nat)] :
      ( nths(A,cons(A,Xb,L),A3) = append(A,
          $ite(member(nat,zero_zero(nat),A3),cons(A,Xb,nil(A)),nil(A)),
          nths(A,L,collect(nat,aTP_Lamp_oq(set(nat),fun(nat,$o),A3)))) ) ).

% nths_Cons
tff(fact_5475_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( upt(I,aa(nat,nat,suc,J)) = append(nat,upt(I,J),cons(nat,J,nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_5476_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( upt(I,aa(nat,nat,suc,J)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J),append(nat,upt(I,J),cons(nat,J,nil(nat))),nil(nat)) ) ).

% upt_Suc
tff(fact_5477_take__Suc__conv__app__nth,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( take(A,aa(nat,nat,suc,I),Xs) = append(A,take(A,I,Xs),cons(A,aa(nat,A,nth(A,Xs),I),nil(A))) ) ) ).

% take_Suc_conv_app_nth
tff(fact_5478_list__encode_Osimps_I2_J,axiom,
    ! [Xb: nat,Xs: list(nat)] : ( aa(list(nat),nat,nat_list_encode,cons(nat,Xb,Xs)) = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xb),aa(list(nat),nat,nat_list_encode,Xs)))) ) ).

% list_encode.simps(2)
tff(fact_5479_nth__repl,axiom,
    ! [A: $tType,Ma: nat,Xs: list(A),Nb: nat,Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => ( ( Ma != Nb )
         => ( aa(nat,A,nth(A,append(A,take(A,Nb,Xs),append(A,cons(A,Xb,nil(A)),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),Xs)))),Ma) = aa(nat,A,nth(A,Xs),Ma) ) ) ) ) ).

% nth_repl
tff(fact_5480_pos__n__replace,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),append(A,take(A,Nb,Xs),append(A,cons(A,Y,nil(A)),drop(A,aa(nat,nat,suc,Nb),Xs)))) ) ) ).

% pos_n_replace
tff(fact_5481_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)),cons(int,J,Js))) ) ).

% upto_aux_rec
tff(fact_5482_drop0,axiom,
    ! [A: $tType,X3: list(A)] : ( drop(A,zero_zero(nat),X3) = X3 ) ).

% drop0
tff(fact_5483_drop__drop,axiom,
    ! [A: $tType,Nb: nat,Ma: nat,Xs: list(A)] : ( drop(A,Nb,drop(A,Ma,Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Ma),Xs) ) ).

% drop_drop
tff(fact_5484_drop__upt,axiom,
    ! [Ma: nat,I: nat,J: nat] : ( drop(nat,Ma,upt(I,J)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Ma),J) ) ).

% drop_upt
tff(fact_5485_drop__Suc__Cons,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : ( drop(A,aa(nat,nat,suc,Nb),cons(A,Xb,Xs)) = drop(A,Nb,Xs) ) ).

% drop_Suc_Cons
tff(fact_5486_length__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : ( aa(list(A),nat,size_size(list(A)),drop(A,Nb,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ) ).

% length_drop
tff(fact_5487_drop__update__cancel,axiom,
    ! [A: $tType,Nb: nat,Ma: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma)
     => ( drop(A,Ma,list_update(A,Xs,Nb,Xb)) = drop(A,Ma,Xs) ) ) ).

% drop_update_cancel
tff(fact_5488_drop__replicate,axiom,
    ! [A: $tType,I: nat,K: nat,Xb: A] : ( drop(A,I,replicate(A,K,Xb)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),I),Xb) ) ).

% drop_replicate
tff(fact_5489_drop__append,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys2: list(A)] : ( drop(A,Nb,append(A,Xs,Ys2)) = append(A,drop(A,Nb,Xs),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),Ys2)) ) ).

% drop_append
tff(fact_5490_drop__Cons__numeral,axiom,
    ! [A: $tType,V2: num,Xb: A,Xs: list(A)] : ( drop(A,aa(num,nat,numeral_numeral(nat),V2),cons(A,Xb,Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat)),Xs) ) ).

% drop_Cons_numeral
tff(fact_5491_nth__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,drop(A,Nb,Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),I)) ) ) ).

% nth_drop
tff(fact_5492_drop__0,axiom,
    ! [A: $tType,Xs: list(A)] : ( drop(A,zero_zero(nat),Xs) = Xs ) ).

% drop_0
tff(fact_5493_in__set__dropD,axiom,
    ! [A: $tType,Xb: A,Nb: nat,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),drop(A,Nb,Xs)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% in_set_dropD
tff(fact_5494_set__drop__subset,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,Nb,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_drop_subset
tff(fact_5495_take__drop,axiom,
    ! [A: $tType,Nb: nat,Ma: nat,Xs: list(A)] : ( take(A,Nb,drop(A,Ma,Xs)) = drop(A,Ma,take(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Ma),Xs)) ) ).

% take_drop
tff(fact_5496_drop__take,axiom,
    ! [A: $tType,Nb: nat,Ma: nat,Xs: list(A)] : ( drop(A,Nb,take(A,Ma,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb),drop(A,Nb,Xs)) ) ).

% drop_take
tff(fact_5497_set__drop__subset__set__drop,axiom,
    ! [A: $tType,Nb: nat,Ma: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,Ma,Xs))),aa(list(A),set(A),set2(A),drop(A,Nb,Xs))) ) ).

% set_drop_subset_set_drop
tff(fact_5498_take__add,axiom,
    ! [A: $tType,I: nat,J: nat,Xs: list(A)] : ( take(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J),Xs) = append(A,take(A,I,Xs),take(A,J,drop(A,I,Xs))) ) ).

% take_add
tff(fact_5499_drop__update__swap,axiom,
    ! [A: $tType,Ma: nat,Nb: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => ( drop(A,Ma,list_update(A,Xs,Nb,Xb)) = list_update(A,drop(A,Ma,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma),Xb) ) ) ).

% drop_update_swap
tff(fact_5500_nths__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),I5: set(nat)] : ( nths(A,drop(A,Nb,Xs),I5) = nths(A,Xs,image(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),I5)) ) ).

% nths_drop
tff(fact_5501_drop__Cons_H,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] :
      ( drop(A,Nb,cons(A,Xb,Xs)) = $ite(Nb = zero_zero(nat),cons(A,Xb,Xs),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Xs)) ) ).

% drop_Cons'
tff(fact_5502_drop__rev,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : ( drop(A,Nb,rev(A,Xs)) = rev(A,take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb),Xs)) ) ).

% drop_rev
tff(fact_5503_rev__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] : ( rev(A,drop(A,I,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),I),rev(A,Xs)) ) ).

% rev_drop
tff(fact_5504_rev__take,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] : ( rev(A,take(A,I,Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),I),rev(A,Xs)) ) ).

% rev_take
tff(fact_5505_take__rev,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : ( take(A,Nb,rev(A,Xs)) = rev(A,drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb),Xs)) ) ).

% take_rev
tff(fact_5506_Cons__nth__drop__Suc,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( cons(A,aa(nat,A,nth(A,Xs),I),drop(A,aa(nat,nat,suc,I),Xs)) = drop(A,I,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_5507_set__take__disj__set__drop__if__distinct,axiom,
    ! [A: $tType,Vs3: list(A),I: nat,J: nat] :
      ( distinct(A,Vs3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),take(A,I,Vs3))),aa(list(A),set(A),set2(A),drop(A,J,Vs3))) = bot_bot(set(A)) ) ) ) ).

% set_take_disj_set_drop_if_distinct
tff(fact_5508_id__take__nth__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( Xs = append(A,take(A,I,Xs),cons(A,aa(nat,A,nth(A,Xs),I),drop(A,aa(nat,nat,suc,I),Xs))) ) ) ).

% id_take_nth_drop
tff(fact_5509_upd__conv__take__nth__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A),A2: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,Xs,I,A2) = append(A,take(A,I,Xs),cons(A,A2,drop(A,aa(nat,nat,suc,I),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_5510_upto_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xb,Xaa) = Y )
     => ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,Xb),Xaa))
       => ~ ( ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),Xaa),cons(int,Xb,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),one_one(int)),Xaa)),nil(int)) )
           => ~ accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,Xb),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_5511_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,I),J))
     => ( upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),cons(int,I,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ) ) ).

% upto.psimps
tff(fact_5512_upto__rec__numeral_I2_J,axiom,
    ! [Ma: num,Nb: num] :
      ( upto(aa(num,int,numeral_numeral(int),Ma),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))),cons(int,aa(num,int,numeral_numeral(int),Ma),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),Ma)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb)))),nil(int)) ) ).

% upto_rec_numeral(2)
tff(fact_5513_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil(int) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I) ) ).

% upto_Nil
tff(fact_5514_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil(int) = upto(I,J) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I) ) ).

% upto_Nil2
tff(fact_5515_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I)
     => ( upto(I,J) = nil(int) ) ) ).

% upto_empty
tff(fact_5516_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K))),J)
     => ( aa(nat,int,nth(int,upto(I,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).

% nth_upto
tff(fact_5517_length__upto,axiom,
    ! [I: int,J: int] : ( aa(list(int),nat,size_size(list(int)),upto(I,J)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),I)),one_one(int))) ) ).

% length_upto
tff(fact_5518_upto__rec__numeral_I1_J,axiom,
    ! [Ma: num,Nb: num] :
      ( upto(aa(num,int,numeral_numeral(int),Ma),aa(num,int,numeral_numeral(int),Nb)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Ma)),aa(num,int,numeral_numeral(int),Nb)),cons(int,aa(num,int,numeral_numeral(int),Ma),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),Ma)),one_one(int)),aa(num,int,numeral_numeral(int),Nb))),nil(int)) ) ).

% upto_rec_numeral(1)
tff(fact_5519_upto__rec__numeral_I4_J,axiom,
    ! [Ma: num,Nb: num] :
      ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))),cons(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb)))),nil(int)) ) ).

% upto_rec_numeral(4)
tff(fact_5520_upto__rec__numeral_I3_J,axiom,
    ! [Ma: num,Nb: num] :
      ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma)),aa(num,int,numeral_numeral(int),Nb)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma))),aa(num,int,numeral_numeral(int),Nb)),cons(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma))),one_one(int)),aa(num,int,numeral_numeral(int),Nb))),nil(int)) ) ).

% upto_rec_numeral(3)
tff(fact_5521_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : ( set_or1337092689740270186AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(I,J)) ) ).

% atLeastAtMost_upto
tff(fact_5522_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : sorted_wrt(int,ord_less(int),upto(I,J)) ).

% sorted_wrt_upto
tff(fact_5523_interv__sum__list__conv__sum__set__int,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(int,A),K: int,L: int] : ( aa(list(A),A,groups8242544230860333062m_list(A),aa(list(int),list(A),map(int,A,F2),upto(K,L))) = aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),F2),aa(list(int),set(int),set2(int),upto(K,L))) ) ) ).

% interv_sum_list_conv_sum_set_int
tff(fact_5524_sum__set__upto__conv__sum__list__int,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(int,A),I: int,J: int] : ( aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),F2),aa(list(int),set(int),set2(int),upto(I,J))) = aa(list(A),A,groups8242544230860333062m_list(A),aa(list(int),list(A),map(int,A,F2),upto(I,J))) ) ) ).

% sum_set_upto_conv_sum_list_int
tff(fact_5525_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = append(int,upto(I,J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K)) ) ) ) ).

% upto_split2
tff(fact_5526_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = append(int,upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int))),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_5527_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : ( set_or7035219750837199246ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ) ).

% atLeastLessThan_upto
tff(fact_5528_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : ( set_or3652927894154168847AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ).

% greaterThanAtMost_upto
tff(fact_5529_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = cons(int,I,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_5530_upto_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xb,Xaa) = Y )
     => ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),Xaa),cons(int,Xb,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),one_one(int)),Xaa)),nil(int)) ) ) ).

% upto.elims
tff(fact_5531_upto_Osimps,axiom,
    ! [I: int,J: int] :
      ( upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),cons(int,I,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ) ).

% upto.simps
tff(fact_5532_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = append(int,upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int))),cons(int,J,nil(int))) ) ) ).

% upto_rec2
tff(fact_5533_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : ( set_or5935395276787703475ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ) ).

% greaterThanLessThan_upto
tff(fact_5534_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = append(int,upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int))),cons(int,J,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K))) ) ) ) ).

% upto_split3
tff(fact_5535_take__hd__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( append(A,take(A,Nb,Xs),cons(A,hd(A,drop(A,Nb,Xs)),nil(A))) = take(A,aa(nat,nat,suc,Nb),Xs) ) ) ).

% take_hd_drop
tff(fact_5536_extract__Some__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys2: list(A),Y: A,Zs3: list(A)] :
      ( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),product_Pair(list(A),product_prod(A,list(A)),Ys2),aa(list(A),product_prod(A,list(A)),product_Pair(A,list(A),Y),Zs3))) )
    <=> ( ( Xs = append(A,Ys2,cons(A,Y,Zs3)) )
        & aa(A,$o,P,Y)
        & ~ ? [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Ys2))
              & aa(A,$o,P,X4) ) ) ) ).

% extract_Some_iff
tff(fact_5537_extract__SomeE,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys2: list(A),Y: A,Zs3: list(A)] :
      ( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),product_Pair(list(A),product_prod(A,list(A)),Ys2),aa(list(A),product_prod(A,list(A)),product_Pair(A,list(A),Y),Zs3))) )
     => ( ( Xs = append(A,Ys2,cons(A,Y,Zs3)) )
        & aa(A,$o,P,Y)
        & ~ ? [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Ys2))
              & aa(A,$o,P,X3) ) ) ) ).

% extract_SomeE
tff(fact_5538_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_5539_hd__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( Nb != zero_zero(nat) )
     => ( hd(A,replicate(A,Nb,Xb)) = Xb ) ) ).

% hd_replicate
tff(fact_5540_hd__take,axiom,
    ! [A: $tType,J: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),J)
     => ( hd(A,take(A,J,Xs)) = hd(A,Xs) ) ) ).

% hd_take
tff(fact_5541_list_Oset__sel_I1_J,axiom,
    ! [A: $tType,A2: list(A)] :
      ( ( A2 != nil(A) )
     => member(A,hd(A,A2),aa(list(A),set(A),set2(A),A2)) ) ).

% list.set_sel(1)
tff(fact_5542_hd__in__set,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => member(A,hd(A,Xs),aa(list(A),set(A),set2(A),Xs)) ) ).

% hd_in_set
tff(fact_5543_extract__None__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( extract(A,P,Xs) = none(product_prod(list(A),product_prod(A,list(A)))) )
    <=> ~ ? [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P,X4) ) ) ).

% extract_None_iff
tff(fact_5544_hd__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( hd(A,Xs) = aa(nat,A,nth(A,Xs),zero_zero(nat)) ) ) ).

% hd_conv_nth
tff(fact_5545_hd__drop__conv__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( hd(A,drop(A,Nb,Xs)) = aa(nat,A,nth(A,Xs),Nb) ) ) ).

% hd_drop_conv_nth
tff(fact_5546_insort__key__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [A2: A,Xs: list(A),F2: fun(A,B)] :
          ( member(A,A2,aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F2),Xs))
           => ( ( hd(A,filter2(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_or(A,fun(fun(A,B),fun(A,$o)),A2),F2),Xs)) = A2 )
             => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),A2),remove1(A,A2,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_5547_sorted__insort__is__snoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A2: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A2) )
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_os(A,A)),A2),Xs) = append(A,Xs,cons(A,A2,nil(A))) ) ) ) ) ).

% sorted_insort_is_snoc
tff(fact_5548_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] :
      ( size_list(A,F2,Xs) = $ite(Xs = nil(A),zero_zero(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,hd(A,Xs))),size_list(A,F2,tl(A,Xs))))) ) ).

% Nitpick.size_list_simp(1)
tff(fact_5549_tl__upt,axiom,
    ! [Ma: nat,Nb: nat] : ( tl(nat,upt(Ma,Nb)) = upt(aa(nat,nat,suc,Ma),Nb) ) ).

% tl_upt
tff(fact_5550_length__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xb: A,Xs: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xb),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% length_insort
tff(fact_5551_length__tl,axiom,
    ! [A: $tType,Xs: list(A)] : ( aa(list(A),nat,size_size(list(A)),tl(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ) ).

% length_tl
tff(fact_5552_tl__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : ( tl(A,replicate(A,Nb,Xb)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Xb) ) ).

% tl_replicate
tff(fact_5553_list_Oset__sel_I2_J,axiom,
    ! [A: $tType,A2: list(A),Xb: A] :
      ( ( A2 != nil(A) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),tl(A,A2)))
       => member(A,Xb,aa(list(A),set(A),set2(A),A2)) ) ) ).

% list.set_sel(2)
tff(fact_5554_set__insort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xb: A,Xs: list(A)] : ( aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xb),Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),aa(list(A),set(A),set2(A),Xs)) ) ) ).

% set_insort_key
tff(fact_5555_take__tl,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : ( take(A,Nb,tl(A,Xs)) = tl(A,take(A,aa(nat,nat,suc,Nb),Xs)) ) ).

% take_tl
tff(fact_5556_drop__Suc,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : ( drop(A,aa(nat,nat,suc,Nb),Xs) = drop(A,Nb,tl(A,Xs)) ) ).

% drop_Suc
tff(fact_5557_distinct__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xb: A,Xs: list(A)] :
          ( distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xb),Xs))
        <=> ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
            & distinct(A,Xs) ) ) ) ).

% distinct_insort
tff(fact_5558_insort__is__Cons,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F2: fun(A,B),A2: A] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A2)),aa(A,B,F2,X)) )
         => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),A2),Xs) = cons(A,A2,Xs) ) ) ) ).

% insort_is_Cons
tff(fact_5559_tl__take,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : ( tl(A,take(A,Nb,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),tl(A,Xs)) ) ).

% tl_take
tff(fact_5560_Nitpick_Osize__list__simp_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(list(A),nat,size_size(list(A)),Xs) = $ite(Xs = nil(A),zero_zero(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),tl(A,Xs)))) ) ).

% Nitpick.size_list_simp(2)
tff(fact_5561_nth__tl,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),tl(A,Xs)))
     => ( aa(nat,A,nth(A,tl(A,Xs)),Nb) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,Nb)) ) ) ).

% nth_tl
tff(fact_5562_distinct__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xb: B,Xs: list(B)] :
          ( distinct(A,aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Xb),Xs)))
        <=> ( ~ member(A,aa(B,A,F2,Xb),image(B,A,F2,aa(list(B),set(B),set2(B),Xs)))
            & distinct(A,aa(list(B),list(A),map(B,A,F2),Xs)) ) ) ) ).

% distinct_insort_key
tff(fact_5563_insort__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,Xs: list(A)] :
          ( member(A,A2,aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_os(A,A)),A2),remove1(A,A2,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_5564_take__Suc,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( Xs != nil(A) )
     => ( take(A,aa(nat,nat,suc,Nb),Xs) = cons(A,hd(A,Xs),take(A,Nb,tl(A,Xs))) ) ) ).

% take_Suc
tff(fact_5565_list__encode_Opelims,axiom,
    ! [Xb: list(nat),Y: nat] :
      ( ( aa(list(nat),nat,nat_list_encode,Xb) = Y )
     => ( accp(list(nat),nat_list_encode_rel,Xb)
       => ( ( ( Xb = nil(nat) )
           => ( ( Y = zero_zero(nat) )
             => ~ accp(list(nat),nat_list_encode_rel,nil(nat)) ) )
         => ~ ! [X: nat,Xs2: list(nat)] :
                ( ( Xb = cons(nat,X,Xs2) )
               => ( ( Y = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),aa(list(nat),nat,nat_list_encode,Xs2)))) )
                 => ~ accp(list(nat),nat_list_encode_rel,cons(nat,X,Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_5566_remdups__adj__singleton__iff,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs)) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( Xs != nil(A) )
        & ( Xs = replicate(A,aa(list(A),nat,size_size(list(A)),Xs),hd(A,Xs)) ) ) ) ).

% remdups_adj_singleton_iff
tff(fact_5567_Code__Target__Nat_ONat_Oabs__eq,axiom,
    ! [Xb: int] : ( aa(code_integer,nat,code_Target_Nat,aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,nat,nat2,Xb) ) ).

% Code_Target_Nat.Nat.abs_eq
tff(fact_5568_remdups__adj__set,axiom,
    ! [A: $tType,Xs: list(A)] : ( aa(list(A),set(A),set2(A),remdups_adj(A,Xs)) = aa(list(A),set(A),set2(A),Xs) ) ).

% remdups_adj_set
tff(fact_5569_remdups__adj__adjacent,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs)))
     => ( aa(nat,A,nth(A,remdups_adj(A,Xs)),I) != aa(nat,A,nth(A,remdups_adj(A,Xs)),aa(nat,nat,suc,I)) ) ) ).

% remdups_adj_adjacent
tff(fact_5570_remdups__adj__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( remdups_adj(A,replicate(A,Nb,Xb)) = $ite(Nb = zero_zero(nat),nil(A),cons(A,Xb,nil(A))) ) ).

% remdups_adj_replicate
tff(fact_5571_remdups__adj__length__ge1,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))) ) ).

% remdups_adj_length_ge1
tff(fact_5572_Code__Target__Nat_ONat_Orep__eq,axiom,
    ! [Xb: code_integer] : ( aa(code_integer,nat,code_Target_Nat,Xb) = aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,Xb)) ) ).

% Code_Target_Nat.Nat.rep_eq
tff(fact_5573_remdups__adj__altdef,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( remdups_adj(A,Xs) = Ys2 )
    <=> ? [F5: fun(nat,nat)] :
          ( order_mono(nat,nat,F5)
          & ( image(nat,nat,F5,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Ys2)) )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
             => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys2),aa(nat,nat,F5,I3)) ) )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xs))
             => ( ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))) )
              <=> ( aa(nat,nat,F5,I3) = aa(nat,nat,F5,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))) ) ) ) ) ) ).

% remdups_adj_altdef
tff(fact_5574_range__mod,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( image(nat,nat,aTP_Lamp_ot(nat,fun(nat,nat),Nb),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb) ) ) ).

% range_mod
tff(fact_5575_prod_Odistinct__set__conv__list,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Xs: list(A),G: fun(A,B)] :
          ( distinct(A,Xs)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(list(A),set(A),set2(A),Xs)) = aa(list(B),B,groups5270119922927024881d_list(B),aa(list(A),list(B),map(A,B,G),Xs)) ) ) ) ).

% prod.distinct_set_conv_list
tff(fact_5576_inf__top__left,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),Xb) = Xb ) ) ).

% inf_top_left
tff(fact_5577_inf__top__right,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),top_top(A)) = Xb ) ) ).

% inf_top_right
tff(fact_5578_inf__eq__top__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = top_top(A) )
        <=> ( ( Xb = top_top(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% inf_eq_top_iff
tff(fact_5579_top__eq__inf__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xb: A,Y: A] :
          ( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) )
        <=> ( ( Xb = top_top(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% top_eq_inf_iff
tff(fact_5580_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_5581_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_5582_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_5583_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_5584_range__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: 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_5585_surj__plus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: 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_5586_range__diff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : ( image(A,A,aa(A,fun(A,A),minus_minus(A),A2),top_top(set(A))) = top_top(set(A)) ) ) ).

% range_diff
tff(fact_5587_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_5588_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_5589_surj__fn,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat] :
      ( ( image(A,A,F2,top_top(set(A))) = top_top(set(A)) )
     => ( image(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_5590_finite__compl,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( aa(set(A),$o,finite_finite(A),aa(set(A),set(A),uminus_uminus(set(A)),A3))
      <=> aa(set(A),$o,finite_finite(A),top_top(set(A))) ) ) ).

% finite_compl
tff(fact_5591_Gcd__UNIV,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,top_top(set(A))) = one_one(A) ) ) ).

% Gcd_UNIV
tff(fact_5592_prod__list_ONil,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aa(list(A),A,groups5270119922927024881d_list(A),nil(A)) = one_one(A) ) ) ).

% prod_list.Nil
tff(fact_5593_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : ( image(A,A,aTP_Lamp_lv(A,fun(A,A),A2),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_diff_right
tff(fact_5594_mono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_inf(A)
        & semilattice_inf(B) )
     => ! [F2: fun(A,B),A3: A,B3: A] :
          ( order_mono(A,B,F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,A3)),aa(A,B,F2,B3))) ) ) ).

% mono_inf
tff(fact_5595_mono__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% mono_invE
tff(fact_5596_incseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: fun(nat,A),I: nat] :
          ( order_mono(nat,A,A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A3,I)),aa(nat,A,A3,aa(nat,nat,suc,I))) ) ) ).

% incseq_SucD
tff(fact_5597_incseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,N)),aa(nat,A,X6,aa(nat,nat,suc,N)))
         => order_mono(nat,A,X6) ) ) ).

% incseq_SucI
tff(fact_5598_incseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_mono(nat,A,F2)
        <=> ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N4)),aa(nat,A,F2,aa(nat,nat,suc,N4))) ) ) ).

% incseq_Suc_iff
tff(fact_5599_mono__pow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),Nb: nat] :
          ( order_mono(A,A,F2)
         => order_mono(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)) ) ) ).

% mono_pow
tff(fact_5600_Kleene__iter__gpfp,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [F2: fun(A,A),P3: A,K: nat] :
          ( order_mono(A,A,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,F2,P3))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),top_top(A))) ) ) ) ).

% Kleene_iter_gpfp
tff(fact_5601_funpow__increasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Ma: nat,Nb: nat,F2: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( order_mono(A,A,F2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),top_top(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Ma),F2),top_top(A))) ) ) ) ).

% funpow_increasing
tff(fact_5602_mono__add,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A] : order_mono(A,A,aa(A,fun(A,A),plus_plus(A),A2)) ) ).

% mono_add
tff(fact_5603_max__of__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),Ma: A,Nb: A] :
          ( order_mono(A,B,F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F2,Ma)),aa(A,B,F2,Nb)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_max(A),Ma),Nb)) ) ) ) ).

% max_of_mono
tff(fact_5604_mono__strict__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ) ).

% mono_strict_invE
tff(fact_5605_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_5606_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_5607_mono__Suc,axiom,
    order_mono(nat,nat,suc) ).

% mono_Suc
tff(fact_5608_mono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Q: fun(A,A)] :
          ( order_mono(A,A,Q)
         => order_mono(nat,A,aTP_Lamp_ou(fun(A,A),fun(nat,A),Q)) ) ) ).

% mono_funpow
tff(fact_5609_min__of__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),Ma: A,Nb: A] :
          ( order_mono(A,B,F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F2,Ma)),aa(A,B,F2,Nb)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_min(A),Ma),Nb)) ) ) ) ).

% min_of_mono
tff(fact_5610_funpow__mono,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(A,A),A3: A,B3: A,Nb: nat] :
          ( order_mono(A,A,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),A3)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),B3)) ) ) ) ).

% funpow_mono
tff(fact_5611_finite__fun__UNIVD1,axiom,
    ! [B: $tType,A: $tType] :
      ( aa(set(fun(A,B)),$o,finite_finite(fun(A,B)),top_top(set(fun(A,B))))
     => ( ( aa(set(B),nat,finite_card(B),top_top(set(B))) != aa(nat,nat,suc,zero_zero(nat)) )
       => aa(set(A),$o,finite_finite(A),top_top(set(A))) ) ) ).

% finite_fun_UNIVD1
tff(fact_5612_bij__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => bij_betw(A,A,uminus_uminus(A),top_top(set(A)),top_top(set(A))) ) ).

% bij_uminus
tff(fact_5613_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_5614_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_5615_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_5616_bij__fn,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat] :
      ( bij_betw(A,A,F2,top_top(set(A)),top_top(set(A)))
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),top_top(set(A)),top_top(set(A))) ) ).

% bij_fn
tff(fact_5617_mono__times__nat,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => order_mono(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb)) ) ).

% mono_times_nat
tff(fact_5618_mono__mult,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => order_mono(A,A,aa(A,fun(A,A),times_times(A),A2)) ) ) ).

% mono_mult
tff(fact_5619_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( aa(set(A),$o,finite_finite(A),image(nat,A,F2,top_top(set(nat))))
         => ( order_mono(nat,A,F2)
           => ( ! [N: nat] :
                  ( ( aa(nat,A,F2,N) = aa(nat,A,F2,aa(nat,nat,suc,N)) )
                 => ( aa(nat,A,F2,aa(nat,nat,suc,N)) = aa(nat,A,F2,aa(nat,nat,suc,aa(nat,nat,suc,N))) ) )
             => ? [N7: nat] :
                  ( ! [N3: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N7)
                     => ! [M4: nat] :
                          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N7)
                         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N3)
                           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,M4)),aa(nat,A,F2,N3)) ) ) )
                  & ! [N3: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N3)
                     => ( aa(nat,A,F2,N7) = aa(nat,A,F2,N3) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_5620_mono__image__least,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [F2: fun(A,B),Ma: A,Nb: A,M5: B,N2: B] :
          ( order_mono(A,B,F2)
         => ( ( image(A,B,F2,set_or7035219750837199246ssThan(A,Ma,Nb)) = set_or7035219750837199246ssThan(B,M5,N2) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ma),Nb)
             => ( aa(A,B,F2,Ma) = M5 ) ) ) ) ) ).

% mono_image_least
tff(fact_5621_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F2: fun(A,A),P3: A,K: nat] :
          ( order_mono(A,A,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,P3)),P3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),bot_bot(A))),P3) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_5622_funpow__mono2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(A,A),I: nat,J: nat,Xb: A,Y: A] :
          ( order_mono(A,A,F2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,F2,Xb))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),I),F2),Xb)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),J),F2),Y)) ) ) ) ) ) ).

% funpow_mono2
tff(fact_5623_prod__list__zero__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [Xs: list(A)] :
          ( ( aa(list(A),A,groups5270119922927024881d_list(A),Xs) = zero_zero(A) )
        <=> member(A,zero_zero(A),aa(list(A),set(A),set2(A),Xs)) ) ) ).

% prod_list_zero_iff
tff(fact_5624_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( ( image(B,A,F2,top_top(set(B))) = top_top(set(A)) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),image(B,A,F2,A3))),image(B,A,F2,aa(set(B),set(B),uminus_uminus(set(B)),A3))) ) ).

% surj_Compl_image_subset
tff(fact_5625_bij__image__Compl__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => ( image(A,B,F2,aa(set(A),set(A),uminus_uminus(set(A)),A3)) = aa(set(B),set(B),uminus_uminus(set(B)),image(A,B,F2,A3)) ) ) ).

% bij_image_Compl_eq
tff(fact_5626_Nats__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( semiring_1_Nats(A) = image(nat,A,semiring_1_of_nat(A),top_top(set(nat))) ) ) ).

% Nats_def
tff(fact_5627_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Ma: nat,Nb: nat,F2: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
         => ( order_mono(A,A,F2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Ma),F2),bot_bot(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),bot_bot(A))) ) ) ) ).

% funpow_decreasing
tff(fact_5628_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => folding_insort_key(A,A,ord_less_eq(A),ord_less(A),top_top(set(A)),aTP_Lamp_os(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_5629_inf__top_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => semila1105856199041335345_order(A,inf_inf(A),top_top(A),ord_less_eq(A),ord_less(A)) ) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_5630_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( aa(set(A),$o,finite_finite(A),top_top(set(A)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ).

% finite_UNIV_card_ge_0
tff(fact_5631_prod__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xs: list(A)] : ( aa(list(A),A,groups5270119922927024881d_list(A),Xs) = aa(A,A,foldr(A,A,times_times(A),Xs),one_one(A)) ) ) ).

% prod_list.eq_foldr
tff(fact_5632_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] :
      ( aa(set(A),$o,finite_finite(A),image(B,A,F2,top_top(set(B))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),image(B,A,F2,top_top(set(B))))) ) ).

% card_range_greater_zero
tff(fact_5633_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => order_mono(nat,nat,aTP_Lamp_ov(nat,fun(nat,nat),K)) ) ).

% mono_ge2_power_minus_self
tff(fact_5634_UNIV__nat__eq,axiom,
    top_top(set(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),image(nat,nat,suc,top_top(set(nat)))) ).

% UNIV_nat_eq
tff(fact_5635_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F2: fun(nat,set(A)),S3: set(A)] :
      ( ! [I2: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F2,I2)),S3)
     => ( aa(set(A),$o,finite_finite(A),S3)
       => ( ? [N8: nat] :
              ( ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N8)
                 => ! [M: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N8)
                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N)
                       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(nat,set(A),F2,M)),aa(nat,set(A),F2,N)) ) ) )
              & ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N)
                 => ( aa(nat,set(A),F2,N8) = aa(nat,set(A),F2,N) ) ) )
         => ( aa(nat,set(A),F2,aa(set(A),nat,finite_card(A),S3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),F2,top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_5636_takeWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ( takeWhile(A,aTP_Lamp_nv(A,fun(A,$o),Xb),rev(A,Xs)) = rev(A,tl(A,dropWhile(A,aTP_Lamp_nv(A,fun(A,$o),Xb),Xs))) ) ) ) ).

% takeWhile_neq_rev
tff(fact_5637_card__UNIV__unit,axiom,
    aa(set(product_unit),nat,finite_card(product_unit),top_top(set(product_unit))) = one_one(nat) ).

% card_UNIV_unit
tff(fact_5638_range__abs__Nats,axiom,
    image(int,int,abs_abs(int),top_top(set(int))) = semiring_1_Nats(int) ).

% range_abs_Nats
tff(fact_5639_Sup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Xb,Y)) = Y ) ) ) ).

% Sup_atLeastLessThan
tff(fact_5640_cSup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Y,Xb)) = Xb ) ) ) ).

% cSup_atLeastLessThan
tff(fact_5641_cSup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Y,Xb)) = Xb ) ) ) ).

% cSup_greaterThanLessThan
tff(fact_5642_Sup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Xb,Y)) = Y ) ) ) ).

% Sup_greaterThanLessThan
tff(fact_5643_Sup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Xb,Y)) = Y ) ) ) ).

% Sup_greaterThanAtMost
tff(fact_5644_cSup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Y,Xb)) = Xb ) ) ) ).

% cSup_greaterThanAtMost
tff(fact_5645_dropWhile__eq__Nil__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( dropWhile(A,P,Xs) = nil(A) )
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X4) ) ) ).

% dropWhile_eq_Nil_conv
tff(fact_5646_dropWhile__append1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o),Ys2: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P,Xb)
       => ( dropWhile(A,P,append(A,Xs,Ys2)) = append(A,dropWhile(A,P,Xs),Ys2) ) ) ) ).

% dropWhile_append1
tff(fact_5647_dropWhile__append2,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Ys2: list(A)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X) )
     => ( dropWhile(A,P,append(A,Xs,Ys2)) = dropWhile(A,P,Ys2) ) ) ).

% dropWhile_append2
tff(fact_5648_set__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] : ( aa(list(A),set(A),set2(A),concat(A,Xs)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(list(A),set(A),set2(A),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))) ) ).

% set_concat
tff(fact_5649_infinite__UNIV__int,axiom,
    ~ aa(set(int),$o,finite_finite(int),top_top(set(int))) ).

% infinite_UNIV_int
tff(fact_5650_less__cSupE,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Y: A,X6: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ~ ! [X: A] :
                  ( member(A,X,X6)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ) ).

% less_cSupE
tff(fact_5651_less__cSupD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Z: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(set(A),A,complete_Sup_Sup(A),X6))
           => ? [X: A] :
                ( member(A,X,X6)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X) ) ) ) ) ).

% less_cSupD
tff(fact_5652_finite__imp__Sup__less,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Xb: A,A2: A] :
          ( aa(set(A),$o,finite_finite(A),X6)
         => ( member(A,Xb,X6)
           => ( ! [X: A] :
                  ( member(A,X,X6)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A2) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A2) ) ) ) ) ).

% finite_imp_Sup_less
tff(fact_5653_Sup__inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B3: set(A),A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B3)),A2) = bot_bot(A) )
        <=> ! [X4: A] :
              ( member(A,X4,B3)
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X4),A2) = bot_bot(A) ) ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_5654_inf__Sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A2: A,B3: set(A)] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,complete_Sup_Sup(A),B3)) = aa(set(A),A,complete_Sup_Sup(A),image(A,A,aa(A,fun(A,A),inf_inf(A),A2),B3)) ) ) ).

% inf_Sup
tff(fact_5655_dropWhile__cong,axiom,
    ! [A: $tType,L: list(A),K: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( L = K )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),L))
           => ( aa(A,$o,P,X)
            <=> aa(A,$o,Q,X) ) )
       => ( dropWhile(A,P,L) = dropWhile(A,Q,K) ) ) ) ).

% dropWhile_cong
tff(fact_5656_set__dropWhileD,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),dropWhile(A,P,Xs)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% set_dropWhileD
tff(fact_5657_finite__subset__Union,axiom,
    ! [A: $tType,A3: set(A),B10: set(set(A))] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B10))
       => ~ ! [F6: set(set(A))] :
              ( aa(set(set(A)),$o,finite_finite(set(A)),F6)
             => ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),F6),B10)
               => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F6)) ) ) ) ) ).

% finite_subset_Union
tff(fact_5658_SUP__inf__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),A3: set(B),G: fun(C,A),B3: set(C)] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A3))),aa(set(A),A,complete_Sup_Sup(A),image(C,A,G,B3))) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_ox(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B3),A3)) ) ) ).

% SUP_inf_distrib2
tff(fact_5659_inf__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A2: A,F2: fun(B,A),B3: set(B)] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,B3))) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_oy(A,fun(fun(B,A),fun(B,A)),A2),F2),B3)) ) ) ).

% inf_SUP
tff(fact_5660_Sup__inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B3: set(A),A2: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B3)),A2) = aa(set(A),A,complete_Sup_Sup(A),image(A,A,aTP_Lamp_oz(A,fun(A,A),A2),B3)) ) ) ).

% Sup_inf
tff(fact_5661_SUP__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),B3: set(B),A2: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,B3))),A2) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_pa(fun(B,A),fun(A,fun(B,A)),F2),A2),B3)) ) ) ).

% SUP_inf
tff(fact_5662_bij__list__encode,axiom,
    bij_betw(list(nat),nat,nat_list_encode,top_top(set(list(nat))),top_top(set(nat))) ).

% bij_list_encode
tff(fact_5663_surj__list__encode,axiom,
    image(list(nat),nat,nat_list_encode,top_top(set(list(nat)))) = top_top(set(nat)) ).

% surj_list_encode
tff(fact_5664_surj__prod__encode,axiom,
    image(product_prod(nat,nat),nat,nat_prod_encode,top_top(set(product_prod(nat,nat)))) = top_top(set(nat)) ).

% surj_prod_encode
tff(fact_5665_bij__prod__encode,axiom,
    bij_betw(product_prod(nat,nat),nat,nat_prod_encode,top_top(set(product_prod(nat,nat))),top_top(set(nat))) ).

% bij_prod_encode
tff(fact_5666_finite__Sup__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite(A),X6)
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A2)
            <=> ! [X4: A] :
                  ( member(A,X4,X6)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),A2) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_5667_sum_OUnion__comp,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B3: set(set(A)),G: fun(A,B)] :
          ( ! [X: set(A)] :
              ( member(set(A),X,B3)
             => aa(set(A),$o,finite_finite(A),X) )
         => ( ! [A12: set(A)] :
                ( member(set(A),A12,B3)
               => ! [A23: set(A)] :
                    ( member(set(A),A23,B3)
                   => ( ( A12 != A23 )
                     => ! [X: A] :
                          ( member(A,X,A12)
                         => ( member(A,X,A23)
                           => ( aa(A,B,G,X) = zero_zero(B) ) ) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7311177749621191930dd_sum(set(A),B)),groups7311177749621191930dd_sum(A,B)),G),B3) ) ) ) ) ).

% sum.Union_comp
tff(fact_5668_prod_OUnion__comp,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B3: set(set(A)),G: fun(A,B)] :
          ( ! [X: set(A)] :
              ( member(set(A),X,B3)
             => aa(set(A),$o,finite_finite(A),X) )
         => ( ! [A12: set(A)] :
                ( member(set(A),A12,B3)
               => ! [A23: set(A)] :
                    ( member(set(A),A23,B3)
                   => ( ( A12 != A23 )
                     => ! [X: A] :
                          ( member(A,X,A12)
                         => ( member(A,X,A23)
                           => ( aa(A,B,G,X) = one_one(B) ) ) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B)),groups7121269368397514597t_prod(A,B)),G),B3) ) ) ) ) ).

% prod.Union_comp
tff(fact_5669_takeWhile__eq__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),dropWhile(A,P,Xs)))
         => ~ aa(A,$o,P,X) )
     => ( takeWhile(A,P,Xs) = filter2(A,P,Xs) ) ) ).

% takeWhile_eq_filter
tff(fact_5670_UN__UN__finite__eq,axiom,
    ! [A: $tType,A3: fun(nat,set(A))] : ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),aTP_Lamp_pb(fun(nat,set(A)),fun(nat,set(A)),A3),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A3,top_top(set(nat)))) ) ).

% UN_UN_finite_eq
tff(fact_5671_UN__le__add__shift__strict,axiom,
    ! [A: $tType,M9: fun(nat,set(A)),K: nat,Nb: nat] : ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_pc(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M9),K),set_ord_lessThan(nat,Nb))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),M9,set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K)))) ) ).

% UN_le_add_shift_strict
tff(fact_5672_UN__le__add__shift,axiom,
    ! [A: $tType,M9: fun(nat,set(A)),K: nat,Nb: nat] : ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_pc(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M9),K),set_ord_atMost(nat,Nb))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),M9,set_or1337092689740270186AtMost(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K)))) ) ).

% UN_le_add_shift
tff(fact_5673_dropWhile__append,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys2: list(A)] :
      ( dropWhile(A,P,append(A,Xs,Ys2)) = $ite(
          ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
           => aa(A,$o,P,X4) ),
          dropWhile(A,P,Ys2),
          append(A,dropWhile(A,P,Xs),Ys2) ) ) ).

% dropWhile_append
tff(fact_5674_Ints__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_Ints(A) = image(int,A,ring_1_of_int(A),top_top(set(int))) ) ) ).

% Ints_def
tff(fact_5675_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),L: A,E2: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,S3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),L))),E2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Sup_Sup(A),S3)),L))),E2) ) ) ) ).

% cSup_asclose
tff(fact_5676_int__in__range__abs,axiom,
    ! [Nb: nat] : member(int,aa(nat,int,semiring_1_of_nat(int),Nb),image(int,int,abs_abs(int),top_top(set(int)))) ).

% int_in_range_abs
tff(fact_5677_lexordp_Omono,axiom,
    ! [A: $tType] :
      ( ord(A)
     => order_mono(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_pd(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)))) ) ).

% lexordp.mono
tff(fact_5678_UN__finite__subset,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),C6: set(A)] :
      ( ! [N: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A3,set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),C6)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A3,top_top(set(nat))))),C6) ) ).

% UN_finite_subset
tff(fact_5679_UN__finite2__eq,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),B3: fun(nat,set(A)),K: nat] :
      ( ! [N: nat] : ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A3,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),B3,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))) )
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A3,top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),B3,top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_5680_dropWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs)))
     => ( aa(nat,A,nth(A,dropWhile(A,P,Xs)),J) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ) ).

% dropWhile_nth
tff(fact_5681_UN__finite2__subset,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),B3: fun(nat,set(A)),K: nat] :
      ( ! [N: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A3,set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),B3,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A3,top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),B3,top_top(set(nat))))) ) ).

% UN_finite2_subset
tff(fact_5682_dropWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ( dropWhile(A,aTP_Lamp_nv(A,fun(A,$o),Xb),rev(A,Xs)) = cons(A,Xb,rev(A,takeWhile(A,aTP_Lamp_nv(A,fun(A,$o),Xb),Xs))) ) ) ) ).

% dropWhile_neq_rev
tff(fact_5683_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A3)) = top_top(A) )
        <=> ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),top_top(A))
             => ? [Xa3: B] :
                  ( member(B,Xa3,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),aa(B,A,F2,Xa3)) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_5684_Sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A3) = top_top(A) )
        <=> ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),top_top(A))
             => ? [Xa3: A] :
                  ( member(A,Xa3,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Xa3) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_5685_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xb: A,F2: fun(B,A),A3: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A3)))
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),Xb)
             => ? [X4: B] :
                  ( member(B,X4,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),aa(B,A,F2,X4)) ) ) ) ) ).

% le_SUP_iff
tff(fact_5686_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_5687_less__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,S3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),S3))
        <=> ? [X4: A] :
              ( member(A,X4,S3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X4) ) ) ) ).

% less_Sup_iff
tff(fact_5688_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xb: A,A3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),A3))
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),Xb)
             => ? [X4: A] :
                  ( member(A,X4,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X4) ) ) ) ) ).

% le_Sup_iff
tff(fact_5689_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: set(B),Y: A,I: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A3))),Y)
         => ( member(B,I,A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,I)),Y) ) ) ) ).

% SUP_lessD
tff(fact_5690_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,F2: fun(B,A),A3: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A3)))
        <=> ? [X4: B] :
              ( member(B,X4,A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,X4)) ) ) ) ).

% less_SUP_iff
tff(fact_5691_card__UNION,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite(set(A)),A3)
     => ( ! [X: set(A)] :
            ( member(set(A),X,A3)
           => aa(set(A),$o,finite_finite(A),X) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)) = aa(int,nat,nat2,aa(set(set(set(A))),int,aa(fun(set(set(A)),int),fun(set(set(set(A))),int),groups7311177749621191930dd_sum(set(set(A)),int),aTP_Lamp_pe(set(set(A)),int)),collect(set(set(A)),aTP_Lamp_pf(set(set(A)),fun(set(set(A)),$o),A3)))) ) ) ) ).

% card_UNION
tff(fact_5692_root__def,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(real,real,root(Nb),Xb) = $ite(Nb = zero_zero(nat),zero_zero(real),the_inv_into(real,real,top_top(set(real)),aTP_Lamp_pg(nat,fun(real,real),Nb),Xb)) ) ).

% root_def
tff(fact_5693_length__remdups__concat,axiom,
    ! [A: $tType,Xss: list(list(A))] : ( aa(list(A),nat,size_size(list(A)),remdups(A,concat(A,Xss))) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(list(A),set(A),set2(A),aa(list(list(A)),set(list(A)),set2(list(A)),Xss)))) ) ).

% length_remdups_concat
tff(fact_5694_set__remdups,axiom,
    ! [A: $tType,Xs: list(A)] : ( aa(list(A),set(A),set2(A),remdups(A,Xs)) = aa(list(A),set(A),set2(A),Xs) ) ).

% set_remdups
tff(fact_5695_Inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),A3) = bot_bot(A) )
        <=> ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X4)
             => ? [Xa3: A] :
                  ( member(A,Xa3,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa3),X4) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_5696_Inf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Xb,Y)) = Xb ) ) ) ).

% Inf_atLeastLessThan
tff(fact_5697_cInf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Y,Xb)) = Y ) ) ) ).

% cInf_atLeastLessThan
tff(fact_5698_cInf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Y,Xb)) = Y ) ) ) ).

% cInf_greaterThanLessThan
tff(fact_5699_Inf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Xb,Y)) = Xb ) ) ) ).

% Inf_greaterThanLessThan
tff(fact_5700_Inf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Xb,Y)) = Xb ) ) ) ).

% Inf_greaterThanAtMost
tff(fact_5701_cInf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Y,Xb)) = Y ) ) ) ).

% cInf_greaterThanAtMost
tff(fact_5702_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A3)) = bot_bot(A) )
        <=> ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X4)
             => ? [Xa3: B] :
                  ( member(B,Xa3,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,Xa3)),X4) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_5703_Compl__INT,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A3: set(B)] : ( aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image(B,set(A),B3,A3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),aTP_Lamp_ph(fun(B,set(A)),fun(B,set(A)),B3),A3)) ) ).

% Compl_INT
tff(fact_5704_Compl__UN,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A3: set(B)] : ( aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),B3,A3))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image(B,set(A),aTP_Lamp_ph(fun(B,set(A)),fun(B,set(A)),B3),A3)) ) ).

% Compl_UN
tff(fact_5705_Inf__atMostLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_ord_lessThan(A,Xb)) = bot_bot(A) ) ) ) ).

% Inf_atMostLessThan
tff(fact_5706_finite__imp__less__Inf,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Xb: A,A2: A] :
          ( aa(set(A),$o,finite_finite(A),X6)
         => ( member(A,Xb,X6)
           => ( ! [X: A] :
                  ( member(A,X,X6)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Inf_Inf(A),X6)) ) ) ) ) ).

% finite_imp_less_Inf
tff(fact_5707_Sup__Inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: set(set(A))] : ( aa(set(A),A,complete_Sup_Sup(A),image(set(A),A,complete_Inf_Inf(A),A3)) = aa(set(A),A,complete_Inf_Inf(A),image(set(A),A,complete_Sup_Sup(A),collect(set(A),aTP_Lamp_pi(set(set(A)),fun(set(A),$o),A3)))) ) ) ).

% Sup_Inf
tff(fact_5708_Inf__less__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [S3: set(A),A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S3)),A2)
        <=> ? [X4: A] :
              ( member(A,X4,S3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),A2) ) ) ) ).

% Inf_less_iff
tff(fact_5709_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),Xb)
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y5)
             => ? [X4: A] :
                  ( member(A,X4,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y5) ) ) ) ) ).

% Inf_le_iff
tff(fact_5710_cInf__lessD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Z: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Z)
           => ? [X: A] :
                ( member(A,X,X6)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z) ) ) ) ) ).

% cInf_lessD
tff(fact_5711_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A,F2: fun(B,A),A3: set(B),I: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A3)))
         => ( member(B,I,A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(B,A,F2,I)) ) ) ) ).

% less_INF_D
tff(fact_5712_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B),A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A3))),A2)
        <=> ? [X4: B] :
              ( member(B,X4,A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X4)),A2) ) ) ) ).

% INF_less_iff
tff(fact_5713_remdups_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( remdups(A,cons(A,Xb,Xs)) = $ite(member(A,Xb,aa(list(A),set(A),set2(A),Xs)),remdups(A,Xs),cons(A,Xb,remdups(A,Xs))) ) ).

% remdups.simps(2)
tff(fact_5714_SUP__INF__set,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [G: fun(B,A),A3: set(set(B))] : ( aa(set(A),A,complete_Sup_Sup(A),image(set(B),A,aTP_Lamp_pj(fun(B,A),fun(set(B),A),G),A3)) = aa(set(A),A,complete_Inf_Inf(A),image(set(B),A,aTP_Lamp_pk(fun(B,A),fun(set(B),A),G),collect(set(B),aTP_Lamp_pl(set(set(B)),fun(set(B),$o),A3)))) ) ) ).

% SUP_INF_set
tff(fact_5715_INF__SUP__set,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [G: fun(B,A),A3: set(set(B))] : ( aa(set(A),A,complete_Inf_Inf(A),image(set(B),A,aTP_Lamp_pk(fun(B,A),fun(set(B),A),G),A3)) = aa(set(A),A,complete_Sup_Sup(A),image(set(B),A,aTP_Lamp_pj(fun(B,A),fun(set(B),A),G),collect(set(B),aTP_Lamp_pl(set(set(B)),fun(set(B),$o),A3)))) ) ) ).

% INF_SUP_set
tff(fact_5716_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B),Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A3))),Xb)
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y5)
             => ? [X4: B] :
                  ( member(B,X4,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X4)),Y5) ) ) ) ) ).

% INF_le_iff
tff(fact_5717_finite__less__Inf__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite(A),X6)
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Inf_Inf(A),X6))
            <=> ! [X4: A] :
                  ( member(A,X4,X6)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X4) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_5718_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),A2: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,S3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Inf_Inf(A),S3))),A2) ) ) ) ).

% cInf_abs_ge
tff(fact_5719_uminus__Sup,axiom,
    ! [A: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [A3: set(A)] : ( aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Sup_Sup(A),A3)) = aa(set(A),A,complete_Inf_Inf(A),image(A,A,uminus_uminus(A),A3)) ) ) ).

% uminus_Sup
tff(fact_5720_uminus__Inf,axiom,
    ! [A: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [A3: set(A)] : ( aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Inf_Inf(A),A3)) = aa(set(A),A,complete_Sup_Sup(A),image(A,A,uminus_uminus(A),A3)) ) ) ).

% uminus_Inf
tff(fact_5721_uminus__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [B3: fun(B,A),A3: set(B)] : ( aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,B3,A3))) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,aTP_Lamp_pm(fun(B,A),fun(B,A),B3),A3)) ) ) ).

% uminus_SUP
tff(fact_5722_uminus__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [B3: fun(B,A),A3: set(B)] : ( aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,B3,A3))) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,aTP_Lamp_pm(fun(B,A),fun(B,A),B3),A3)) ) ) ).

% uminus_INF
tff(fact_5723_length__remdups__card__conv,axiom,
    ! [A: $tType,Xs: list(A)] : ( aa(list(A),nat,size_size(list(A)),remdups(A,Xs)) = aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs)) ) ).

% length_remdups_card_conv
tff(fact_5724_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),L: A,E2: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,S3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),L))),E2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Inf_Inf(A),S3)),L))),E2) ) ) ) ).

% cInf_asclose
tff(fact_5725_mono__bij__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & comple5582772986160207858norder(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F2)
         => ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
           => ( aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A3)) = aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,A3)) ) ) ) ) ).

% mono_bij_Inf
tff(fact_5726_sum__code,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),Xs: list(B)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(list(B),set(B),set2(B),Xs)) = aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,G),remdups(B,Xs))) ) ) ).

% sum_code
tff(fact_5727_SUP__INF,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P: fun(C,fun(B,A))] : ( aa(set(A),A,complete_Sup_Sup(A),image(B,A,aTP_Lamp_po(fun(C,fun(B,A)),fun(B,A),P),top_top(set(B)))) = aa(set(A),A,complete_Inf_Inf(A),image(fun(B,C),A,aTP_Lamp_pq(fun(C,fun(B,A)),fun(fun(B,C),A),P),top_top(set(fun(B,C))))) ) ) ).

% SUP_INF
tff(fact_5728_INF__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P: fun(C,fun(B,A))] : ( aa(set(A),A,complete_Inf_Inf(A),image(B,A,aTP_Lamp_pr(fun(C,fun(B,A)),fun(B,A),P),top_top(set(B)))) = aa(set(A),A,complete_Sup_Sup(A),image(fun(B,C),A,aTP_Lamp_ps(fun(C,fun(B,A)),fun(fun(B,C),A),P),top_top(set(fun(B,C))))) ) ) ).

% INF_SUP
tff(fact_5729_prod_Oset__conv__list,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),Xs: list(B)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(list(B),set(B),set2(B),Xs)) = aa(list(A),A,groups5270119922927024881d_list(A),aa(list(B),list(A),map(B,A,G),remdups(B,Xs))) ) ) ).

% prod.set_conv_list
tff(fact_5730_INF__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),image(nat,A,aTP_Lamp_pt(A,fun(nat,A),B3),collect(nat,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) ) ) ).

% INF_nat_binary
tff(fact_5731_DERIV__real__root__generic,axiom,
    ! [Nb: nat,Xb: real,D7: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( Xb != zero_zero(real) )
       => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
             => ( D7 = 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),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))) ) ) )
         => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
               => ( D7 = 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),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat))))))) ) ) )
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
               => ( D7 = 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),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(Nb),D7,topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_5732_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
         => has_field_derivative(real,root(Nb),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),Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_even_real_root
tff(fact_5733_DERIV__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : has_field_derivative(A,cos(A),aa(A,A,uminus_uminus(A),sin(A,Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ).

% DERIV_cos
tff(fact_5734_Inf__real__def,axiom,
    ! [X6: set(real)] : ( aa(set(real),real,complete_Inf_Inf(real),X6) = aa(real,real,uminus_uminus(real),aa(set(real),real,complete_Sup_Sup(real),image(real,real,uminus_uminus(real),X6))) ) ).

% Inf_real_def
tff(fact_5735_DERIV__ln,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => has_field_derivative(real,ln_ln(real),aa(real,real,inverse_inverse(real),Xb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_ln
tff(fact_5736_DERIV__fun__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Ma: A,Xb: A] :
          ( has_field_derivative(A,G,Ma,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_pu(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,G,Xb)))),Ma),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_fun_cos
tff(fact_5737_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K: A,Xaa: A] : has_field_derivative(A,aTP_Lamp_pv(A,fun(A,A),K),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xaa),K))),topolo174197925503356063within(A,Xaa,top_top(set(A)))) ) ).

% DERIV_cos_add
tff(fact_5738_Inf__int__def,axiom,
    ! [X6: set(int)] : ( aa(set(int),int,complete_Inf_Inf(int),X6) = aa(int,int,uminus_uminus(int),aa(set(int),int,complete_Sup_Sup(int),image(int,int,uminus_uminus(int),X6))) ) ).

% Inf_int_def
tff(fact_5739_DERIV__ln__divide,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => has_field_derivative(real,ln_ln(real),divide_divide(real,one_one(real),Xb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_ln_divide
tff(fact_5740_at__within__Icc__at,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,Xb: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B2)
           => ( topolo174197925503356063within(A,Xb,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,Xb,top_top(set(A))) ) ) ) ) ).

% at_within_Icc_at
tff(fact_5741_DERIV__fun__pow,axiom,
    ! [G: fun(real,real),Ma: real,Xb: real,Nb: nat] :
      ( has_field_derivative(real,G,Ma,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_pw(fun(real,real),fun(nat,fun(real,real)),G),Nb),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),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,G,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))))),Ma),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_fun_pow
tff(fact_5742_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_5743_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Z)
     => has_field_derivative(real,aTP_Lamp_px(real,fun(real,real),R2),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,Z,aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),one_one(real)))),topolo174197925503356063within(real,Z,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_5744_DERIV__log,axiom,
    ! [Xb: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => has_field_derivative(real,log2(B2),divide_divide(real,one_one(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,ln_ln(real),B2)),Xb)),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_5745_DERIV__fun__powr,axiom,
    ! [G: fun(real,real),Ma: real,Xb: real,R2: real] :
      ( has_field_derivative(real,G,Ma,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G,Xb))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_py(fun(real,real),fun(real,fun(real,real)),G),R2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,aa(real,real,G,Xb),aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),Ma),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_5746_DERIV__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( aa(A,A,cos(A),Xb) != 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),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_tan
tff(fact_5747_DERIV__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => has_field_derivative(real,sqrt,divide_divide(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb)),aa(num,real,numeral_numeral(real),bit0(one2))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_real_sqrt
tff(fact_5748_DERIV__arctan,axiom,
    ! [Xb: 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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ).

% DERIV_arctan
tff(fact_5749_arsinh__real__has__field__derivative,axiom,
    ! [Xb: real,A3: set(real)] : has_field_derivative(real,arsinh(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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,Xb,A3)) ).

% arsinh_real_has_field_derivative
tff(fact_5750_DERIV__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sin(A,Xb) != zero_zero(A) )
         => has_field_derivative(A,cot(A),aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_cot
tff(fact_5751_has__field__derivative__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Xb: A,Db: A,S: set(A)] :
          ( ( cosh(A,aa(A,A,G,Xb)) != zero_zero(A) )
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aTP_Lamp_pz(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,Xb))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Db),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_5752_DERIV__real__sqrt__generic,axiom,
    ! [Xb: real,D7: real] :
      ( ( Xb != zero_zero(real) )
     => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( D7 = divide_divide(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb)),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
       => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
           => ( D7 = divide_divide(real,aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb))),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
         => has_field_derivative(real,sqrt,D7,topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_5753_arcosh__real__has__field__derivative,axiom,
    ! [Xb: real,A3: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => has_field_derivative(real,arcosh(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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,Xb,A3)) ) ).

% arcosh_real_has_field_derivative
tff(fact_5754_artanh__real__has__field__derivative,axiom,
    ! [Xb: real,A3: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => has_field_derivative(real,artanh(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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,Xb,A3)) ) ).

% artanh_real_has_field_derivative
tff(fact_5755_DERIV__power__series_H,axiom,
    ! [R: real,F2: fun(nat,real),X0: real] :
      ( ! [X: real] :
          ( member(real,X,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_qa(fun(nat,real),fun(real,fun(nat,real)),F2),X)) )
     => ( member(real,X0,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
         => has_field_derivative(real,aTP_Lamp_qc(fun(nat,real),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),aTP_Lamp_qa(fun(nat,real),fun(real,fun(nat,real)),F2),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_5756_DERIV__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => has_field_derivative(real,root(Nb),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),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_real_root
tff(fact_5757_DERIV__arccos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_arccos
tff(fact_5758_DERIV__arcsin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),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),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_arcsin
tff(fact_5759_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Xb: real,Nb: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
        & ! [M: nat,X: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),X),topolo174197925503356063within(real,X,top_top(set(real)))) )
     => ? [T2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),Xb))
          & ( aa(real,real,F2,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_qd(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T2),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_5760_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Xb: real,Nb: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M: nat,X: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),X),topolo174197925503356063within(real,X,top_top(set(real))))
       => ? [T2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),Xb))
            & ( aa(real,real,F2,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_qd(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T2),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_5761_DERIV__odd__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( ( Xb != zero_zero(real) )
       => has_field_derivative(real,root(Nb),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),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_odd_real_root
tff(fact_5762_Maclaurin,axiom,
    ! [H: real,Nb: 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)),Nb)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M: nat,T2: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),H) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
           => ? [T2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),H)
                & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_qe(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T2),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_5763_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),F2: fun(real,real),Nb: 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 )
       => ( ! [M: nat,T2: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),H) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
         => ? [T2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T2)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),H)
              & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_qe(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T2),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_5764_Maclaurin__minus,axiom,
    ! [H: real,Nb: 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)),Nb)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M: nat,T2: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),H),T2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),zero_zero(real)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
           => ? [T2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H),T2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),zero_zero(real))
                & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_qe(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T2),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_5765_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Nb: nat,Xb: 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)),Nb)
       => ( ( Xb != zero_zero(real) )
         => ( ! [M: nat,X: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),X),topolo174197925503356063within(real,X,top_top(set(real))))
           => ? [T2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T2))
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),Xb))
                & ( aa(real,real,F2,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_qd(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T2),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_5766_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Nb: nat,Xb: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M: nat,T2: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),Xb)) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
       => ? [T2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),Xb))
            & ( aa(real,real,F2,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_qd(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T2),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_5767_Taylor,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M: nat,T2: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T2),topolo174197925503356063within(real,T2,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),Xb)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),B2)
                 => ( ( Xb != C2 )
                   => ? [T2: real] :
                        ( $ite(
                            aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),C2),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),T2)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),C2) ),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),T2)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),Xb) ) )
                        & ( aa(real,real,F2,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_qf(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C2),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T2),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),C2)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_5768_Taylor__up,axiom,
    ! [Nb: 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)),Nb)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M: nat,T2: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T2),topolo174197925503356063within(real,T2,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)
             => ? [T2: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),T2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),B2)
                  & ( aa(real,real,F2,B2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_qg(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B2),C2)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T2),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),C2)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_5769_Taylor__down,axiom,
    ! [Nb: 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)),Nb)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M: nat,T2: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T2),topolo174197925503356063within(real,T2,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)
             => ? [T2: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),T2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),C2)
                  & ( aa(real,real,F2,A2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_qg(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A2),C2)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T2),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),C2)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_5770_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun(nat,fun(real,real)),K: nat,B3: real] :
      ( ! [M: nat,T2: real] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),H) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
     => ( ( Nb = aa(nat,nat,suc,K) )
       => ! [M4: nat,T5: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Nb)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T5)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),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_qi(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Nb),Diff),B3),M4),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T5)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_qj(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M4),T5)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,M4))))),aa(real,real,aa(real,fun(real,real),times_times(real),B3),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),T5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,M4))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,M4))))))),topolo174197925503356063within(real,T5,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_5771_DERIV__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => has_field_derivative(real,aTP_Lamp_qk(real,real),suminf(real,aTP_Lamp_ql(real,fun(nat,real),Xb)),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_5772_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,A,F2,Xb) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_qm(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D3),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_5773_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xb: A,S: set(A),G: fun(A,A),E2: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E2,topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,A,G,Xb) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qn(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),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),D3),aa(A,A,G,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),E2),aa(A,A,F2,Xb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,G,Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% DERIV_quotient
tff(fact_5774_DERIV__pow,axiom,
    ! [Nb: nat,Xb: real,S: set(real)] : has_field_derivative(real,aTP_Lamp_qo(nat,fun(real,real),Nb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(real,Xb,S)) ).

% DERIV_pow
tff(fact_5775_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K: A,F4: filter(A)] : has_field_derivative(A,aTP_Lamp_qp(A,fun(A,A),K),zero_zero(A),F4) ) ).

% DERIV_const
tff(fact_5776_DERIV__ident,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F4: filter(A)] : has_field_derivative(A,aTP_Lamp_qq(A,A),one_one(A),F4) ) ).

% DERIV_ident
tff(fact_5777_field__differentiable__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,F4: filter(A),G: fun(A,A),G2: A] :
          ( has_field_derivative(A,F2,F7,F4)
         => ( has_field_derivative(A,G,G2,F4)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qr(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),F7),G2),F4) ) ) ) ).

% field_differentiable_add
tff(fact_5778_field__differentiable__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,F4: filter(A),G: fun(A,A),G2: A] :
          ( has_field_derivative(A,F2,F7,F4)
         => ( has_field_derivative(A,G,G2,F4)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qs(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),minus_minus(A),F7),G2),F4) ) ) ) ).

% field_differentiable_diff
tff(fact_5779_field__differentiable__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,F4: filter(A)] :
          ( has_field_derivative(A,F2,F7,F4)
         => has_field_derivative(A,aTP_Lamp_qt(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),F7),F4) ) ) ).

% field_differentiable_minus
tff(fact_5780_DERIV__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D7: A,Xb: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D7,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qr(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),D7),E4),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_add
tff(fact_5781_DERIV__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D7: A,Xb: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D7,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qs(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),minus_minus(A),D7),E4),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_diff
tff(fact_5782_DERIV__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D7: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,F2,D7,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aTP_Lamp_qt(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),D7),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_minus
tff(fact_5783_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,Xb: A,S: set(A),G: fun(A,A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qu(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,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),Db),aa(A,A,F2,Xb))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_mult
tff(fact_5784_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D7: A,Xb: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D7,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qu(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,Xb)),E4)),aa(A,A,aa(A,fun(A,A),times_times(A),D7),aa(A,A,G,Xb))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_mult'
tff(fact_5785_DERIV__at__within__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Xb),image(A,A,aa(A,fun(A,A),plus_plus(A),Z),S3)))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_qv(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_at_within_shift
tff(fact_5786_DERIV__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Xb: A,Z: A] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z),top_top(set(A))))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_qw(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_shift
tff(fact_5787_DERIV__mirror,axiom,
    ! [F2: fun(real,real),Y: real,Xb: real] :
      ( has_field_derivative(real,F2,Y,topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),Xb),top_top(set(real))))
    <=> has_field_derivative(real,aTP_Lamp_qx(fun(real,real),fun(real,real),F2),aa(real,real,uminus_uminus(real),Y),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_mirror
tff(fact_5788_DERIV__at__within__shift__lemma,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Xb),image(A,A,aa(A,fun(A,A),plus_plus(A),Z),S3)))
         => has_field_derivative(A,aa(fun(A,A),fun(A,A),comp(A,A,A,F2),aa(A,fun(A,A),plus_plus(A),Z)),Y,topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_at_within_shift_lemma
tff(fact_5789_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D7: A,Xb: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D7,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,A,G,Xb) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qn(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),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),D7),aa(A,A,G,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,Xb)),E4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G,Xb)),aa(A,A,G,Xb))),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% DERIV_divide
tff(fact_5790_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D7: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,F2,D7,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,A,F2,Xb) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_qm(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(A,A,F2,Xb))),D7)),aa(A,A,inverse_inverse(A),aa(A,A,F2,Xb)))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_inverse'
tff(fact_5791_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D7: A,Xb: A,S: set(A),Nb: nat] :
          ( has_field_derivative(A,F2,D7,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_qy(fun(A,A),fun(nat,fun(A,A)),F2),Nb),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),Nb))),aa(A,A,aa(A,fun(A,A),times_times(A),D7),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,Xb)),Nb))),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_power_Suc
tff(fact_5792_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xb: A,S: set(A)] :
          ( ( Xb != 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),Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_inverse
tff(fact_5793_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D7: A,Xb: A,S: set(A),Nb: nat] :
          ( has_field_derivative(A,F2,D7,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_qz(fun(A,A),fun(nat,fun(A,A)),F2),Nb),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),times_times(A),D7),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_power
tff(fact_5794_DERIV__power__int,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xb: A,S: set(A),Nb: int] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,A,F2,Xb) != zero_zero(A) )
           => has_field_derivative(A,aa(int,fun(A,A),aTP_Lamp_ra(fun(A,A),fun(int,fun(A,A)),F2),Nb),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Nb)),power_int(A,aa(A,A,F2,Xb),aa(int,int,aa(int,fun(int,int),minus_minus(int),Nb),one_one(int))))),D3),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_power_int
tff(fact_5795_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G2: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G,Xb)),one_one(real))
           => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xb,S))
             => has_derivative(A,real,aTP_Lamp_rb(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_rc(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G2),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_5796_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G2: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G,Xb)),one_one(real))
           => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xb,S))
             => has_derivative(A,real,aTP_Lamp_rd(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_re(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G2),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_5797_has__derivative__tan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G2: fun(A,real),S: set(A)] :
          ( ( aa(real,real,cos(real),aa(A,real,G,Xb)) != zero_zero(real) )
         => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,real,aTP_Lamp_rf(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_rg(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G2),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_tan
tff(fact_5798_has__derivative__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,real),F7: fun(A,real),Xb: A,S: set(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,real,F2,F7,topolo174197925503356063within(A,Xb,S))
         => ( has_derivative(A,B,G,G2,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rh(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_ri(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F7),Xb),G),G2),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_scaleR
tff(fact_5799_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C2: B,F4: filter(A)] : has_derivative(A,B,aTP_Lamp_rj(B,fun(A,B),C2),aTP_Lamp_rk(A,B),F4) ) ).

% has_derivative_const
tff(fact_5800_has__derivative__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F4: filter(A)] :
          ( has_derivative(A,B,F2,F7,F4)
         => has_derivative(A,B,aTP_Lamp_rl(fun(A,B),fun(A,B),F2),aTP_Lamp_rl(fun(A,B),fun(A,B),F7),F4) ) ) ).

% has_derivative_minus
tff(fact_5801_has__derivative__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F4: filter(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,F4)
         => ( has_derivative(A,B,G,G2,F4)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_rm(fun(A,B),fun(fun(A,B),fun(A,B)),F7),G2),F4) ) ) ) ).

% has_derivative_add
tff(fact_5802_has__derivative__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F4: filter(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,F4)
         => ( has_derivative(A,B,G,G2,F4)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rn(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_rn(fun(A,B),fun(fun(A,B),fun(A,B)),F7),G2),F4) ) ) ) ).

% has_derivative_diff
tff(fact_5803_has__derivative__zero__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F4: fun(A,B),Xb: A] :
          ( has_derivative(A,B,aTP_Lamp_rk(A,B),F4,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => ! [X3: A] : ( aa(A,B,F4,X3) = zero_zero(B) ) ) ) ).

% has_derivative_zero_unique
tff(fact_5804_has__derivative__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xb: A,S: set(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xb,S))
         => ( has_derivative(A,B,G,G2,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ro(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_rp(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F7),Xb),G),G2),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_mult
tff(fact_5805_has__derivative__divide_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xb: A,S3: set(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xb,S3))
         => ( has_derivative(A,B,G,G2,topolo174197925503356063within(A,Xb,S3))
           => ( ( aa(A,B,G,Xb) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rq(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_rr(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F7),Xb),G),G2),topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_5806_has__derivative__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(B,A),Xb: B,F7: fun(B,A),S3: set(B)] :
          ( ( aa(B,A,F2,Xb) != zero_zero(A) )
         => ( has_derivative(B,A,F2,F7,topolo174197925503356063within(B,Xb,S3))
           => has_derivative(B,A,aTP_Lamp_rs(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_rt(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),F2),Xb),F7),topolo174197925503356063within(B,Xb,S3)) ) ) ) ).

% has_derivative_inverse
tff(fact_5807_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A,S3: set(A)] :
          ( ( Xb != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_ru(A,fun(A,A),Xb),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_inverse'
tff(fact_5808_has__derivative__cos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),Xb: A,S: set(A)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xb,S))
         => has_derivative(A,real,aTP_Lamp_rv(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_rw(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G2),Xb),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_derivative_cos
tff(fact_5809_has__derivative__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xb: A,S3: set(A),Nb: nat] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xb,S3))
         => has_derivative(A,B,aa(nat,fun(A,B),aTP_Lamp_rx(fun(A,B),fun(nat,fun(A,B)),F2),Nb),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_ry(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),F2),F7),Xb),Nb),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_power
tff(fact_5810_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G2: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xb))
         => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,real,aTP_Lamp_rz(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_sa(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G2),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_ln
tff(fact_5811_has__derivative__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xb: A,S3: set(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xb,S3))
         => ( has_derivative(A,B,G,G2,topolo174197925503356063within(A,Xb,S3))
           => ( ( aa(A,B,G,Xb) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sb(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_sc(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F7),Xb),G),G2),topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ).

% has_derivative_divide
tff(fact_5812_has__derivative__power__int_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xb: A,Nb: int,S3: set(A)] :
          ( ( Xb != zero_zero(A) )
         => has_derivative(A,A,aTP_Lamp_sd(int,fun(A,A),Nb),aa(int,fun(A,A),aTP_Lamp_se(A,fun(int,fun(A,A)),Xb),Nb),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_power_int'
tff(fact_5813_has__derivative__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(B,A),Xb: B,F7: fun(B,A),S3: set(B),Nb: int] :
          ( ( aa(B,A,F2,Xb) != zero_zero(A) )
         => ( has_derivative(B,A,F2,F7,topolo174197925503356063within(B,Xb,S3))
           => has_derivative(B,A,aa(int,fun(B,A),aTP_Lamp_sf(fun(B,A),fun(int,fun(B,A)),F2),Nb),aa(int,fun(B,A),aa(fun(B,A),fun(int,fun(B,A)),aa(B,fun(fun(B,A),fun(int,fun(B,A))),aTP_Lamp_sg(fun(B,A),fun(B,fun(fun(B,A),fun(int,fun(B,A)))),F2),Xb),F7),Nb),topolo174197925503356063within(B,Xb,S3)) ) ) ) ).

% has_derivative_power_int
tff(fact_5814_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G2: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xb))
         => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,real,aTP_Lamp_sh(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_si(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G2),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_5815_has__derivative__arctan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),Xb: A,S: set(A)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,Xb,S))
         => has_derivative(A,real,aTP_Lamp_sj(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_sk(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G2),Xb),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_derivative_arctan
tff(fact_5816_has__derivative__floor,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [G: fun(B,real),Xb: B,F2: fun(real,A),G2: fun(B,real),S: set(B)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,aa(B,real,G,Xb),top_top(set(real))),F2)
         => ( ~ member(A,aa(real,A,F2,aa(B,real,G,Xb)),ring_1_Ints(A))
           => ( has_derivative(B,real,G,G2,topolo174197925503356063within(B,Xb,S))
             => has_derivative(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_sl(fun(B,real),fun(fun(real,A),fun(B,real)),G),F2),aTP_Lamp_sm(fun(B,real),fun(B,real),G2),topolo174197925503356063within(B,Xb,S)) ) ) ) ) ).

% has_derivative_floor
tff(fact_5817_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_sn(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K6))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_sp(fun(nat,A),fun(A,fun(A,A)),C2),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_5818_surj__int__encode,axiom,
    image(int,nat,nat_int_encode,top_top(set(int))) = top_top(set(nat)) ).

% surj_int_encode
tff(fact_5819_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F4: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_sq(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C2)),F4)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_5820_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F4: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_sr(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),L)),F4)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_5821_power__tendsto__0__iff,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,real),F4: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ss(nat,fun(fun(A,real),fun(A,real)),Nb),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% power_tendsto_0_iff
tff(fact_5822_tendsto__one__prod_H,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(A),F2: fun(B,fun(A,C)),F4: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_st(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I2),topolo7230453075368039082e_nhds(C,one_one(C)),F4) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_su(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,one_one(C)),F4) ) ) ).

% tendsto_one_prod'
tff(fact_5823_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,B),F4: filter(A),Nb: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_sv(fun(A,B),fun(nat,fun(A,B)),F2),Nb),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_null_power
tff(fact_5824_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F4: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( ( sin(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_sw(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),A2)),F4) ) ) ) ).

% tendsto_cot
tff(fact_5825_tendsto__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( cosh(B,A2) != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_sx(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,tanh(B),A2)),F4) ) ) ) ).

% tendsto_tanh
tff(fact_5826_tendsto__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ( L != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_sy(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,sgn_sgn(B),L)),F4) ) ) ) ).

% tendsto_sgn
tff(fact_5827_tendsto__norm__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,real,aTP_Lamp_sz(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_norm_zero
tff(fact_5828_tendsto__power__int,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A),Nb: int] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( A2 != zero_zero(B) )
           => filterlim(A,B,aa(int,fun(A,B),aTP_Lamp_ta(fun(A,B),fun(int,fun(A,B)),F2),Nb),topolo7230453075368039082e_nhds(B,power_int(B,A2,Nb)),F4) ) ) ) ).

% tendsto_power_int
tff(fact_5829_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_sz(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_iff
tff(fact_5830_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_sz(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_cancel
tff(fact_5831_tendsto__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( A2 != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_tb(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,inverse_inverse(B),A2)),F4) ) ) ) ).

% tendsto_inverse
tff(fact_5832_continuous__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_tc(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_minus
tff(fact_5833_tendsto__minus__cancel__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),Y: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),Y)),F4)
        <=> filterlim(A,B,aTP_Lamp_td(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,Y),F4) ) ) ).

% tendsto_minus_cancel_left
tff(fact_5834_tendsto__minus__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_td(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),A2)),F4)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ).

% tendsto_minus_cancel
tff(fact_5835_tendsto__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => filterlim(A,B,aTP_Lamp_td(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),A2)),F4) ) ) ).

% tendsto_minus
tff(fact_5836_tendsto__uminus__nhds,axiom,
    ! [A: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [A2: A] : filterlim(A,A,uminus_uminus(A),topolo7230453075368039082e_nhds(A,aa(A,A,uminus_uminus(A),A2)),topolo7230453075368039082e_nhds(A,A2)) ) ).

% tendsto_uminus_nhds
tff(fact_5837_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F4: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( ( aa(A,A,cos(A),A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_te(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),A2)),F4) ) ) ) ).

% tendsto_tan
tff(fact_5838_int__encode__eq,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa(int,nat,nat_int_encode,Xb) = aa(int,nat,nat_int_encode,Y) )
    <=> ( Xb = Y ) ) ).

% int_encode_eq
tff(fact_5839_tendsto__add__const__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [C2: B,F2: fun(A,B),D3: B,F4: filter(A)] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tf(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),D3)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,D3),F4) ) ) ).

% tendsto_add_const_iff
tff(fact_5840_tendsto__add,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tg(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)),F4) ) ) ) ).

% tendsto_add
tff(fact_5841_continuous__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_th(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_add
tff(fact_5842_continuous__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_ti(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_diff
tff(fact_5843_tendsto__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tj(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)),F4) ) ) ) ).

% tendsto_diff
tff(fact_5844_tendsto__divide__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_tk(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_divide_zero
tff(fact_5845_tendsto__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F4)
           => ( ( B2 != zero_zero(B) )
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tl(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,divide_divide(B,A2,B2)),F4) ) ) ) ) ).

% tendsto_divide
tff(fact_5846_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_tm(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% LIM_zero
tff(fact_5847_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_tm(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_iff
tff(fact_5848_Lim__transform,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [G: fun(A,B),A2: B,F4: filter(A),F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tn(fun(A,B),fun(fun(A,B),fun(A,B)),G),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform
tff(fact_5849_Lim__transform2,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_to(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform2
tff(fact_5850_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_tm(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_cancel
tff(fact_5851_Lim__transform__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),G: fun(A,B),F4: filter(A),A2: B] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_to(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
          <=> filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform_eq
tff(fact_5852_tendsto__add__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tg(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_add_zero
tff(fact_5853_tendsto__mult__right__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_tp(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_mult_right_zero
tff(fact_5854_tendsto__mult__left__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_tq(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_mult_left_zero
tff(fact_5855_tendsto__mult__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_mult_zero
tff(fact_5856_tendsto__mult__one,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,one_one(B)),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,one_one(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ts(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,one_one(B)),F4) ) ) ) ).

% tendsto_mult_one
tff(fact_5857_tendsto__of__int__ceiling,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( ring_1(C)
        & topolo4958980785337419405_space(C)
        & archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => filterlim(A,C,aTP_Lamp_tt(fun(A,B),fun(A,C),F2),topolo7230453075368039082e_nhds(C,aa(int,C,ring_1_of_int(C),archimedean_ceiling(B,L))),F4) ) ) ) ).

% tendsto_of_int_ceiling
tff(fact_5858_tendsto__of__int__floor,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( ring_1(C)
        & topolo4958980785337419405_space(C)
        & archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => filterlim(A,C,aTP_Lamp_tu(fun(A,B),fun(A,C),F2),topolo7230453075368039082e_nhds(C,aa(int,C,ring_1_of_int(C),aa(B,int,archim6421214686448440834_floor(B),L))),F4) ) ) ) ).

% tendsto_of_int_floor
tff(fact_5859_tendsto__arcosh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
       => filterlim(A,real,aTP_Lamp_tv(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F4) ) ) ).

% tendsto_arcosh
tff(fact_5860_tendsto__log,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
         => ( ( A2 != one_one(real) )
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
             => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_tw(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,aa(real,real,log2(A2),B2)),F4) ) ) ) ) ) ).

% tendsto_log
tff(fact_5861_tendsto__null__sum,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),F2: fun(B,fun(A,C)),F4: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_tx(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_ty(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ).

% tendsto_null_sum
tff(fact_5862_has__field__derivative__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D7: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F2,D7,topolo174197925503356063within(A,Xb,S3))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_tz(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D7),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_field_derivative_iff
tff(fact_5863_has__field__derivativeD,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D7: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F2,D7,topolo174197925503356063within(A,Xb,S3))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_tz(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D7),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_field_derivativeD
tff(fact_5864_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_ua(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(B,A2,top_top(set(B)))) ) ) ).

% LIM_not_zero
tff(fact_5865_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_ub(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_5866_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_uc(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_5867_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_uc(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_5868_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_uc(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_5869_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Xb: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,top_top(set(A))),F2)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ud(A,fun(fun(A,B),fun(A,B)),Xb),F2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,Xb)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_5870_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_ue(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_5871_tendsto__artanh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),A2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),one_one(real))
         => filterlim(A,real,aTP_Lamp_uf(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A2)),F4) ) ) ) ).

% tendsto_artanh
tff(fact_5872_LIM__imp__LIM,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L: B,A2: A,G: fun(A,C),Ma: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X: A] :
                ( ( X != A2 )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(C,C,aa(C,fun(C,C),minus_minus(C),aa(A,C,G,X)),Ma))),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X)),L))) )
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,Ma),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ).

% LIM_imp_LIM
tff(fact_5873_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_ug(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_5874_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R: real,A2: A,F2: fun(A,B),G: fun(A,B),L: B] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
         => ( ! [X: A] :
                ( ( X != A2 )
               => ( aa(real,$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))),R)
                 => ( 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_5875_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,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S5: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S5)
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),A2))),S5) )
                     => aa(real,$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_5876_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,$o),ord_less(real),zero_zero(real)),R4)
             => ? [S6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S6)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A2))),S6) )
                     => aa(real,$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,X)),L5))),R4) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_5877_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,R2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [S2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S2)
                & ! [X3: A] :
                    ( ( ( X3 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),A2))),S2) )
                   => aa(real,$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,X3)),L5))),R2) ) ) ) ) ) ).

% LIM_D
tff(fact_5878_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,A),A2: A,D7: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_uh(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D7),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_ui(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D7),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_5879_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_ue(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_5880_continuous__at__within__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),G)
           => ( ( aa(A,B,G,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aa(fun(A,B),fun(A,B),aTP_Lamp_uj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_5881_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_th(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_add
tff(fact_5882_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_uk(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_diff
tff(fact_5883_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_ul(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_minus
tff(fact_5884_continuous__at__within__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [A2: A,S: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aTP_Lamp_um(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_inverse
tff(fact_5885_continuous__at__within__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,S: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aTP_Lamp_un(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_sgn
tff(fact_5886_continuous__at__within__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [A2: A,S: set(A),F2: fun(A,B),Nb: int] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aa(int,fun(A,B),aTP_Lamp_uo(fun(A,B),fun(int,fun(A,B)),F2),Nb)) ) ) ) ).

% continuous_at_within_power_int
tff(fact_5887_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D7: A,Xb: A] :
          ( has_field_derivative(A,F2,D7,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_up(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D7),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_5888_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D7: A,Xb: A] :
          ( has_field_derivative(A,F2,D7,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_up(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D7),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_5889_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_uq(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_5890_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_5891_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D7: A,Xb: A] :
          ( has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D7),topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_up(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D7),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_5892_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_uj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% isCont_divide
tff(fact_5893_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_un(fun(A,B),fun(A,B),F2)) ) ) ) ).

% isCont_sgn
tff(fact_5894_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F4: filter(B),A2: A] :
          ( filterlim(A,B,F2,F4,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_ur(fun(A,B),fun(A,fun(A,B)),F2),A2),F4,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_5895_continuous__within__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,S: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),F2)
         => ( ( aa(A,A,cos(A),aa(A,A,F2,Xb)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),aTP_Lamp_te(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_tan
tff(fact_5896_continuous__within__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,S: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),F2)
         => ( ( sin(A,aa(A,A,F2,Xb)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),aTP_Lamp_sw(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_cot
tff(fact_5897_continuous__at__within__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Xb: A,A3: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,A3),F2)
         => ( ( cosh(B,aa(A,B,F2,Xb)) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,A3),aTP_Lamp_us(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_tanh
tff(fact_5898_CARAT__DERIV,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A,Xb: A] :
          ( has_field_derivative(A,F2,L,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ? [G3: fun(A,A)] :
              ( ! [Z5: A] : ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,F2,Z5)),aa(A,A,F2,Xb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G3,Z5)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z5),Xb)) )
              & topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),G3)
              & ( aa(A,A,G3,Xb) = L ) ) ) ) ).

% CARAT_DERIV
tff(fact_5899_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,$o),ord_less_eq(real),X),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X,top_top(set(real))),F2) )
           => ? [M8: A] :
                ( ! [X3: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X3)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F2,X3)),M8) )
                & ! [N8: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N8),M8)
                   => ? [X: real] :
                        ( aa(real,$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),N8),aa(real,A,F2,X)) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_5900_isCont__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( aa(A,A,cos(A),Xb) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),tan(A)) ) ) ).

% isCont_tan
tff(fact_5901_filterlim__shift__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),D3: A,F4: filter(B),A2: A] :
          ( filterlim(A,B,aa(fun(A,A),fun(A,B),comp(A,B,A,F2),aa(A,fun(A,A),plus_plus(A),D3)),F4,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D3),top_top(set(A))))
        <=> filterlim(A,B,F2,F4,topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% filterlim_shift_iff
tff(fact_5902_filterlim__shift,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F4: filter(B),A2: A,D3: A] :
          ( filterlim(A,B,F2,F4,topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(fun(A,A),fun(A,B),comp(A,B,A,F2),aa(A,fun(A,A),plus_plus(A),D3)),F4,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D3),top_top(set(A)))) ) ) ).

% filterlim_shift
tff(fact_5903_isCont__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sin(A,Xb) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),cot(A)) ) ) ).

% isCont_cot
tff(fact_5904_isCont__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cosh(A,Xb) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),tanh(A)) ) ) ).

% isCont_tanh
tff(fact_5905_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A2: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S)
         => ( ! [X: A] :
                ( ( X != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),S)
                 => sums(A,aa(A,fun(nat,A),aTP_Lamp_ut(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_5906_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A2: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S)
         => ( ! [X: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),S)
               => sums(A,aa(A,fun(nat,A),aTP_Lamp_ut(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_5907_bij__int__encode,axiom,
    bij_betw(int,nat,nat_int_encode,top_top(set(int)),top_top(set(nat))) ).

% bij_int_encode
tff(fact_5908_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_uu(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_5909_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_te(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_tan'
tff(fact_5910_isCont__arcosh,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),arcosh(real)) ) ).

% isCont_arcosh
tff(fact_5911_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)
         => ( ( sin(A,aa(A,A,F2,A2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_sw(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_cot'
tff(fact_5912_continuous__floor,axiom,
    ! [Xb: real] :
      ( ~ member(real,Xb,ring_1_Ints(real))
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),aa(fun(real,int),fun(real,real),comp(int,real,real,ring_1_of_int(real)),archim6421214686448440834_floor(real))) ) ).

% continuous_floor
tff(fact_5913_DERIV__inverse__function,axiom,
    ! [F2: fun(real,real),D7: real,G: fun(real,real),Xb: real,A2: real,B2: real] :
      ( has_field_derivative(real,F2,D7,topolo174197925503356063within(real,aa(real,real,G,Xb),top_top(set(real))))
     => ( ( D7 != zero_zero(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),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,Xb,top_top(set(real))),G)
               => has_field_derivative(real,G,aa(real,real,inverse_inverse(real),D7),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_5914_isCont__arccos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),arccos) ) ) ).

% isCont_arccos
tff(fact_5915_isCont__arcsin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),arcsin) ) ) ).

% isCont_arcsin
tff(fact_5916_isCont__artanh,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),artanh(real)) ) ) ).

% isCont_artanh
tff(fact_5917_floor__has__real__derivative,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Xb: real,F2: fun(real,A)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,Xb,top_top(set(real))),F2)
         => ( ~ member(A,aa(real,A,F2,Xb),ring_1_Ints(A))
           => has_field_derivative(real,aTP_Lamp_uv(fun(real,A),fun(real,real),F2),zero_zero(real),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% floor_has_real_derivative
tff(fact_5918_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)))
         => ! [N3: nat] : member(real,suminf(real,aTP_Lamp_uw(fun(nat,real),fun(nat,real),A2)),set_or1337092689740270186AtMost(real,aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_uw(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))),N3))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_uw(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))),N3)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_5919_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))
         => ! [N3: nat] : member(real,suminf(real,aTP_Lamp_uw(fun(nat,real),fun(nat,real),A2)),set_or1337092689740270186AtMost(real,aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_uw(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))),N3)),one_one(nat)))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_uw(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))),N3))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_5920_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_ux(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_uw(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_5921_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_uy(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_5922_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_uz(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_5923_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_va(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_5924_approx__from__above__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ? [U2: fun(nat,A)] :
              ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(nat,A,U2,N3))
              & filterlim(nat,A,U2,topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ) ).

% approx_from_above_dense_linorder
tff(fact_5925_approx__from__below__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ? [U2: fun(nat,A)] :
              ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,U2,N3)),Xb)
              & filterlim(nat,A,U2,topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ) ).

% approx_from_below_dense_linorder
tff(fact_5926_LIMSEQ__imp__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A] :
          ( filterlim(nat,A,aTP_Lamp_vb(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_5927_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_vb(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_Suc
tff(fact_5928_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_vc(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_5929_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_vc(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_ignore_initial_segment
tff(fact_5930_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_vd(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% seq_offset_neg
tff(fact_5931_summable__LIMSEQ__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% summable_LIMSEQ_zero
tff(fact_5932_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_5933_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_ve(nat,fun(nat,nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_5934_LIMSEQ__root,axiom,
    filterlim(nat,real,aTP_Lamp_vf(nat,real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ).

% LIMSEQ_root
tff(fact_5935_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A] : filterlim(nat,A,aTP_Lamp_vg(A,fun(nat,A),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_5936_lim__inverse__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_vh(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_inverse_n
tff(fact_5937_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X6: fun(nat,A),Xb: A,L: nat] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),L)
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_vi(fun(nat,A),fun(nat,fun(nat,A)),X6),L),topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_5938_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_vj(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable
tff(fact_5939_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_vk(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable'
tff(fact_5940_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_vl(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => ? [L2: real] :
                ( ! [N3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N3)),L2)
                & filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L2),at_top(nat))
                & ! [N3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L2),aa(nat,real,G,N3))
                & filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_5941_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun(nat,real)] :
      ( ! [R4: real] :
        ? [N8: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),R4),aa(nat,real,X6,N)) )
     => filterlim(nat,real,aTP_Lamp_vm(fun(nat,real),fun(nat,real),X6),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_5942_lim__inverse__n_H,axiom,
    filterlim(nat,real,aTP_Lamp_vn(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% lim_inverse_n'
tff(fact_5943_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_vo(real,fun(nat,real),C2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_5944_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim(nat,real,aTP_Lamp_vp(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_5945_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_vq(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_5946_continuous__at__within__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),G)
           => ( aa(real,$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,S),aa(fun(A,real),fun(A,real),aTP_Lamp_vr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_5947_increasing__LIMSEQ,axiom,
    ! [F2: fun(nat,real),L: real] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N)),aa(nat,real,F2,aa(nat,nat,suc,N)))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N)),L)
       => ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [N3: 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,N3)),E)) )
         => filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_5948_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_vs(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_5949_LIMSEQ__n__over__Suc__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_vt(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_n_over_Suc_n
tff(fact_5950_LIMSEQ__Suc__n__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_vu(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_Suc_n_over_n
tff(fact_5951_LIMSEQ__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_5952_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))
         => sums(A,aTP_Lamp_vk(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_5953_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))
         => sums(A,aTP_Lamp_vj(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_5954_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xb: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_vv(real,fun(real,fun(nat,real)),Xb),A2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_5955_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_5956_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_5957_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => filterlim(nat,real,aTP_Lamp_vw(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_5958_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A] :
          ( sums(A,F2,S)
        <=> filterlim(nat,A,aTP_Lamp_vx(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S),at_top(nat)) ) ) ).

% sums_def'
tff(fact_5959_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),Xb: real] :
          ( filterlim(nat,real,aTP_Lamp_vy(fun(nat,A),fun(nat,real),F2),topolo7230453075368039082e_nhds(real,Xb),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
           => summable(A,F2) ) ) ) ).

% root_test_convergence
tff(fact_5960_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_vz(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_5961_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_vr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_5962_LIMSEQ__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No: nat] :
                ! [N4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N4)
                 => aa(real,$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,X6,N4)),L5))),R5) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_5963_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( ! [R4: real] :
              ( aa(real,$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,X6,N)),L5))),R4) ) )
         => filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_5964_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L5: A,R2: real] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [No3: nat] :
              ! [N3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N3)
               => aa(real,$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,X6,N3)),L5))),R2) ) ) ) ) ).

% LIMSEQ_D
tff(fact_5965_LIMSEQ__power__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),one_one(real))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_power_zero
tff(fact_5966_tendsto__at__left__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [B2: A,A2: A,X6: fun(A,B),L5: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( ! [S7: fun(nat,A)] :
                ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,S7,N3)),A2)
               => ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(nat,A,S7,N3))
                 => ( order_mono(nat,A,S7)
                   => ( filterlim(nat,A,S7,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_wa(fun(A,B),fun(fun(nat,A),fun(nat,B)),X6),S7),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) ) ) ) )
           => filterlim(A,B,X6,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2))) ) ) ) ).

% tendsto_at_left_sequentially
tff(fact_5967_tendsto__exp__limit__sequentially,axiom,
    ! [Xb: real] : filterlim(nat,real,aTP_Lamp_wb(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,exp(real,Xb)),at_top(nat)) ).

% tendsto_exp_limit_sequentially
tff(fact_5968_LIMSEQ__iff__nz,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),No)
                  & ! [N4: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N4)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,N4),L5)),R5) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_5969_tendsto__power__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,nat),F4: filter(A),Xb: B] :
          ( filterlim(A,nat,F2,at_top(nat),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,Xb)),one_one(real))
           => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_wc(fun(A,nat),fun(B,fun(A,B)),F2),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_power_zero
tff(fact_5970_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_wd(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_5971_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))),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_5972_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_uw(fun(nat,real),fun(nat,real),A2)) ) ) ).

% summable_Leibniz(1)
tff(fact_5973_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Df: A,Z: A,S: fun(nat,A),A2: A] :
          ( has_field_derivative(A,F2,Df,topolo174197925503356063within(A,Z,top_top(set(A))))
         => ( filterlim(nat,A,S,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
           => ( ! [N: nat] : ( aa(nat,A,S,N) != zero_zero(A) )
             => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_we(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),F2),Z),S),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
               => ( Df = A2 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_5974_powser__times__n__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),one_one(real))
         => filterlim(nat,A,aTP_Lamp_wf(A,fun(nat,A),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_5975_lim__n__over__pown,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,Xb))
         => filterlim(nat,A,aTP_Lamp_wg(A,fun(nat,A),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_5976_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_uw(fun(nat,real),fun(nat,real),A2)) ) ) ) ).

% summable
tff(fact_5977_cos__diff__limit__1,axiom,
    ! [Theta: fun(nat,real),Theta2: real] :
      ( filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_wh(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_wi(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_5978_cos__limit__1,axiom,
    ! [Theta: fun(nat,real)] :
      ( filterlim(nat,real,aTP_Lamp_wj(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_wi(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_5979_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_wk(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_uw(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(4)
tff(fact_5980_zeroseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => filterlim(nat,real,aTP_Lamp_ap(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_5981_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_wk(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_uw(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_5982_summable__Leibniz_H_I2_J,axiom,
    ! [A2: fun(nat,real),Nb: nat] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_uw(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))),Nb)))),suminf(real,aTP_Lamp_uw(fun(nat,real),fun(nat,real),A2))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_5983_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] :
              ( ! [N3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_uw(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))),N3)))),L2)
              & filterlim(nat,real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L2),at_top(nat))
              & ! [N3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L2),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_uw(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))),N3)),one_one(nat)))))
              & filterlim(nat,real,aTP_Lamp_ux(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_5984_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_ux(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_uw(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_5985_summable__Leibniz_H_I4_J,axiom,
    ! [A2: fun(nat,real),Nb: 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_uw(fun(nat,real),fun(nat,real),A2))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_uw(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))),Nb)),one_one(nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_5986_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),F4: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_on(fun(nat,A),fun(nat,A),F2),F4,at_top(nat))
    <=> filterlim(nat,A,F2,F4,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_5987_summable__bounded__partials,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_wl(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F2),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_bounded_partials
tff(fact_5988_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_wm(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F7),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_5989_eventually__sequentially__Suc,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,aTP_Lamp_wn(fun(nat,$o),fun(nat,$o),P),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_5990_eventually__sequentially__seg,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_wo(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_5991_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_wp(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ).

% filterlim_at_top_dense
tff(fact_5992_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N6: A] :
            ! [N4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N6),N4)
             => aa(A,$o,P,N4) ) ) ) ).

% eventually_at_top_dense
tff(fact_5993_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_5994_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_rk(A,B)) ) ).

% bounded_linear_zero
tff(fact_5995_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_rl(fun(A,B),fun(A,B),F2)) ) ) ).

% bounded_linear_minus
tff(fact_5996_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_rn(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_sub
tff(fact_5997_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_rm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_add
tff(fact_5998_sequentially__offset,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,P,at_top(nat))
     => eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_wo(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat)) ) ).

% sequentially_offset
tff(fact_5999_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)))
          <=> ? [B5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),top_top(A))
                & ! [Z5: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),Z5)
                   => aa(A,$o,P,Z5) ) ) ) ) ) ).

% eventually_nhds_top
tff(fact_6000_eventually__at__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,Xb: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( eventually(A,P,topolo174197925503356063within(A,Xb,set_ord_lessThan(A,Xb)))
          <=> ? [B5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),Xb)
                & ! [Y5: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),Y5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),Xb)
                     => aa(A,$o,P,Y5) ) ) ) ) ) ) ).

% eventually_at_left
tff(fact_6001_eventually__at__left__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),Xb: A] :
          ( eventually(A,P,topolo174197925503356063within(A,Xb,set_ord_lessThan(A,Xb)))
        <=> ? [B5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),Xb)
              & ! [Y5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),Y5)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),Xb)
                   => aa(A,$o,P,Y5) ) ) ) ) ) ).

% eventually_at_left_field
tff(fact_6002_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),Z6)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_wq(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ) ).

% filterlim_at_top_gt
tff(fact_6003_order__tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Xb: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xb),F4)
        <=> ( ! [L3: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L3),Xb)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_wr(fun(A,B),fun(B,fun(A,$o)),F2),L3),F4) )
            & ! [U3: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Xb),U3)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ws(fun(A,B),fun(B,fun(A,$o)),F2),U3),F4) ) ) ) ) ).

% order_tendsto_iff
tff(fact_6004_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Y: A,F2: fun(B,A),F4: filter(B)] :
          ( ! [A4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),Y)
             => eventually(B,aa(A,fun(B,$o),aTP_Lamp_wt(fun(B,A),fun(A,fun(B,$o)),F2),A4),F4) )
         => ( ! [A4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A4)
               => eventually(B,aa(A,fun(B,$o),aTP_Lamp_wu(fun(B,A),fun(A,fun(B,$o)),F2),A4),F4) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y),F4) ) ) ) ).

% order_tendstoI
tff(fact_6005_order__tendstoD_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),Y)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_wr(fun(A,B),fun(B,fun(A,$o)),F2),A2),F4) ) ) ) ).

% order_tendstoD(1)
tff(fact_6006_order__tendstoD_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),A2)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ws(fun(A,B),fun(B,fun(A,$o)),F2),A2),F4) ) ) ) ).

% order_tendstoD(2)
tff(fact_6007_bounded__linear_Otendsto__zero,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(C,A),F4: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_wv(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% bounded_linear.tendsto_zero
tff(fact_6008_sequentially__imp__eventually__at__left,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,A2: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( ! [F3: fun(nat,A)] :
                ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(nat,A,F3,N3))
               => ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,N3)),A2)
                 => ( order_mono(nat,A,F3)
                   => ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ww(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F3),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2))) ) ) ) ).

% sequentially_imp_eventually_at_left
tff(fact_6009_eventually__at__leftI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( ! [X: A] :
              ( member(A,X,set_or5935395276787703475ssThan(A,A2,B2))
             => aa(A,$o,P,X) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => eventually(A,P,topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2))) ) ) ) ).

% eventually_at_leftI
tff(fact_6010_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,$o),A2: A] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> eventually(A,aa(A,fun(A,$o),aTP_Lamp_wx(fun(A,$o),fun(A,fun(A,$o)),P),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_6011_increasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_wy(fun(A,B),fun(B,fun(A,$o)),F2),L),F4)
         => ( ! [X: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),L)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_wr(fun(A,B),fun(B,fun(A,$o)),F2),X),F4) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% increasing_tendsto
tff(fact_6012_decreasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [L: B,F2: fun(A,B),F4: filter(A)] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_wz(B,fun(fun(A,B),fun(A,$o)),L),F2),F4)
         => ( ! [X: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),X)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ws(fun(A,B),fun(B,fun(A,$o)),F2),X),F4) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% decreasing_tendsto
tff(fact_6013_eventually__ceiling__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_xa(fun(A,B),fun(B,fun(A,$o)),F2),L),F4) ) ) ) ).

% eventually_ceiling_eq
tff(fact_6014_continuous__arcosh__strong,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( eventually(A,aTP_Lamp_xb(fun(A,real),fun(A,$o),F2),F4)
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_xc(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh_strong
tff(fact_6015_bounded__linear__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),K6: real] :
          ( ! [X: A,Y3: A] : ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X)),aa(A,B,F2,Y3)) )
         => ( ! [R4: real,X: A] : ( aa(A,B,F2,real_V8093663219630862766scaleR(A,R4,X)) = real_V8093663219630862766scaleR(B,R4,aa(A,B,F2,X)) )
           => ( ! [X: A] : aa(real,$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_6016_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_ws(fun(A,B),fun(B,fun(A,$o)),F2),L5),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,set_ord_lessThan(B,L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_6017_tendsto__arcosh__strong,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),A2)
       => ( eventually(A,aTP_Lamp_xd(fun(A,real),fun(A,$o),F2),F4)
         => filterlim(A,real,aTP_Lamp_tv(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F4) ) ) ) ).

% tendsto_arcosh_strong
tff(fact_6018_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A),A2: A] :
          ( ! [X: A,Y3: A] :
              ( aa(A,$o,Q,X)
             => ( aa(A,$o,Q,Y3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y3)) ) ) )
         => ( ! [X: B] :
                ( aa(B,$o,P,X)
               => ( aa(A,B,F2,aa(B,A,G,X)) = X ) )
           => ( ! [X: B] :
                  ( aa(B,$o,P,X)
                 => aa(A,$o,Q,aa(B,A,G,X)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2)))
               => ( ! [B4: A] :
                      ( aa(A,$o,Q,B4)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),A2) )
                 => ( eventually(B,P,at_top(B))
                   => filterlim(A,B,F2,at_top(B),topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_6019_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,C),K6: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( eventually(A,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_xe(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),F2),G),K6),F4)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ).

% tendsto_0_le
tff(fact_6020_eventually__floor__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_xf(fun(A,B),fun(B,fun(A,$o)),F2),L),F4) ) ) ) ).

% eventually_floor_less
tff(fact_6021_eventually__less__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_xg(fun(A,B),fun(B,fun(A,$o)),F2),L),F4) ) ) ) ).

% eventually_less_ceiling
tff(fact_6022_filterlim__Suc,axiom,
    filterlim(nat,nat,suc,at_top(nat),at_top(nat)) ).

% filterlim_Suc
tff(fact_6023_has__derivative__iff__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xb: A,S: set(A)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xb,S))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_xh(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),F7),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_iff_norm
tff(fact_6024_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F7: fun(A,B),Xb: A,F2: fun(A,B),S: set(A)] :
          ( real_V3181309239436604168linear(A,B,F7)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_xi(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F7),Xb),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivativeI
tff(fact_6025_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xb: A,S: set(A)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xb,S))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_xj(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F7),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_at_within
tff(fact_6026_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & ? [E3: fun(A,B)] :
                ( ! [H3: A] : ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),H3)) = 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,Xb)),aa(A,B,F7,H3))),aa(A,B,E3,H3)) )
                & filterlim(A,real,aTP_Lamp_xk(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_6027_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E2: real,F7: fun(A,B),S: set(A),Xb: A,F2: fun(A,B),H4: fun(A,real)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => ( real_V3181309239436604168linear(A,B,F7)
           => ( ! [Y3: A] :
                  ( member(A,Y3,S)
                 => ( ( Y3 != Xb )
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y3,Xb)),E2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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,Xb))),aa(A,B,F7,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y3),Xb)))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y3),Xb)))),aa(A,real,H4,Y3)) ) ) )
             => ( filterlim(A,real,H4,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xb,S))
               => has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xb,S)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_6028_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),Xb: A,S: set(A)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xb,S))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_wm(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F7),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_within
tff(fact_6029_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),D7: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F2,D7,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D7)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_xl(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),D7),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_6030_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C2: fun(nat,A),K: nat,Nb: nat,B3: real] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
             => eventually(A,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_xm(fun(nat,A),fun(nat,fun(real,fun(A,$o))),C2),Nb),B3),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_6031_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F4: filter(A)] :
          ( has_derivative(A,B,F2,F7,F4)
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_xo(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F2),F7),F4),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% has_derivative_def
tff(fact_6032_filterlim__real__of__int__at__top,axiom,
    filterlim(int,real,ring_1_of_int(real),at_top(real),at_top(int)) ).

% filterlim_real_of_int_at_top
tff(fact_6033_filterlim__int__sequentially,axiom,
    filterlim(nat,int,semiring_1_of_nat(int),at_top(int),at_top(nat)) ).

% filterlim_int_sequentially
tff(fact_6034_filterlim__nat__sequentially,axiom,
    filterlim(int,nat,nat2,at_top(nat),at_top(int)) ).

% filterlim_nat_sequentially
tff(fact_6035_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_6036_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_6037_tendsto__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => filterlim(nat,A,semiring_1_of_nat(A),at_infinity(A),at_top(nat)) ) ).

% tendsto_of_nat
tff(fact_6038_filterlim__pow__at__top,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,real),F4: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(A,real,F2,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ss(nat,fun(fun(A,real),fun(A,real)),Nb),F2),at_top(real),F4) ) ) ).

% filterlim_pow_at_top
tff(fact_6039_tendsto__add__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( filterlim(A,B,G,at_infinity(B),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xp(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F4) ) ) ) ).

% tendsto_add_filterlim_at_infinity
tff(fact_6040_tendsto__add__filterlim__at__infinity_H,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B),C2: B] :
          ( filterlim(A,B,F2,at_infinity(B),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xp(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F4) ) ) ) ).

% tendsto_add_filterlim_at_infinity'
tff(fact_6041_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => ( ( aa(A,B,G,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_uj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_divide
tff(fact_6042_continuous__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_um(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_inverse
tff(fact_6043_tendsto__inverse__0__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F4)
     => filterlim(A,real,aTP_Lamp_xr(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ).

% tendsto_inverse_0_at_top
tff(fact_6044_continuous__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_un(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_sgn
tff(fact_6045_continuous__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F2: fun(A,B),Nb: int] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aa(int,fun(A,B),aTP_Lamp_uo(fun(A,B),fun(int,fun(A,B)),F2),Nb)) ) ) ) ).

% continuous_power_int
tff(fact_6046_filterlim__int__of__nat__at__topD,axiom,
    ! [A: $tType,F2: fun(int,A),F4: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_xs(fun(int,A),fun(nat,A),F2),F4,at_top(nat))
     => filterlim(int,A,F2,F4,at_top(int)) ) ).

% filterlim_int_of_nat_at_topD
tff(fact_6047_tendsto__inverse__0,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => filterlim(A,A,inverse_inverse(A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_infinity(A)) ) ).

% tendsto_inverse_0
tff(fact_6048_tendsto__mult__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( ( C2 != zero_zero(B) )
           => ( filterlim(A,B,G,at_infinity(B),F4)
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xt(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F4) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_6049_tendsto__divide__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( filterlim(A,B,G,at_infinity(B),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xu(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_divide_0
tff(fact_6050_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),Nb: nat] :
          ( filterlim(A,B,F2,at_infinity(B),F4)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_xv(fun(A,B),fun(nat,fun(A,B)),F2),Nb),at_infinity(B),F4) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_6051_continuous__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F4,F2)
         => ( ( aa(A,A,cos(A),aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xw(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F4,aTP_Lamp_te(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_tan
tff(fact_6052_continuous__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F4,F2)
         => ( ( sin(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xw(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F4,aTP_Lamp_sw(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_cot
tff(fact_6053_continuous__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( cosh(B,aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A)))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_us(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_tanh
tff(fact_6054_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A))))
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_xc(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh
tff(fact_6055_filterlim__inverse__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => filterlim(A,A,inverse_inverse(A),at_infinity(A),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% filterlim_inverse_at_infinity
tff(fact_6056_filterlim__at__top__iff__inverse__0,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_xx(fun(A,real),fun(A,$o),F2),F4)
     => ( filterlim(A,real,F2,at_top(real),F4)
      <=> filterlim(A,real,aa(fun(A,real),fun(A,real),comp(real,real,A,inverse_inverse(real)),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% filterlim_at_top_iff_inverse_0
tff(fact_6057_filterlim__inverse__at__top__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_xx(fun(A,real),fun(A,$o),F2),F4)
     => ( filterlim(A,real,aTP_Lamp_xr(fun(A,real),fun(A,real),F2),at_top(real),F4)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% filterlim_inverse_at_top_iff
tff(fact_6058_filterlim__inverse__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( eventually(A,aTP_Lamp_xx(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,aTP_Lamp_xr(fun(A,real),fun(A,real),F2),at_top(real),F4) ) ) ).

% filterlim_inverse_at_top
tff(fact_6059_tendsto__exp__limit__at__top,axiom,
    ! [Xb: real] : filterlim(real,real,aTP_Lamp_xy(real,fun(real,real),Xb),topolo7230453075368039082e_nhds(real,exp(real,Xb)),at_top(real)) ).

% tendsto_exp_limit_at_top
tff(fact_6060_filterlim__inverse__at__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [G: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_tb(fun(A,B),fun(A,B),G),topolo174197925503356063within(B,zero_zero(B),top_top(set(B))),F4)
        <=> filterlim(A,B,G,at_infinity(B),F4) ) ) ).

% filterlim_inverse_at_iff
tff(fact_6061_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( topolo3448309680560233919inuous(A,real,F4,G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A))))
             => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A))) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A))))
                 => topolo3448309680560233919inuous(A,real,F4,aa(fun(A,real),fun(A,real),aTP_Lamp_vr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_6062_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),C2: A,F4: filter(A),G: fun(A,A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,C2),F4)
         => ( filterlim(A,A,G,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F4)
           => ( ( C2 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_qn(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),at_infinity(A),F4) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_6063_continuous__artanh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( member(real,aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_xq(A,A))),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_xz(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_artanh
tff(fact_6064_filterlim__realpow__sequentially__gt1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,Xb))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),Xb),at_infinity(A),at_top(nat)) ) ) ).

% filterlim_realpow_sequentially_gt1
tff(fact_6065_lim__at__infinity__0,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,L),at_infinity(A))
        <=> filterlim(A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,F2),inverse_inverse(A)),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% lim_at_infinity_0
tff(fact_6066_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_ya(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,L),at_infinity(A)) ) ) ).

% lim_zero_infinity
tff(fact_6067_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Xb: A,S3: set(A),F2: fun(A,B),F7: fun(A,B)] :
          ( member(A,Xb,S3)
         => ( topolo1002775350975398744n_open(A,S3)
           => ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,Xb,S3))
            <=> ( real_V3181309239436604168linear(A,B,F7)
                & ? [E3: fun(A,B)] :
                    ( ! [H3: A] :
                        ( member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),H3),S3)
                       => ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),H3)) = 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,Xb)),aa(A,B,F7,H3))),aa(A,B,E3,H3)) ) )
                    & filterlim(A,real,aTP_Lamp_xk(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_6068_tendsto__exp__limit__at__right,axiom,
    ! [Xb: real] : filterlim(real,real,aTP_Lamp_yb(real,fun(real,real),Xb),topolo7230453075368039082e_nhds(real,exp(real,Xb)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% tendsto_exp_limit_at_right
tff(fact_6069_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_6070_Compl__atMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : ( aa(set(A),set(A),uminus_uminus(set(A)),set_ord_atMost(A,K)) = set_ord_greaterThan(A,K) ) ) ).

% Compl_atMost
tff(fact_6071_Compl__greaterThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : ( aa(set(A),set(A),uminus_uminus(set(A)),set_ord_greaterThan(A,K)) = set_ord_atMost(A,K) ) ) ).

% Compl_greaterThan
tff(fact_6072_Sup__greaterThanAtLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),top_top(A))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_ord_greaterThan(A,Xb)) = top_top(A) ) ) ) ).

% Sup_greaterThanAtLeast
tff(fact_6073_image__uminus__greaterThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A] : ( image(A,A,uminus_uminus(A),set_ord_greaterThan(A,Xb)) = set_ord_lessThan(A,aa(A,A,uminus_uminus(A),Xb)) ) ) ).

% image_uminus_greaterThan
tff(fact_6074_image__uminus__lessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A] : ( image(A,A,uminus_uminus(A),set_ord_lessThan(A,Xb)) = set_ord_greaterThan(A,aa(A,A,uminus_uminus(A),Xb)) ) ) ).

% image_uminus_lessThan
tff(fact_6075_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_6076_Sup__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A3: set(A),Xb: A] :
          ( topolo1002775350975398744n_open(A,A3)
         => ( ! [X: A] :
                ( member(A,X,A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Xb) )
           => ~ member(A,aa(set(A),A,complete_Sup_Sup(A),A3),A3) ) ) ) ).

% Sup_notin_open
tff(fact_6077_Inf__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A3: set(A),Xb: A] :
          ( topolo1002775350975398744n_open(A,A3)
         => ( ! [X: A] :
                ( member(A,X,A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X) )
           => ~ member(A,aa(set(A),A,complete_Inf_Inf(A),A3),A3) ) ) ) ).

% Inf_notin_open
tff(fact_6078_filterlim__at__left__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A),A2: real] :
      ( filterlim(real,A,F2,F4,topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
    <=> filterlim(real,A,aTP_Lamp_yc(fun(real,A),fun(real,A),F2),F4,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_6079_eventually__at__left__to__right,axiom,
    ! [P: fun(real,$o),A2: real] :
      ( eventually(real,P,topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
    <=> eventually(real,aTP_Lamp_yd(fun(real,$o),fun(real,$o),P),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A2),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),A2)))) ) ).

% eventually_at_left_to_right
tff(fact_6080_open__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S3: set(A),Xb: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S3)
         => ( member(A,Xb,S3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
             => ? [B4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B4)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,Xb,B4)),S3) ) ) ) ) ) ).

% open_right
tff(fact_6081_open__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S3: set(A),Xb: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S3)
         => ( member(A,Xb,S3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
             => ? [B4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),Xb)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,B4,Xb)),S3) ) ) ) ) ) ).

% open_left
tff(fact_6082_eventually__at__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Xb: A,Y: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( eventually(A,P,topolo174197925503356063within(A,Xb,set_ord_greaterThan(A,Xb)))
          <=> ? [B5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B5)
                & ! [Y5: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),B5)
                     => aa(A,$o,P,Y5) ) ) ) ) ) ) ).

% eventually_at_right
tff(fact_6083_eventually__at__right__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),Xb: A] :
          ( eventually(A,P,topolo174197925503356063within(A,Xb,set_ord_greaterThan(A,Xb)))
        <=> ? [B5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B5)
              & ! [Y5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y5)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),B5)
                   => aa(A,$o,P,Y5) ) ) ) ) ) ).

% eventually_at_right_field
tff(fact_6084_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_6085_open__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_ye(product_prod(A,A),$o))) ) ).

% open_subdiagonal
tff(fact_6086_open__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_yf(product_prod(A,A),$o))) ) ).

% open_superdiagonal
tff(fact_6087_eventually__at__right__to__top,axiom,
    ! [P: fun(real,$o)] :
      ( eventually(real,P,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
    <=> eventually(real,aTP_Lamp_yg(fun(real,$o),fun(real,$o),P),at_top(real)) ) ).

% eventually_at_right_to_top
tff(fact_6088_eventually__at__top__to__right,axiom,
    ! [P: fun(real,$o)] :
      ( eventually(real,P,at_top(real))
    <=> eventually(real,aTP_Lamp_yg(fun(real,$o),fun(real,$o),P),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% eventually_at_top_to_right
tff(fact_6089_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_6090_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_6091_filterlim__at__top__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A)] :
      ( filterlim(real,A,F2,F4,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_yh(fun(real,A),fun(real,A),F2),F4,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% filterlim_at_top_to_right
tff(fact_6092_filterlim__at__right__to__top,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A)] :
      ( filterlim(real,A,F2,F4,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
    <=> filterlim(real,A,aTP_Lamp_yh(fun(real,A),fun(real,A),F2),F4,at_top(real)) ) ).

% filterlim_at_right_to_top
tff(fact_6093_eventually__at__right__less,axiom,
    ! [A: $tType] :
      ( ( no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xb: A] : eventually(A,aa(A,fun(A,$o),ord_less(A),Xb),topolo174197925503356063within(A,Xb,set_ord_greaterThan(A,Xb))) ) ).

% eventually_at_right_less
tff(fact_6094_less__separate,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ? [A4: A,B4: A] :
              ( member(A,Xb,set_ord_lessThan(A,A4))
              & member(A,Y,set_ord_greaterThan(A,B4))
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,A4)),set_ord_greaterThan(A,B4)) = bot_bot(set(A)) ) ) ) ) ).

% less_separate
tff(fact_6095_eventually__at__rightI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( ! [X: A] :
              ( member(A,X,set_or5935395276787703475ssThan(A,A2,B2))
             => aa(A,$o,P,X) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => eventually(A,P,topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ).

% eventually_at_rightI
tff(fact_6096_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_6097_filterlim__times__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),P3: B,F12: filter(A),C2: B,L: B] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,P3,set_ord_greaterThan(B,P3)),F12)
         => ( aa(B,$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),P3) )
             => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_yi(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo174197925503356063within(B,L,set_ord_greaterThan(B,L)),F12) ) ) ) ) ).

% filterlim_times_pos
tff(fact_6098_tendsto__imp__filterlim__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_wr(fun(A,B),fun(B,fun(A,$o)),F2),L5),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,set_ord_greaterThan(B,L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_right
tff(fact_6099_tendsto__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A2: B,S3: set(B),F2: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A2)
         => ( member(B,A2,S3)
           => ( topolo1002775350975398744n_open(B,S3)
             => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,S3))
              <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_ug(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_6100_filterlim__tan__at__right,axiom,
    filterlim(real,real,tan(real),at_bot(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))))) ).

% filterlim_tan_at_right
tff(fact_6101_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F2: fun(real,real),F4: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(real,real,F2,at_bot(real),F4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yj(nat,fun(fun(real,real),fun(real,real)),Nb),F2),at_top(real),F4) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_6102_eventually__at__bot__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N6: A] :
            ! [N4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N4),N6)
             => aa(A,$o,P,N4) ) ) ) ).

% eventually_at_bot_dense
tff(fact_6103_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_yk(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_6104_filterlim__uminus__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F4)
    <=> filterlim(A,real,aTP_Lamp_yl(fun(A,real),fun(A,real),F2),at_bot(real),F4) ) ).

% filterlim_uminus_at_top
tff(fact_6105_filterlim__uminus__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_bot(real),F4)
    <=> filterlim(A,real,aTP_Lamp_yl(fun(A,real),fun(A,real),F2),at_top(real),F4) ) ).

% filterlim_uminus_at_bot
tff(fact_6106_filterlim__at__bot__mirror,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A)] :
      ( filterlim(real,A,F2,F4,at_bot(real))
    <=> filterlim(real,A,aTP_Lamp_yc(fun(real,A),fun(real,A),F2),F4,at_top(real)) ) ).

% filterlim_at_bot_mirror
tff(fact_6107_filterlim__at__top__mirror,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A)] :
      ( filterlim(real,A,F2,F4,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_yc(fun(real,A),fun(real,A),F2),F4,at_bot(real)) ) ).

% filterlim_at_top_mirror
tff(fact_6108_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_6109_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_6110_greaterThan__0,axiom,
    set_ord_greaterThan(nat,zero_zero(nat)) = image(nat,nat,suc,top_top(set(nat))) ).

% greaterThan_0
tff(fact_6111_greaterThan__Suc,axiom,
    ! [K: nat] : ( set_ord_greaterThan(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_greaterThan(nat,K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(nat,nat,suc,K)),bot_bot(set(nat)))) ) ).

% greaterThan_Suc
tff(fact_6112_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_ym(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ).

% filterlim_at_bot_dense
tff(fact_6113_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_6114_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_6115_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z6),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_yn(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_6116_filterlim__inverse__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( eventually(A,aTP_Lamp_yo(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,aTP_Lamp_xr(fun(A,real),fun(A,real),F2),at_bot(real),F4) ) ) ).

% filterlim_inverse_at_bot
tff(fact_6117_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_6118_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F2: fun(real,real),F4: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(real,real,F2,at_bot(real),F4)
       => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yj(nat,fun(fun(real,real),fun(real,real)),Nb),F2),at_bot(real),F4) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_6119_filterlim__at__bot__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A),A2: A] :
          ( ! [X: A,Y3: A] :
              ( aa(A,$o,Q,X)
             => ( aa(A,$o,Q,Y3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y3)) ) ) )
         => ( ! [X: B] :
                ( aa(B,$o,P,X)
               => ( aa(A,B,F2,aa(B,A,G,X)) = X ) )
           => ( ! [X: B] :
                  ( aa(B,$o,P,X)
                 => aa(A,$o,Q,aa(B,A,G,X)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2)))
               => ( ! [B4: A] :
                      ( aa(A,$o,Q,B4)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B4) )
                 => ( eventually(B,P,at_bot(B))
                   => filterlim(A,B,F2,at_bot(B),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ) ) ) ) ).

% filterlim_at_bot_at_right
tff(fact_6120_tendsto__arctan__at__bot,axiom,
    filterlim(real,real,arctan,topolo7230453075368039082e_nhds(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),at_bot(real)) ).

% tendsto_arctan_at_bot
tff(fact_6121_sequentially__imp__eventually__at__right,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ! [F3: fun(nat,A)] :
                ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(nat,A,F3,N3))
               => ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,N3)),B2)
                 => ( order_antimono(nat,A,F3)
                   => ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ww(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F3),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ).

% sequentially_imp_eventually_at_right
tff(fact_6122_tendsto__at__right__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,X6: fun(A,B),L5: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ! [S7: fun(nat,A)] :
                ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(nat,A,S7,N3))
               => ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,S7,N3)),B2)
                 => ( order_antimono(nat,A,S7)
                   => ( filterlim(nat,A,S7,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_wa(fun(A,B),fun(fun(nat,A),fun(nat,B)),X6),S7),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) ) ) ) )
           => filterlim(A,B,X6,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ).

% tendsto_at_right_sequentially
tff(fact_6123_decseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_antimono(nat,A,F2)
        <=> ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,aa(nat,nat,suc,N4))),aa(nat,A,F2,N4)) ) ) ).

% decseq_Suc_iff
tff(fact_6124_decseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N))),aa(nat,A,X6,N))
         => order_antimono(nat,A,X6) ) ) ).

% decseq_SucI
tff(fact_6125_decseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: fun(nat,A),I: nat] :
          ( order_antimono(nat,A,A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A3,aa(nat,nat,suc,I))),aa(nat,A,A3,I)) ) ) ).

% decseq_SucD
tff(fact_6126_decseq__eq__incseq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X6: fun(nat,A)] :
          ( order_antimono(nat,A,X6)
        <=> order_mono(nat,A,aTP_Lamp_hd(fun(nat,A),fun(nat,A),X6)) ) ) ).

% decseq_eq_incseq
tff(fact_6127_min__of__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)) ) ) ) ).

% min_of_antimono
tff(fact_6128_max__of__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)) ) ) ) ).

% max_of_antimono
tff(fact_6129_antimono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Q: fun(A,A)] :
          ( order_mono(A,A,Q)
         => order_antimono(nat,A,aTP_Lamp_yp(fun(A,A),fun(nat,A),Q)) ) ) ).

% antimono_funpow
tff(fact_6130_Bfun__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( A2 != zero_zero(B) )
           => bfun(A,B,aTP_Lamp_tb(fun(A,B),fun(A,B),F2),F4) ) ) ) ).

% Bfun_inverse
tff(fact_6131_subset__subseqs,axiom,
    ! [A: $tType,X6: set(A),Xs: list(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),aa(list(A),set(A),set2(A),Xs))
     => member(set(A),X6,image(list(A),set(A),set2(A),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))) ) ).

% subset_subseqs
tff(fact_6132_subseqs__refl,axiom,
    ! [A: $tType,Xs: list(A)] : member(list(A),Xs,aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))) ).

% subseqs_refl
tff(fact_6133_Bseq__add,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( bfun(nat,A,F2,at_top(nat))
         => bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_yq(fun(nat,A),fun(A,fun(nat,A)),F2),C2),at_top(nat)) ) ) ).

% Bseq_add
tff(fact_6134_Bseq__add__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_yq(fun(nat,A),fun(A,fun(nat,A)),F2),C2),at_top(nat))
        <=> bfun(nat,A,F2,at_top(nat)) ) ) ).

% Bseq_add_iff
tff(fact_6135_Bseq__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_bs(fun(nat,A),fun(nat,A),X6),at_top(nat))
        <=> bfun(nat,A,X6,at_top(nat)) ) ) ).

% Bseq_minus_iff
tff(fact_6136_Bseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_bt(fun(nat,A),fun(nat,A),F2),at_top(nat))
        <=> bfun(nat,A,F2,at_top(nat)) ) ) ).

% Bseq_Suc_iff
tff(fact_6137_Bseq__offset,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),K: nat] :
          ( bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_yr(fun(nat,A),fun(nat,fun(nat,A)),X6),K),at_top(nat))
         => bfun(nat,A,X6,at_top(nat)) ) ) ).

% Bseq_offset
tff(fact_6138_Bseq__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),K: nat] :
          ( bfun(nat,A,X6,at_top(nat))
         => bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_yr(fun(nat,A),fun(nat,fun(nat,A)),X6),K),at_top(nat)) ) ) ).

% Bseq_ignore_initial_segment
tff(fact_6139_Bseq__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bn(A,fun(fun(nat,A),fun(nat,A)),C2),F2),at_top(nat))
          <=> bfun(nat,A,F2,at_top(nat)) ) ) ) ).

% Bseq_cmult_iff
tff(fact_6140_distinct__set__subseqs,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => distinct(set(A),aa(list(list(A)),list(set(A)),map(list(A),set(A),set2(A)),subseqs(A,Xs))) ) ).

% distinct_set_subseqs
tff(fact_6141_Cons__in__subseqsD,axiom,
    ! [A: $tType,Y: A,Ys2: list(A),Xs: list(A)] :
      ( member(list(A),cons(A,Y,Ys2),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))
     => member(list(A),Ys2,aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))) ) ).

% Cons_in_subseqsD
tff(fact_6142_subseqs__distinctD,axiom,
    ! [A: $tType,Ys2: list(A),Xs: list(A)] :
      ( member(list(A),Ys2,aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))
     => ( distinct(A,Xs)
       => distinct(A,Ys2) ) ) ).

% subseqs_distinctD
tff(fact_6143_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [N6: nat] :
            ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N4))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6))) ) ) ).

% Bseq_iff1a
tff(fact_6144_Bseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [N6: nat] :
            ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N4))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6))) ) ) ).

% Bseq_iff
tff(fact_6145_Bseq__realpow,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => bfun(nat,real,aa(real,fun(nat,real),power_power(real),Xb),at_top(nat)) ) ) ).

% Bseq_realpow
tff(fact_6146_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K3)
              & ? [X4: A] :
                ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X6,N4)),aa(A,A,uminus_uminus(A),X4)))),K3) ) ) ) ).

% Bseq_iff2
tff(fact_6147_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K3)
              & ? [N6: nat] :
                ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X6,N4)),aa(A,A,uminus_uminus(A),aa(nat,A,X6,N6))))),K3) ) ) ) ).

% Bseq_iff3
tff(fact_6148_inj__sgn__power,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => inj_on(real,real,aTP_Lamp_pg(nat,fun(real,real),Nb),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_6149_continuous__on__IccI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A,B2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2)))
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,B2)),topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2)))
           => ( ! [X: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),B2)
                   => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) )
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
               => topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2) ) ) ) ) ) ).

% continuous_on_IccI
tff(fact_6150_inj__uminus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: set(A)] : inj_on(A,A,uminus_uminus(A),A3) ) ).

% inj_uminus
tff(fact_6151_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_6152_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( inj_on(A,A,aTP_Lamp_ys(A,fun(A,A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_6153_continuous__inj__imp__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo8458572112393995274pology(A)
        & topolo1944317154257567458pology(B) )
     => ! [A2: A,Xb: A,B2: A,F2: fun(A,B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B2)
           => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
             => ( inj_on(A,B,F2,set_or1337092689740270186AtMost(A,A2,B2))
               => ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),aa(A,B,F2,Xb))
                    & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,B2)) )
                  | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B2)),aa(A,B,F2,Xb))
                    & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,A2)) ) ) ) ) ) ) ) ).

% continuous_inj_imp_mono
tff(fact_6154_continuous__on__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( ! [X: A] :
                ( member(A,X,S)
               => ( aa(A,B,F2,X) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aTP_Lamp_yt(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_sgn
tff(fact_6155_continuous__on__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S: set(A),F2: fun(A,B),Nb: int] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( ! [X: A] :
                ( member(A,X,S)
               => ( aa(A,B,F2,X) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aa(int,fun(A,B),aTP_Lamp_yu(fun(A,B),fun(int,fun(A,B)),F2),Nb)) ) ) ) ).

% continuous_on_power_int
tff(fact_6156_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : inj_on(A,A,aTP_Lamp_lv(A,fun(A,A),A2),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_6157_continuous__on__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( topolo81223032696312382ous_on(A,B,S,G)
           => ( ! [X: A] :
                  ( member(A,X,S)
                 => ( aa(A,B,G,X) != zero_zero(B) ) )
             => topolo81223032696312382ous_on(A,B,S,aa(fun(A,B),fun(A,B),aTP_Lamp_yv(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_on_divide
tff(fact_6158_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_6159_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_6160_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_6161_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A)] : inj_on(A,A,aTP_Lamp_yw(A,fun(A,A),A2),A3) ) ).

% inj_on_add'
tff(fact_6162_continuous__on__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => topolo81223032696312382ous_on(A,B,S,aTP_Lamp_yx(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_minus
tff(fact_6163_continuous__on__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( topolo81223032696312382ous_on(A,B,S,G)
           => topolo81223032696312382ous_on(A,B,S,aa(fun(A,B),fun(A,B),aTP_Lamp_yy(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_add
tff(fact_6164_continuous__on__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( topolo81223032696312382ous_on(A,B,S,G)
           => topolo81223032696312382ous_on(A,B,S,aa(fun(A,B),fun(A,B),aTP_Lamp_yz(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_diff
tff(fact_6165_continuous__on__op__minus,axiom,
    ! [A: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [S: set(A),Xb: A] : topolo81223032696312382ous_on(A,A,S,aa(A,fun(A,A),minus_minus(A),Xb)) ) ).

% continuous_on_op_minus
tff(fact_6166_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_6167_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_6168_inj__fn,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat] :
      ( inj_on(A,A,F2,top_top(set(A)))
     => inj_on(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),top_top(set(A))) ) ).

% inj_fn
tff(fact_6169_subset__image__inj,axiom,
    ! [A: $tType,B: $tType,S3: set(A),F2: fun(B,A),T4: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),image(B,A,F2,T4))
    <=> ? [U4: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),U4),T4)
          & inj_on(B,A,F2,U4)
          & ( S3 = image(B,A,F2,U4) ) ) ) ).

% subset_image_inj
tff(fact_6170_inj__on__mapI,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(list(A))] :
      ( inj_on(A,B,F2,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(list(A),set(A),set2(A),A3)))
     => inj_on(list(A),list(B),map(A,B,F2),A3) ) ).

% inj_on_mapI
tff(fact_6171_continuous__on__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( ! [X: A] :
                ( member(A,X,S)
               => ( aa(A,B,F2,X) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aTP_Lamp_za(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_inverse
tff(fact_6172_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A3: set(A),A9: set(B)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ? [F5: fun(A,B)] :
            ( inj_on(A,B,F5,A3)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,F5,A3)),A9) )
      <=> ? [G3: fun(B,A)] : ( image(B,A,G3,A9) = A3 ) ) ) ).

% inj_on_iff_surj
tff(fact_6173_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),image(B,A,F2,A3))),aa(set(B),nat,finite_card(B),A3))
     => ~ inj_on(B,A,F2,A3) ) ).

% pigeonhole
tff(fact_6174_distinct__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F2),Xs))
    <=> ( distinct(B,Xs)
        & inj_on(B,A,F2,aa(list(B),set(B),set2(B),Xs)) ) ) ).

% distinct_map
tff(fact_6175_open__Collect__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo1002775350975398744n_open(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_zb(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G))) ) ) ) ).

% open_Collect_less
tff(fact_6176_continuous__on__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S,F2)
         => ( ! [X: A] :
                ( member(A,X,S)
               => ( aa(A,A,cos(A),aa(A,A,F2,X)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S,aTP_Lamp_te(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_tan
tff(fact_6177_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Y: A,Xs: list(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),aa(list(A),set(A),set2(A),Xs)))
     => ( filter2(A,aa(A,fun(A,$o),aTP_Lamp_zc(fun(A,B),fun(A,fun(A,$o)),F2),Y),Xs) = filter2(A,aa(A,fun(A,$o),fequal(A),Y),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_6178_continuous__on__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S,F2)
         => ( ! [X: A] :
                ( member(A,X,S)
               => ( sin(A,aa(A,A,F2,X)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S,aTP_Lamp_sw(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_cot
tff(fact_6179_continuous__on__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A3: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A3,F2)
         => ( ! [X: A] :
                ( member(A,X,A3)
               => ( cosh(B,aa(A,B,F2,X)) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,A3,aTP_Lamp_zd(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_tanh
tff(fact_6180_continuous__on__arcosh_H,axiom,
    ! [A3: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A3,F2)
     => ( ! [X: real] :
            ( member(real,X,A3)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,F2,X)) )
       => topolo81223032696312382ous_on(real,real,A3,aTP_Lamp_ze(fun(real,real),fun(real,real),F2)) ) ) ).

% continuous_on_arcosh'
tff(fact_6181_inj__image__Compl__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,F2,aa(set(A),set(A),uminus_uminus(set(A)),A3))),aa(set(B),set(B),uminus_uminus(set(B)),image(A,B,F2,A3))) ) ).

% inj_image_Compl_subset
tff(fact_6182_map__removeAll__inj__on,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xb: A,Xs: list(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(B),map(A,B,F2),removeAll(A,Xb,Xs)) = removeAll(B,aa(A,B,F2,Xb),aa(list(A),list(B),map(A,B,F2),Xs)) ) ) ).

% map_removeAll_inj_on
tff(fact_6183_continuous__on__log,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( topolo81223032696312382ous_on(A,real,S,G)
           => ( ! [X: A] :
                  ( member(A,X,S)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,X)) )
             => ( ! [X: A] :
                    ( member(A,X,S)
                   => ( aa(A,real,F2,X) != one_one(real) ) )
               => ( ! [X: A] :
                      ( member(A,X,S)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,X)) )
                 => topolo81223032696312382ous_on(A,real,S,aa(fun(A,real),fun(A,real),aTP_Lamp_zf(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_on_log
tff(fact_6184_continuous__on__arccos_H,axiom,
    topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)),arccos) ).

% continuous_on_arccos'
tff(fact_6185_continuous__on__arcsin_H,axiom,
    topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)),arcsin) ).

% continuous_on_arcsin'
tff(fact_6186_continuous__on__arccos,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( ! [X: A] :
                ( member(A,X,S)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F2,X))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F2,X)),one_one(real)) ) )
           => topolo81223032696312382ous_on(A,real,S,aTP_Lamp_zg(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_arccos
tff(fact_6187_continuous__on__arcsin,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( ! [X: A] :
                ( member(A,X,S)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F2,X))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F2,X)),one_one(real)) ) )
           => topolo81223032696312382ous_on(A,real,S,aTP_Lamp_zh(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_arcsin
tff(fact_6188_continuous__on__artanh,axiom,
    ! [A3: set(real)] :
      ( aa(set(real),$o,aa(set(real),fun(set(real),$o),ord_less_eq(set(real)),A3),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))
     => topolo81223032696312382ous_on(real,real,A3,artanh(real)) ) ).

% continuous_on_artanh
tff(fact_6189_continuous__on__artanh_H,axiom,
    ! [A3: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A3,F2)
     => ( ! [X: real] :
            ( member(real,X,A3)
           => member(real,aa(real,real,F2,X),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real))) )
       => topolo81223032696312382ous_on(real,real,A3,aTP_Lamp_zi(fun(real,real),fun(real,real),F2)) ) ) ).

% continuous_on_artanh'
tff(fact_6190_log__inj,axiom,
    ! [B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => inj_on(real,real,log2(B2),set_ord_greaterThan(real,zero_zero(real))) ) ).

% log_inj
tff(fact_6191_continuous__on__of__int__floor,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => topolo81223032696312382ous_on(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),ring_1_Ints(A)),aTP_Lamp_zj(A,B)) ) ).

% continuous_on_of_int_floor
tff(fact_6192_continuous__on__of__int__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => topolo81223032696312382ous_on(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),ring_1_Ints(A)),aTP_Lamp_zk(A,B)) ) ).

% continuous_on_of_int_ceiling
tff(fact_6193_continuous__on__Icc__at__leftD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,B2)),topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2))) ) ) ) ).

% continuous_on_Icc_at_leftD
tff(fact_6194_continuous__on__Icc__at__rightD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ).

% continuous_on_Icc_at_rightD
tff(fact_6195_funpow__inj__finite,axiom,
    ! [A: $tType,P3: fun(A,A),Xb: A] :
      ( inj_on(A,A,P3,top_top(set(A)))
     => ( aa(set(A),$o,finite_finite(A),collect(A,aa(A,fun(A,$o),aTP_Lamp_zl(fun(A,A),fun(A,fun(A,$o)),P3),Xb)))
       => ~ ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),P3),Xb) != Xb ) ) ) ) ).

% funpow_inj_finite
tff(fact_6196_surj__int__decode,axiom,
    image(nat,int,nat_int_decode,top_top(set(nat))) = top_top(set(int)) ).

% surj_int_decode
tff(fact_6197_set__list__bind,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),F2: fun(B,list(A))] : ( aa(list(A),set(A),set2(A),bind(B,A,Xs,F2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),aTP_Lamp_zm(fun(B,list(A)),fun(B,set(A)),F2),aa(list(B),set(B),set2(B),Xs))) ) ).

% set_list_bind
tff(fact_6198_int__decode__inverse,axiom,
    ! [Nb: nat] : ( aa(int,nat,nat_int_encode,aa(nat,int,nat_int_decode,Nb)) = Nb ) ).

% int_decode_inverse
tff(fact_6199_int__encode__inverse,axiom,
    ! [Xb: int] : ( aa(nat,int,nat_int_decode,aa(int,nat,nat_int_encode,Xb)) = Xb ) ).

% int_encode_inverse
tff(fact_6200_inj__list__encode,axiom,
    ! [A3: set(list(nat))] : inj_on(list(nat),nat,nat_list_encode,A3) ).

% inj_list_encode
tff(fact_6201_inj__on__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N5: set(nat)] : inj_on(nat,A,semiring_1_of_nat(A),N5) ) ).

% inj_on_of_nat
tff(fact_6202_inj__prod__encode,axiom,
    ! [A3: set(product_prod(nat,nat))] : inj_on(product_prod(nat,nat),nat,nat_prod_encode,A3) ).

% inj_prod_encode
tff(fact_6203_inj__setminus,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: set(set(A))] : inj_on(set(A),set(A),uminus_uminus(set(A)),A3) ) ).

% inj_setminus
tff(fact_6204_inj__Suc,axiom,
    ! [N5: set(nat)] : inj_on(nat,nat,suc,N5) ).

% inj_Suc
tff(fact_6205_inj__int__decode,axiom,
    ! [A3: set(nat)] : inj_on(nat,int,nat_int_decode,A3) ).

% inj_int_decode
tff(fact_6206_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_6207_inj__int__encode,axiom,
    ! [A3: set(int)] : inj_on(int,nat,nat_int_encode,A3) ).

% inj_int_encode
tff(fact_6208_inj__on__diff__nat,axiom,
    ! [N5: set(nat),K: nat] :
      ( ! [N: nat] :
          ( member(nat,N,N5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N) )
     => inj_on(nat,nat,aTP_Lamp_lr(nat,fun(nat,nat),K),N5) ) ).

% inj_on_diff_nat
tff(fact_6209_int__decode__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa(nat,int,nat_int_decode,Xb) = aa(nat,int,nat_int_decode,Y) )
    <=> ( Xb = Y ) ) ).

% int_decode_eq
tff(fact_6210_inj__on__set__encode,axiom,
    inj_on(set(nat),nat,nat_set_encode,collect(set(nat),finite_finite(nat))) ).

% inj_on_set_encode
tff(fact_6211_list__bind__cong,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(A),F2: fun(A,list(B)),G: fun(A,list(B))] :
      ( ( Xs = Ys2 )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Xs))
           => ( aa(A,list(B),F2,X) = aa(A,list(B),G,X) ) )
       => ( bind(A,B,Xs,F2) = bind(A,B,Ys2,G) ) ) ) ).

% list_bind_cong
tff(fact_6212_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ? [N: nat,F3: fun(nat,A)] :
          ( ( A3 = image(nat,A,F3,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aa(nat,fun(nat,$o)),N))) )
          & inj_on(nat,A,F3,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aa(nat,fun(nat,$o)),N))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_6213_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ? [F3: fun(A,nat),N: nat] :
          ( ( image(A,nat,F3,A3) = collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aa(nat,fun(nat,$o)),N)) )
          & inj_on(A,nat,F3,A3) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_6214_inj__on__nth,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( distinct(A,Xs)
     => ( ! [X: nat] :
            ( member(nat,X,I5)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(list(A),nat,size_size(list(A)),Xs)) )
       => inj_on(nat,A,nth(A,Xs),I5) ) ) ).

% inj_on_nth
tff(fact_6215_infinite__countable__subset,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ~ aa(set(A),$o,finite_finite(A),S3)
     => ? [F3: fun(nat,A)] :
          ( inj_on(nat,A,F3,top_top(set(nat)))
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(nat,A,F3,top_top(set(nat)))),S3) ) ) ).

% infinite_countable_subset
tff(fact_6216_infinite__iff__countable__subset,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ~ aa(set(A),$o,finite_finite(A),S3)
    <=> ? [F5: fun(nat,A)] :
          ( inj_on(nat,A,F5,top_top(set(nat)))
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(nat,A,F5,top_top(set(nat)))),S3) ) ) ).

% infinite_iff_countable_subset
tff(fact_6217_inj__on__funpow__least,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,A),S: A] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),S) = S )
     => ( ! [M: nat] :
            ( aa(nat,$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),Nb)
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F2),S) != S ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_zn(fun(A,A),fun(A,fun(nat,A)),F2),S),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ) ).

% inj_on_funpow_least
tff(fact_6218_bij__int__decode,axiom,
    bij_betw(nat,int,nat_int_decode,top_top(set(nat)),top_top(set(int))) ).

% bij_int_decode
tff(fact_6219_UN__le__eq__Un0,axiom,
    ! [A: $tType,M9: fun(nat,set(A)),Nb: nat] : ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),M9,set_ord_atMost(nat,Nb))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),M9,set_or1337092689740270186AtMost(nat,one_one(nat),Nb)))),aa(nat,set(A),M9,zero_zero(nat))) ) ).

% UN_le_eq_Un0
tff(fact_6220_INF__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),Xs: list(B)] : ( aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,aa(list(B),set(B),set2(B),Xs))) = aa(A,A,fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,inf_inf(A)),F2),Xs),top_top(A)) ) ) ).

% INF_set_fold
tff(fact_6221_sup__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(B,A),G: fun(B,A),Xb: 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),Xb) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F2,Xb)),aa(B,A,G,Xb)) ) ) ).

% sup_apply
tff(fact_6222_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_6223_sup__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) ) ) ).

% sup_left_idem
tff(fact_6224_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_6225_sup__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Xb) = Xb ) ) ).

% sup_idem
tff(fact_6226_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_6227_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% le_sup_iff
tff(fact_6228_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_6229_sup__top__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),top_top(A)) = top_top(A) ) ) ).

% sup_top_right
tff(fact_6230_sup__top__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),Xb) = top_top(A) ) ) ).

% sup_top_left
tff(fact_6231_sup__bot__left,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),Xb) = Xb ) ) ).

% sup_bot_left
tff(fact_6232_sup__bot__right,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),bot_bot(A)) = Xb ) ) ).

% sup_bot_right
tff(fact_6233_bot__eq__sup__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xb: A,Y: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) )
        <=> ( ( Xb = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% bot_eq_sup_iff
tff(fact_6234_sup__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = bot_bot(A) )
        <=> ( ( Xb = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% sup_eq_bot_iff
tff(fact_6235_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_6236_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_6237_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_6238_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_6239_sup__inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)) = Xb ) ) ).

% sup_inf_absorb
tff(fact_6240_inf__sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) = Xb ) ) ).

% inf_sup_absorb
tff(fact_6241_sup__compl__top__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) = top_top(A) ) ) ).

% sup_compl_top_left1
tff(fact_6242_sup__compl__top__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xb)),Y)) = top_top(A) ) ) ).

% sup_compl_top_left2
tff(fact_6243_boolean__algebra_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xb)),Xb) = top_top(A) ) ) ).

% boolean_algebra.disj_cancel_left
tff(fact_6244_boolean__algebra_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,uminus_uminus(A),Xb)) = top_top(A) ) ) ).

% boolean_algebra.disj_cancel_right
tff(fact_6245_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% boolean_algebra.de_Morgan_conj
tff(fact_6246_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% boolean_algebra.de_Morgan_disj
tff(fact_6247_set__append,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : ( aa(list(A),set(A),set2(A),append(A,Xs,Ys2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) ) ).

% set_append
tff(fact_6248_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_6249_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y: A,Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) ) ).

% inf_sup_ord(4)
tff(fact_6250_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) ) ).

% inf_sup_ord(3)
tff(fact_6251_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),Xb)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xb)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xb) ) ) ) ).

% le_supE
tff(fact_6252_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,Xb: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),Xb) ) ) ) ).

% le_supI
tff(fact_6253_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) ) ).

% sup_ge1
tff(fact_6254_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) ) ).

% sup_ge2
tff(fact_6255_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% le_supI1
tff(fact_6256_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% le_supI2
tff(fact_6257_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,D3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ) ).

% sup.mono
tff(fact_6258_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
           => aa(A,$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),D3)) ) ) ) ).

% sup_mono
tff(fact_6259_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)),Xb) ) ) ) ).

% sup_least
tff(fact_6260_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = Y ) ) ) ).

% le_iff_sup
tff(fact_6261_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_6262_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_6263_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(A,fun(A,A)),Xb: A,Y: A] :
          ( ! [X: A,Y3: A] : aa(A,$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,$o),ord_less_eq(A),Y3),aa(A,A,aa(A,fun(A,A),F2,X),Y3))
           => ( ! [X: A,Y3: A,Z2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,Y3),Z2)),X) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),F2,Xb),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_6264_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_6265_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_6266_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = Xb ) ) ) ).

% sup_absorb1
tff(fact_6267_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = Y ) ) ) ).

% sup_absorb2
tff(fact_6268_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_6269_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_6270_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_6271_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_6272_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_6273_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_6274_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_6275_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_6276_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_6277_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),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),Xb),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Z))) ) ).

% distrib_sup_le
tff(fact_6278_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,$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),Xb),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z))) ) ).

% distrib_inf_le
tff(fact_6279_mono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_sup(A)
        & semilattice_sup(B) )
     => ! [F2: fun(A,B),A3: A,B3: A] :
          ( order_mono(A,B,F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,A3)),aa(A,B,F2,B3))),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3))) ) ) ).

% mono_sup
tff(fact_6280_INF__sup__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),A3: set(B),G: fun(C,A),B3: set(C)] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A3))),aa(set(A),A,complete_Inf_Inf(A),image(C,A,G,B3))) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_zp(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B3),A3)) ) ) ).

% INF_sup_distrib2
tff(fact_6281_sup__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A2: A,F2: fun(B,A),B3: set(B)] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,B3))) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_zq(A,fun(fun(B,A),fun(B,A)),A2),F2),B3)) ) ) ).

% sup_INF
tff(fact_6282_Inf__sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B3: set(A),A2: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B3)),A2) = aa(set(A),A,complete_Inf_Inf(A),image(A,A,aTP_Lamp_zr(A,fun(A,A),A2),B3)) ) ) ).

% Inf_sup
tff(fact_6283_INF__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),B3: set(B),A2: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,B3))),A2) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_zs(fun(B,A),fun(A,fun(B,A)),F2),A2),B3)) ) ) ).

% INF_sup
tff(fact_6284_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_6285_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_6286_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_6287_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_6288_fold__id,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),F2: fun(A,fun(B,B))] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ( aa(A,fun(B,B),F2,X) = id(B) ) )
     => ( fold(A,B,F2,Xs) = id(B) ) ) ).

% fold_id
tff(fact_6289_Collect__imp__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : ( collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_zt(fun(A,$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_6290_sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_sup(B)
     => ! [F2: fun(A,B),G: fun(A,B),X3: 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),X3) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) ) ) ).

% sup_fun_def
tff(fact_6291_sup__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Y: A,Z: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),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),Xb),Z)) ) ) ).

% sup_left_commute
tff(fact_6292_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_6293_sup__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Xb) ) ) ).

% sup_commute
tff(fact_6294_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_6295_sup__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: 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),Xb),Y)),Z) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) ) ) ).

% sup_assoc
tff(fact_6296_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_6297_inf__sup__aci_I5_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Xb) ) ) ).

% inf_sup_aci(5)
tff(fact_6298_inf__sup__aci_I6_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: 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),Xb),Y)),Z) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) ) ) ).

% inf_sup_aci(6)
tff(fact_6299_inf__sup__aci_I7_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A,Z: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),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),Xb),Z)) ) ) ).

% inf_sup_aci(7)
tff(fact_6300_inf__sup__aci_I8_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) ) ) ).

% inf_sup_aci(8)
tff(fact_6301_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% less_supI1
tff(fact_6302_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% less_supI2
tff(fact_6303_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_6304_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_6305_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_6306_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_6307_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_6308_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_6309_sup__max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( sup_sup(A) = ord_max(A) ) ) ).

% sup_max
tff(fact_6310_sup__inf__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Y: A,Z: A,Xb: 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)),Xb) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Xb)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z),Xb)) ) ) ).

% sup_inf_distrib2
tff(fact_6311_sup__inf__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Xb: A,Y: A,Z: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),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),Xb),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Z)) ) ) ).

% sup_inf_distrib1
tff(fact_6312_inf__sup__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Y: A,Z: A,Xb: 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)),Xb) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Xb)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z),Xb)) ) ) ).

% inf_sup_distrib2
tff(fact_6313_inf__sup__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Xb: A,Y: A,Z: A] : ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),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),Xb),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Z)) ) ) ).

% inf_sup_distrib1
tff(fact_6314_distrib__imp2,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( ! [X: A,Y3: A,Z2: 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),Z2)) = 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),Z2)) )
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),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),Xb),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Z)) ) ) ) ).

% distrib_imp2
tff(fact_6315_distrib__imp1,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( ! [X: A,Y3: A,Z2: 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),Z2)) = 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),Z2)) )
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),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),Xb),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Z)) ) ) ) ).

% distrib_imp1
tff(fact_6316_union__set__fold,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),A3) = aa(set(A),set(A),fold(A,set(A),insert2(A),Xs),A3) ) ).

% union_set_fold
tff(fact_6317_List_Ofold__cong,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: A,Xs: list(B),Ys2: list(B),F2: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
      ( ( A2 = B2 )
     => ( ( Xs = Ys2 )
       => ( ! [X: B] :
              ( member(B,X,aa(list(B),set(B),set2(B),Xs))
             => ( aa(B,fun(A,A),F2,X) = aa(B,fun(A,A),G,X) ) )
         => ( aa(A,A,fold(B,A,F2,Xs),A2) = aa(A,A,fold(B,A,G,Ys2),B2) ) ) ) ) ).

% List.fold_cong
tff(fact_6318_fold__invariant,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Q: fun(A,$o),P: fun(B,$o),S: B,F2: fun(A,fun(B,B))] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,Q,X) )
     => ( aa(B,$o,P,S)
       => ( ! [X: A,S2: B] :
              ( aa(A,$o,Q,X)
             => ( aa(B,$o,P,S2)
               => aa(B,$o,P,aa(B,B,aa(A,fun(B,B),F2,X),S2)) ) )
         => aa(B,$o,P,aa(B,B,fold(A,B,F2,Xs),S)) ) ) ) ).

% fold_invariant
tff(fact_6319_sup__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: 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),Xb)),A2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),B2)) = top_top(A) ) ) ).

% sup_cancel_left2
tff(fact_6320_sup__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: 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),Xb),A2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xb)),B2)) = top_top(A) ) ) ).

% sup_cancel_left1
tff(fact_6321_Sup__set__fold,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xs: list(A)] : ( aa(set(A),A,complete_Sup_Sup(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,sup_sup(A),Xs),bot_bot(A)) ) ) ).

% Sup_set_fold
tff(fact_6322_sup__Inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A2: A,B3: set(A)] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),aa(set(A),A,complete_Inf_Inf(A),B3)) = aa(set(A),A,complete_Inf_Inf(A),image(A,A,aa(A,fun(A,A),sup_sup(A),A2),B3)) ) ) ).

% sup_Inf
tff(fact_6323_Inf__sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B3: set(A),A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B3)),A2) = top_top(A) )
        <=> ! [X4: A] :
              ( member(A,X4,B3)
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),A2) = top_top(A) ) ) ) ) ).

% Inf_sup_eq_top_iff
tff(fact_6324_set__shuffles,axiom,
    ! [A: $tType,Zs3: list(A),Xs: list(A),Ys2: list(A)] :
      ( member(list(A),Zs3,shuffles(A,Xs,Ys2))
     => ( aa(list(A),set(A),set2(A),Zs3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) ) ) ).

% set_shuffles
tff(fact_6325_fold__commute,axiom,
    ! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G: fun(A,fun(B,B)),F2: fun(A,fun(C,C))] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ( aa(fun(B,B),fun(B,C),comp(B,C,B,H),aa(A,fun(B,B),G,X)) = aa(fun(B,C),fun(B,C),comp(C,C,B,aa(A,fun(C,C),F2,X)),H) ) )
     => ( aa(fun(B,B),fun(B,C),comp(B,C,B,H),fold(A,B,G,Xs)) = aa(fun(B,C),fun(B,C),comp(C,C,B,fold(A,C,F2,Xs)),H) ) ) ).

% fold_commute
tff(fact_6326_fold__commute__apply,axiom,
    ! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G: fun(A,fun(B,B)),F2: fun(A,fun(C,C)),S: B] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ( aa(fun(B,B),fun(B,C),comp(B,C,B,H),aa(A,fun(B,B),G,X)) = aa(fun(B,C),fun(B,C),comp(C,C,B,aa(A,fun(C,C),F2,X)),H) ) )
     => ( aa(B,C,H,aa(B,B,fold(A,B,G,Xs),S)) = aa(C,C,fold(A,C,F2,Xs),aa(B,C,H,S)) ) ) ).

% fold_commute_apply
tff(fact_6327_SUP__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),Xs: list(B)] : ( aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,aa(list(B),set(B),set2(B),Xs))) = aa(A,A,fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,sup_sup(A)),F2),Xs),bot_bot(A)) ) ) ).

% SUP_set_fold
tff(fact_6328_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xb)),Y) ) ) ).

% sup_shunt
tff(fact_6329_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P3: A,Q4: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q4),R2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P3),aa(A,A,uminus_uminus(A),Q4))),R2) ) ) ).

% sup_neg_inf
tff(fact_6330_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,uminus_uminus(A),Y))),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) ) ) ).

% shunt2
tff(fact_6331_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Y)),Z)) ) ) ).

% shunt1
tff(fact_6332_card__Un__le,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3))) ).

% card_Un_le
tff(fact_6333_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)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)))
     => ( ( aa(list(A),list(B),map(A,B,F2),Xs) = aa(list(A),list(B),map(A,B,F2),Ys2) )
      <=> ( Xs = Ys2 ) ) ) ).

% inj_on_map_eq_map
tff(fact_6334_map__inj__on,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),Ys2: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs) = aa(list(B),list(A),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)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys2)))
       => ( Xs = Ys2 ) ) ) ).

% map_inj_on
tff(fact_6335_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_6336_fold__rev,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,fun(B,B))] :
      ( ! [X: A,Y3: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ( member(A,Y3,aa(list(A),set(A),set2(A),Xs))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y3)),aa(A,fun(B,B),F2,X)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X)),aa(A,fun(B,B),F2,Y3)) ) ) )
     => ( fold(A,B,F2,rev(A,Xs)) = fold(A,B,F2,Xs) ) ) ).

% fold_rev
tff(fact_6337_fold__plus__sum__list__rev,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A)] : ( fold(A,A,plus_plus(A),Xs) = aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),rev(A,Xs))) ) ) ).

% fold_plus_sum_list_rev
tff(fact_6338_fold__remove1__split,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,fun(B,B)),Xb: A] :
      ( ! [X: A,Y3: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ( member(A,Y3,aa(list(A),set(A),set2(A),Xs))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X)),aa(A,fun(B,B),F2,Y3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y3)),aa(A,fun(B,B),F2,X)) ) ) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ( fold(A,B,F2,Xs) = aa(fun(B,B),fun(B,B),comp(B,B,B,fold(A,B,F2,remove1(A,Xb,Xs))),aa(A,fun(B,B),F2,Xb)) ) ) ) ).

% fold_remove1_split
tff(fact_6339_foldr__fold,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,fun(B,B))] :
      ( ! [X: A,Y3: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ( member(A,Y3,aa(list(A),set(A),set2(A),Xs))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y3)),aa(A,fun(B,B),F2,X)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X)),aa(A,fun(B,B),F2,Y3)) ) ) )
     => ( foldr(A,B,F2,Xs) = fold(A,B,F2,Xs) ) ) ).

% foldr_fold
tff(fact_6340_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = top_top(A) )
           => ( aa(A,A,uminus_uminus(A),Xb) = Y ) ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_6341_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Ma: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Ma)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ma),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,Ma)),set_or7035219750837199246ssThan(A,Ma,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_6342_sum_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),$o,finite_finite(A),B3)
           => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))),aa(set(A),B,aa(fun(A,B),fun(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,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B3)) ) ) ) ) ).

% sum.union_inter
tff(fact_6343_infinite__imp__bij__betw2,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ~ aa(set(A),$o,finite_finite(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)),insert2(A),A2),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw2
tff(fact_6344_card__Un__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( aa(set(A),$o,finite_finite(A),B3)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ).

% card_Un_Int
tff(fact_6345_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Ma: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),Ma)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ma),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,Ma)),set_or1337092689740270186AtMost(A,Ma,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(3)
tff(fact_6346_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Ma: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),Ma)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ma),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,Ma)),set_or7035219750837199246ssThan(A,Ma,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(1)
tff(fact_6347_Inf__set__fold,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xs: list(A)] : ( aa(set(A),A,complete_Inf_Inf(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,inf_inf(A),Xs),top_top(A)) ) ) ).

% Inf_set_fold
tff(fact_6348_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Ma: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Ma)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ma),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,Ma)),set_or5935395276787703475ssThan(A,Ma,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(2)
tff(fact_6349_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_6350_ivl__disj__un__two__touch_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Ma: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),Ma)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ma),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,Ma)),set_or7035219750837199246ssThan(A,Ma,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(1)
tff(fact_6351_SUP__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: A,B3: A] : ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),image(nat,A,aTP_Lamp_pt(A,fun(nat,A),B3),collect(nat,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ) ).

% SUP_nat_binary
tff(fact_6352_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_zu(A,fun(A,$o)),aTP_Lamp_zv(A,fun(A,$o))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_6353_sum_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),$o,finite_finite(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,aa(fun(A,B),fun(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,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B3)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_6354_sum__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A3: set(A),B3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),$o,finite_finite(A),B3)
           => ( aa(set(A),B,aa(fun(A,B),fun(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,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B3))),aa(set(A),B,aa(fun(A,B),fun(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_6355_sum_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),$o,finite_finite(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,aa(fun(A,B),fun(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,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B3)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_6356_prod_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),$o,finite_finite(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) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),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),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B3)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6357_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => boolea2506097494486148201lgebra(A,inf_inf(A),sup_sup(A),uminus_uminus(A),bot_bot(A),top_top(A)) ) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_6358_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
         => ( aa(set(A),B,aa(fun(A,B),fun(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,aa(fun(A,B),fun(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,aa(fun(A,B),fun(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,aa(fun(A,B),fun(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_6359_sum_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),$o,finite_finite(A),B3)
           => ( aa(set(A),B,aa(fun(A,B),fun(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,aa(fun(A,B),fun(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,aa(fun(A,B),fun(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,aa(fun(A,B),fun(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_6360_card__Un__disjoint,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( aa(set(A),$o,finite_finite(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),nat,finite_card(A),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),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_6361_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Ma: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Ma)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ma),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,Ma)),set_or5935395276787703475ssThan(A,Ma,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(4)
tff(fact_6362_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)),insert2(A),L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(3)
tff(fact_6363_sum__Un__nat,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( aa(set(A),$o,finite_finite(A),B3)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(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,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),B3))),aa(set(A),nat,aa(fun(A,nat),fun(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_6364_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Ma: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),Ma)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ma),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,Ma)),set_or1337092689740270186AtMost(A,Ma,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(5)
tff(fact_6365_ivl__disj__un__singleton_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(4)
tff(fact_6366_prod__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( field(B)
     => ! [A3: set(A),B3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),$o,finite_finite(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) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),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)) = divide_divide(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),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_6367_map__sorted__distinct__set__unique,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [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)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)))
         => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F2),Xs))
           => ( distinct(B,aa(list(A),list(B),map(A,B,F2),Xs))
             => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F2),Ys2))
               => ( distinct(B,aa(list(A),list(B),map(A,B,F2),Ys2))
                 => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys2) )
                   => ( Xs = Ys2 ) ) ) ) ) ) ) ) ).

% map_sorted_distinct_set_unique
tff(fact_6368_set__union,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : ( aa(list(A),set(A),set2(A),union(A,Xs,Ys2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) ) ).

% set_union
tff(fact_6369_Gcd__eq__Max,axiom,
    ! [M9: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),M9)
     => ( ( M9 != bot_bot(set(nat)) )
       => ( ~ member(nat,zero_zero(nat),M9)
         => ( gcd_Gcd(nat,M9) = lattic643756798349783984er_Max(nat,aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),image(nat,set(nat),aTP_Lamp_zw(nat,set(nat)),M9))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_6370_Max__divisors__self__nat,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => ( lattic643756798349783984er_Max(nat,collect(nat,aTP_Lamp_iy(nat,fun(nat,$o),Nb))) = Nb ) ) ).

% Max_divisors_self_nat
tff(fact_6371_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),lattic643756798349783984er_Max(A,A3)),Xb)
            <=> ! [X4: A] :
                  ( member(A,X4,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Xb) ) ) ) ) ) ).

% Max_less_iff
tff(fact_6372_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_6373_sup__int__def,axiom,
    sup_sup(int) = ord_max(int) ).

% sup_int_def
tff(fact_6374_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),lattic643756798349783984er_Max(A,A3))
            <=> ? [X4: A] :
                  ( member(A,X4,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X4) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_6375_Max_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Xs: list(A)] : ( lattic643756798349783984er_Max(A,aa(list(A),set(A),set2(A),cons(A,Xb,Xs))) = aa(A,A,fold(A,A,ord_max(A),Xs),Xb) ) ) ).

% Max.set_eq_fold
tff(fact_6376_card__le__Suc__Max,axiom,
    ! [S3: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),S3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),S3)),aa(nat,nat,suc,lattic643756798349783984er_Max(nat,S3))) ) ).

% card_le_Suc_Max
tff(fact_6377_Sup__nat__def,axiom,
    ! [X6: set(nat)] :
      ( aa(set(nat),nat,complete_Sup_Sup(nat),X6) = $ite(X6 = bot_bot(set(nat)),zero_zero(nat),lattic643756798349783984er_Max(nat,X6)) ) ).

% Sup_nat_def
tff(fact_6378_divide__nat__def,axiom,
    ! [Ma: nat,Nb: nat] :
      ( divide_divide(nat,Ma,Nb) = $ite(Nb = zero_zero(nat),zero_zero(nat),lattic643756798349783984er_Max(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_zx(nat,fun(nat,fun(nat,$o)),Ma),Nb)))) ) ).

% divide_nat_def
tff(fact_6379_Max__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S3: set(A),F2: fun(A,B),K: B] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ( lattic643756798349783984er_Max(B,image(A,B,aa(B,fun(A,B),aTP_Lamp_zy(fun(A,B),fun(B,fun(A,B)),F2),K),S3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),lattic643756798349783984er_Max(B,image(A,B,F2,S3))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_6380_minus__set__fold,axiom,
    ! [A: $tType,A3: set(A),Xs: list(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(list(A),set(A),set2(A),Xs)) = aa(set(A),set(A),fold(A,set(A),remove(A),Xs),A3) ) ).

% minus_set_fold
tff(fact_6381_minus__Min__eq__Max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),lattic643756798350308766er_Min(A,S3)) = lattic643756798349783984er_Max(A,image(A,A,uminus_uminus(A),S3)) ) ) ) ) ).

% minus_Min_eq_Max
tff(fact_6382_Max__divisors__self__int,axiom,
    ! [Nb: int] :
      ( ( Nb != zero_zero(int) )
     => ( lattic643756798349783984er_Max(int,collect(int,aTP_Lamp_iv(int,fun(int,$o),Nb))) = aa(int,int,abs_abs(int),Nb) ) ) ).

% Max_divisors_self_int
tff(fact_6383_Min__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),lattic643756798350308766er_Min(A,A3))
            <=> ! [X4: A] :
                  ( member(A,X4,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X4) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_6384_minus__Max__eq__Min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),lattic643756798349783984er_Max(A,S3)) = lattic643756798350308766er_Min(A,image(A,A,uminus_uminus(A),S3)) ) ) ) ) ).

% minus_Max_eq_Min
tff(fact_6385_Min__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),lattic643756798350308766er_Min(A,A3)),Xb)
            <=> ? [X4: A] :
                  ( member(A,X4,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Xb) ) ) ) ) ) ).

% Min_less_iff
tff(fact_6386_Min_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Xs: list(A)] : ( lattic643756798350308766er_Min(A,aa(list(A),set(A),set2(A),cons(A,Xb,Xs))) = aa(A,A,fold(A,A,ord_min(A),Xs),Xb) ) ) ).

% Min.set_eq_fold
tff(fact_6387_Min__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S3: set(A),F2: fun(A,B),K: B] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ( lattic643756798350308766er_Min(B,image(A,B,aa(B,fun(A,B),aTP_Lamp_zy(fun(A,B),fun(B,fun(A,B)),F2),K),S3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),lattic643756798350308766er_Min(B,image(A,B,F2,S3))),K) ) ) ) ) ).

% Min_add_commute
tff(fact_6388_remove__code_I1_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Xb),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),removeAll(A,Xb,Xs)) ) ).

% remove_code(1)
tff(fact_6389_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ? [X3: A] :
                ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
                & aa(A,$o,P,X3) )
           => ( find(A,P,Xs) = aa(A,option(A),some(A),lattic643756798350308766er_Min(A,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_zz(list(A),fun(fun(A,$o),fun(A,$o)),Xs),P)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_6390_f__arg__min__list__f,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F2: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( aa(A,B,F2,arg_min_list(A,B,F2,Xs)) = lattic643756798350308766er_Min(B,image(A,B,F2,aa(list(A),set(A),set2(A),Xs))) ) ) ) ).

% f_arg_min_list_f
tff(fact_6391_min__list__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( ( Xs != nil(A) )
         => ( min_list(A,Xs) = lattic643756798350308766er_Min(A,aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% min_list_Min
tff(fact_6392_arg__min__list__in,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F2: fun(A,B)] :
          ( ( Xs != nil(A) )
         => member(A,arg_min_list(A,B,F2,Xs),aa(list(A),set(A),set2(A),Xs)) ) ) ).

% arg_min_list_in
tff(fact_6393_cauchy__def,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
    <=> ! [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
         => ? [K3: nat] :
            ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),M3)
             => ! [N4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N4)
                 => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,X6,M3)),aa(nat,rat,X6,N4)))),R5) ) ) ) ) ).

% cauchy_def
tff(fact_6394_cauchyI,axiom,
    ! [X6: fun(nat,rat)] :
      ( ! [R4: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R4)
         => ? [K4: nat] :
            ! [M: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),M)
             => ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),N)
                 => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,X6,M)),aa(nat,rat,X6,N)))),R4) ) ) )
     => cauchy(X6) ) ).

% cauchyI
tff(fact_6395_cauchy__add,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => cauchy(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_mv(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y6)) ) ) ).

% cauchy_add
tff(fact_6396_cauchy__mult,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => cauchy(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_ms(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y6)) ) ) ).

% cauchy_mult
tff(fact_6397_cauchy__diff,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => cauchy(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_mw(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y6)) ) ) ).

% cauchy_diff
tff(fact_6398_cauchy__const,axiom,
    ! [Xb: rat] : cauchy(aTP_Lamp_mt(rat,fun(nat,rat),Xb)) ).

% cauchy_const
tff(fact_6399_cauchy__minus,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => cauchy(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_mu(fun(nat,rat),fun(nat,rat)),X6)) ) ).

% cauchy_minus
tff(fact_6400_cauchy__inverse,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => cauchy(aTP_Lamp_aaa(fun(nat,rat),fun(nat,rat),X6)) ) ) ).

% cauchy_inverse
tff(fact_6401_cauchy__imp__bounded,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ? [B4: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),B4)
          & ! [N3: nat] : aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N3))),B4) ) ) ).

% cauchy_imp_bounded
tff(fact_6402_vanishes__diff__inverse,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ( cauchy(Y6)
         => ( ~ vanishes(Y6)
           => ( vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_mw(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y6))
             => vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aab(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y6)) ) ) ) ) ) ).

% vanishes_diff_inverse
tff(fact_6403_cauchy__not__vanishes__cases,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ? [B4: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),B4)
            & ? [K2: nat] :
                ( ! [N3: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N3)
                   => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),B4),aa(rat,rat,uminus_uminus(rat),aa(nat,rat,X6,N3))) )
                | ! [N3: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N3)
                   => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),B4),aa(nat,rat,X6,N3)) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_6404_cauchy__not__vanishes,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ? [B4: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),B4)
            & ? [K2: nat] :
              ! [N3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N3)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),B4),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N3))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_6405_cauchyD,axiom,
    ! [X6: fun(nat,rat),R2: rat] :
      ( cauchy(X6)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R2)
       => ? [K2: nat] :
          ! [M4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),M4)
           => ! [N3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N3)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,X6,M4)),aa(nat,rat,X6,N3)))),R2) ) ) ) ) ).

% cauchyD
tff(fact_6406_le__Real,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(fun(nat,rat),real,real2,X6)),aa(fun(nat,rat),real,real2,Y6))
        <=> ! [R5: rat] :
              ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
             => ? [K3: nat] :
                ! [N4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N4)
                 => aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(nat,rat,X6,N4)),aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(nat,rat,Y6,N4)),R5)) ) ) ) ) ) ).

% le_Real
tff(fact_6407_inverse__Real,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( aa(real,real,inverse_inverse(real),aa(fun(nat,rat),real,real2,X6)) = $ite(vanishes(X6),zero_zero(real),aa(fun(nat,rat),real,real2,aTP_Lamp_aaa(fun(nat,rat),fun(nat,rat),X6))) ) ) ).

% inverse_Real
tff(fact_6408_Real__induct,axiom,
    ! [P: fun(real,$o),Xb: real] :
      ( ! [X8: fun(nat,rat)] :
          ( cauchy(X8)
         => aa(real,$o,P,aa(fun(nat,rat),real,real2,X8)) )
     => aa(real,$o,P,Xb) ) ).

% Real_induct
tff(fact_6409_zero__real__def,axiom,
    zero_zero(real) = aa(fun(nat,rat),real,real2,aTP_Lamp_aac(nat,rat)) ).

% zero_real_def
tff(fact_6410_one__real__def,axiom,
    one_one(real) = aa(fun(nat,rat),real,real2,aTP_Lamp_aad(nat,rat)) ).

% one_real_def
tff(fact_6411_of__int__Real,axiom,
    ! [Xb: int] : ( aa(int,real,ring_1_of_int(real),Xb) = aa(fun(nat,rat),real,real2,aTP_Lamp_aae(int,fun(nat,rat),Xb)) ) ).

% of_int_Real
tff(fact_6412_of__nat__Real,axiom,
    ! [Xb: nat] : ( aa(nat,real,semiring_1_of_nat(real),Xb) = aa(fun(nat,rat),real,real2,aTP_Lamp_aaf(nat,fun(nat,rat),Xb)) ) ).

% of_nat_Real
tff(fact_6413_minus__Real,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( aa(real,real,uminus_uminus(real),aa(fun(nat,rat),real,real2,X6)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_mu(fun(nat,rat),fun(nat,rat)),X6)) ) ) ).

% minus_Real
tff(fact_6414_add__Real,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(fun(nat,rat),real,real2,X6)),aa(fun(nat,rat),real,real2,Y6)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_mv(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y6)) ) ) ) ).

% add_Real
tff(fact_6415_mult__Real,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(fun(nat,rat),real,real2,X6)),aa(fun(nat,rat),real,real2,Y6)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_ms(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y6)) ) ) ) ).

% mult_Real
tff(fact_6416_diff__Real,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(fun(nat,rat),real,real2,X6)),aa(fun(nat,rat),real,real2,Y6)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_mw(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y6)) ) ) ) ).

% diff_Real
tff(fact_6417_eq__Real,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => ( ( aa(fun(nat,rat),real,real2,X6) = aa(fun(nat,rat),real,real2,Y6) )
        <=> vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_mw(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y6)) ) ) ) ).

% eq_Real
tff(fact_6418_not__positive__Real,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ~ aa(real,$o,positive2,aa(fun(nat,rat),real,real2,X6))
      <=> ! [R5: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
           => ? [K3: nat] :
              ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N4)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(nat,rat,X6,N4)),R5) ) ) ) ) ).

% not_positive_Real
tff(fact_6419_positive__Real,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( aa(real,$o,positive2,aa(fun(nat,rat),real,real2,X6))
      <=> ? [R5: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
            & ? [K3: nat] :
              ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N4)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,X6,N4)) ) ) ) ) ).

% positive_Real
tff(fact_6420_Real_Opositive__add,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,positive2,Xb)
     => ( aa(real,$o,positive2,Y)
       => aa(real,$o,positive2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)) ) ) ).

% Real.positive_add
tff(fact_6421_Real_Opositive__mult,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,positive2,Xb)
     => ( aa(real,$o,positive2,Y)
       => aa(real,$o,positive2,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)) ) ) ).

% Real.positive_mult
tff(fact_6422_Real_Opositive__zero,axiom,
    ~ aa(real,$o,positive2,zero_zero(real)) ).

% Real.positive_zero
tff(fact_6423_Real_Opositive__minus,axiom,
    ! [Xb: real] :
      ( ~ aa(real,$o,positive2,Xb)
     => ( ( Xb != zero_zero(real) )
       => aa(real,$o,positive2,aa(real,real,uminus_uminus(real),Xb)) ) ) ).

% Real.positive_minus
tff(fact_6424_less__real__def,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
    <=> aa(real,$o,positive2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Y),Xb)) ) ).

% less_real_def
tff(fact_6425_Real_Opositive_Orep__eq,axiom,
    ! [Xb: real] :
      ( aa(real,$o,positive2,Xb)
    <=> ? [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
          & ? [K3: nat] :
            ! [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N4)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,aa(real,fun(nat,rat),rep_real,Xb),N4)) ) ) ) ).

% Real.positive.rep_eq
tff(fact_6426_inverse__real_Oabs__eq,axiom,
    ! [Xb: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Xb),Xb)
     => ( aa(real,real,inverse_inverse(real),aa(fun(nat,rat),real,real2,Xb)) = aa(fun(nat,rat),real,real2,
            $ite(vanishes(Xb),aTP_Lamp_aac(nat,rat),aTP_Lamp_aaa(fun(nat,rat),fun(nat,rat),Xb))) ) ) ).

% inverse_real.abs_eq
tff(fact_6427_realrel__refl,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,X6),X6) ) ).

% realrel_refl
tff(fact_6428_one__real_Orsp,axiom,
    aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,aTP_Lamp_aad(nat,rat)),aTP_Lamp_aad(nat,rat)) ).

% one_real.rsp
tff(fact_6429_zero__real_Orsp,axiom,
    aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,aTP_Lamp_aac(nat,rat)),aTP_Lamp_aac(nat,rat)) ).

% zero_real.rsp
tff(fact_6430_real_Oabs__induct,axiom,
    ! [P: fun(real,$o),Xb: real] :
      ( ! [Y3: fun(nat,rat)] :
          ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Y3),Y3)
         => aa(real,$o,P,aa(fun(nat,rat),real,real2,Y3)) )
     => aa(real,$o,P,Xb) ) ).

% real.abs_induct
tff(fact_6431_uminus__real_Oabs__eq,axiom,
    ! [Xb: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Xb),Xb)
     => ( aa(real,real,uminus_uminus(real),aa(fun(nat,rat),real,real2,Xb)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_mu(fun(nat,rat),fun(nat,rat)),Xb)) ) ) ).

% uminus_real.abs_eq
tff(fact_6432_plus__real_Oabs__eq,axiom,
    ! [Xaa: fun(nat,rat),Xb: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Xaa),Xaa)
     => ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Xb),Xb)
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(fun(nat,rat),real,real2,Xaa)),aa(fun(nat,rat),real,real2,Xb)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_mv(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),Xaa),Xb)) ) ) ) ).

% plus_real.abs_eq
tff(fact_6433_times__real_Oabs__eq,axiom,
    ! [Xaa: fun(nat,rat),Xb: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Xaa),Xaa)
     => ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Xb),Xb)
       => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(fun(nat,rat),real,real2,Xaa)),aa(fun(nat,rat),real,real2,Xb)) = aa(fun(nat,rat),real,real2,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_ms(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),Xaa),Xb)) ) ) ) ).

% times_real.abs_eq
tff(fact_6434_realrelI,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => ( vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_mw(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y6))
         => aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,X6),Y6) ) ) ) ).

% realrelI
tff(fact_6435_realrel__def,axiom,
    ! [X3: fun(nat,rat),Xa: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,X3),Xa)
    <=> ( cauchy(X3)
        & cauchy(Xa)
        & vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_mw(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X3),Xa)) ) ) ).

% realrel_def
tff(fact_6436_Real_Opositive_Oabs__eq,axiom,
    ! [Xb: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Xb),Xb)
     => ( aa(real,$o,positive2,aa(fun(nat,rat),real,real2,Xb))
      <=> ? [R5: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
            & ? [K3: nat] :
              ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N4)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,Xb,N4)) ) ) ) ) ).

% Real.positive.abs_eq
tff(fact_6437_Real_Opositive__def,axiom,
    positive2 = aa(fun(fun(nat,rat),$o),fun(real,$o),map_fun(real,fun(nat,rat),$o,$o,rep_real,id($o)),aTP_Lamp_aag(fun(nat,rat),$o)) ).

% Real.positive_def
tff(fact_6438_inverse__real__def,axiom,
    inverse_inverse(real) = aa(fun(fun(nat,rat),fun(nat,rat)),fun(real,real),map_fun(real,fun(nat,rat),fun(nat,rat),real,rep_real,real2),aTP_Lamp_aah(fun(nat,rat),fun(nat,rat))) ).

% inverse_real_def
tff(fact_6439_uminus__real__def,axiom,
    uminus_uminus(real) = aa(fun(fun(nat,rat),fun(nat,rat)),fun(real,real),map_fun(real,fun(nat,rat),fun(nat,rat),real,rep_real,real2),aTP_Lamp_mu(fun(nat,rat),fun(nat,rat))) ).

% uminus_real_def
tff(fact_6440_times__real__def,axiom,
    times_times(real) = aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(real,fun(real,real)),map_fun(real,fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),fun(real,real),rep_real,map_fun(real,fun(nat,rat),fun(nat,rat),real,rep_real,real2)),aTP_Lamp_ms(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))) ).

% times_real_def
tff(fact_6441_plus__real__def,axiom,
    plus_plus(real) = aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(real,fun(real,real)),map_fun(real,fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),fun(real,real),rep_real,map_fun(real,fun(nat,rat),fun(nat,rat),real,rep_real,real2)),aTP_Lamp_mv(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))) ).

% plus_real_def
tff(fact_6442_Real_Opositive_Orsp,axiom,
    aa(fun(fun(nat,rat),$o),$o,aa(fun(fun(nat,rat),$o),fun(fun(fun(nat,rat),$o),$o),bNF_rel_fun(fun(nat,rat),fun(nat,rat),$o,$o,realrel,fequal($o)),aTP_Lamp_aag(fun(nat,rat),$o)),aTP_Lamp_aag(fun(nat,rat),$o)) ).

% Real.positive.rsp
tff(fact_6443_cr__real__def,axiom,
    ! [X3: fun(nat,rat),Xa: real] :
      ( aa(real,$o,aa(fun(nat,rat),fun(real,$o),cr_real,X3),Xa)
    <=> ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,X3),X3)
        & ( aa(fun(nat,rat),real,real2,X3) = Xa ) ) ) ).

% cr_real_def
tff(fact_6444_power__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & power(A) )
     => ! [R: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R,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),R,bNF_rel_fun(A,B,A,B,R,R)),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),R,bNF_rel_fun(nat,nat,A,B,fequal(nat),R)),power_power(A)),power_power(B)) ) ) ) ).

% power_transfer
tff(fact_6445_transfer__rule__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & semiring_numeral(B)
        & monoid_add(A)
        & semiring_numeral(A) )
     => ! [R: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R,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),R,bNF_rel_fun(A,B,A,B,R,R)),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),R),numeral_numeral(A)),numeral_numeral(B)) ) ) ) ) ).

% transfer_rule_numeral
tff(fact_6446_transfer__rule__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ring_1(B)
        & ring_1(A) )
     => ! [R: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R,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),R,bNF_rel_fun(A,B,A,B,R,R)),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,R,R),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),R),ring_1_of_int(A)),ring_1_of_int(B)) ) ) ) ) ) ).

% transfer_rule_of_int
tff(fact_6447_transfer__rule__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semiring_1(B)
        & semiring_1(A) )
     => ! [R: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R,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),R,bNF_rel_fun(A,B,A,B,R,R)),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),R),semiring_1_of_nat(A)),semiring_1_of_nat(B)) ) ) ) ) ).

% transfer_rule_of_nat
tff(fact_6448_butlast__take,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),list(A),butlast(A),take(A,Nb,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Xs) ) ) ).

% butlast_take
tff(fact_6449_length__butlast,axiom,
    ! [A: $tType,Xs: list(A)] : ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),butlast(A),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ) ).

% length_butlast
tff(fact_6450_times__real_Orsp,axiom,
    aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),$o,aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),$o),bNF_rel_fun(fun(nat,rat),fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),fun(fun(nat,rat),fun(nat,rat)),realrel,bNF_rel_fun(fun(nat,rat),fun(nat,rat),fun(nat,rat),fun(nat,rat),realrel,realrel)),aTP_Lamp_ms(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),aTP_Lamp_ms(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))) ).

% times_real.rsp
tff(fact_6451_plus__real_Orsp,axiom,
    aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),$o,aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),$o),bNF_rel_fun(fun(nat,rat),fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),fun(fun(nat,rat),fun(nat,rat)),realrel,bNF_rel_fun(fun(nat,rat),fun(nat,rat),fun(nat,rat),fun(nat,rat),realrel,realrel)),aTP_Lamp_mv(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),aTP_Lamp_mv(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))) ).

% plus_real.rsp
tff(fact_6452_transfer__rule__of__bool,axiom,
    ! [A: $tType,B: $tType] :
      ( ( zero_neq_one(B)
        & zero_neq_one(A) )
     => ! [R: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R,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),R),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B)) ) ) ) ).

% transfer_rule_of_bool
tff(fact_6453_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_6454_uminus__integer_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),uminus_uminus(int)),uminus_uminus(int)) ).

% uminus_integer.rsp
tff(fact_6455_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_6456_Suc_Orsp,axiom,
    aa(fun(nat,nat),$o,aa(fun(nat,nat),fun(fun(nat,nat),$o),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat)),suc),suc) ).

% Suc.rsp
tff(fact_6457_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_6458_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_6459_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_aai(int,int)),aTP_Lamp_aai(int,int)) ).

% dup.rsp
tff(fact_6460_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_aaj(num,fun(num,int))),aTP_Lamp_aaj(num,fun(num,int))) ).

% sub.rsp
tff(fact_6461_minus__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),minus_minus(int)),minus_minus(int)) ).

% minus_integer.rsp
tff(fact_6462_minus__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),minus_minus(nat)),minus_minus(nat)) ).

% minus_natural.rsp
tff(fact_6463_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_6464_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_6465_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)),aa(fun(int,nat),fun(int,num),comp(nat,num,int,num_of_nat),nat2)),aa(fun(int,nat),fun(int,num),comp(nat,num,int,num_of_nat),nat2)) ).

% num_of_integer.rsp
tff(fact_6466_in__set__butlastD,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Xs)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% in_set_butlastD
tff(fact_6467_in__set__butlast__appendI,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys2: list(A)] :
      ( ( member(A,Xb,aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Xs)))
        | member(A,Xb,aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Ys2))) )
     => member(A,Xb,aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),append(A,Xs,Ys2)))) ) ).

% in_set_butlast_appendI
tff(fact_6468_uminus__real_Orsp,axiom,
    aa(fun(fun(nat,rat),fun(nat,rat)),$o,aa(fun(fun(nat,rat),fun(nat,rat)),fun(fun(fun(nat,rat),fun(nat,rat)),$o),bNF_rel_fun(fun(nat,rat),fun(nat,rat),fun(nat,rat),fun(nat,rat),realrel,realrel),aTP_Lamp_mu(fun(nat,rat),fun(nat,rat))),aTP_Lamp_mu(fun(nat,rat),fun(nat,rat))) ).

% uminus_real.rsp
tff(fact_6469_nth__butlast,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),butlast(A),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),butlast(A),Xs)),Nb) = aa(nat,A,nth(A,Xs),Nb) ) ) ).

% nth_butlast
tff(fact_6470_take__butlast,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( take(A,Nb,aa(list(A),list(A),butlast(A),Xs)) = take(A,Nb,Xs) ) ) ).

% take_butlast
tff(fact_6471_butlast__power,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : ( aa(list(A),list(A),aa(fun(list(A),list(A)),fun(list(A),list(A)),aa(nat,fun(fun(list(A),list(A)),fun(list(A),list(A))),compow(fun(list(A),list(A))),Nb),butlast(A)),Xs) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb),Xs) ) ).

% butlast_power
tff(fact_6472_inverse__real_Orsp,axiom,
    aa(fun(fun(nat,rat),fun(nat,rat)),$o,aa(fun(fun(nat,rat),fun(nat,rat)),fun(fun(fun(nat,rat),fun(nat,rat)),$o),bNF_rel_fun(fun(nat,rat),fun(nat,rat),fun(nat,rat),fun(nat,rat),realrel,realrel),aTP_Lamp_aah(fun(nat,rat),fun(nat,rat))),aTP_Lamp_aah(fun(nat,rat),fun(nat,rat))) ).

% inverse_real.rsp
tff(fact_6473_butlast__conv__take,axiom,
    ! [A: $tType,Xs: list(A)] : ( aa(list(A),list(A),butlast(A),Xs) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),Xs) ) ).

% butlast_conv_take
tff(fact_6474_butlast__list__update,axiom,
    ! [A: $tType,Xs: list(A),K: nat,Xb: A] :
      ( aa(list(A),list(A),butlast(A),list_update(A,Xs,K,Xb)) = $ite(K = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),aa(list(A),list(A),butlast(A),Xs),list_update(A,aa(list(A),list(A),butlast(A),Xs),K,Xb)) ) ).

% butlast_list_update
tff(fact_6475_Real_Opositive_Otransfer,axiom,
    aa(fun(real,$o),$o,aa(fun(fun(nat,rat),$o),fun(fun(real,$o),$o),bNF_rel_fun(fun(nat,rat),real,$o,$o,pcr_real,fequal($o)),aTP_Lamp_aag(fun(nat,rat),$o)),positive2) ).

% Real.positive.transfer
tff(fact_6476_times__int_Otransfer,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_kz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int)) ).

% times_int.transfer
tff(fact_6477_real_Orel__eq__transfer,axiom,
    aa(fun(real,fun(real,$o)),$o,aa(fun(fun(nat,rat),fun(fun(nat,rat),$o)),fun(fun(real,fun(real,$o)),$o),bNF_rel_fun(fun(nat,rat),real,fun(fun(nat,rat),$o),fun(real,$o),pcr_real,bNF_rel_fun(fun(nat,rat),real,$o,$o,pcr_real,fequal($o))),realrel),fequal(real)) ).

% real.rel_eq_transfer
tff(fact_6478_real_Opcr__cr__eq,axiom,
    pcr_real = cr_real ).

% real.pcr_cr_eq
tff(fact_6479_zero__real_Otransfer,axiom,
    aa(real,$o,aa(fun(nat,rat),fun(real,$o),pcr_real,aTP_Lamp_aac(nat,rat)),zero_zero(real)) ).

% zero_real.transfer
tff(fact_6480_one__real_Otransfer,axiom,
    aa(real,$o,aa(fun(nat,rat),fun(real,$o),pcr_real,aTP_Lamp_aad(nat,rat)),one_one(real)) ).

% one_real.transfer
tff(fact_6481_cr__real__eq,axiom,
    ! [X3: fun(nat,rat),Xa: real] :
      ( aa(real,$o,aa(fun(nat,rat),fun(real,$o),pcr_real,X3),Xa)
    <=> ( cauchy(X3)
        & ( aa(fun(nat,rat),real,real2,X3) = Xa ) ) ) ).

% cr_real_eq
tff(fact_6482_uminus__real_Otransfer,axiom,
    aa(fun(real,real),$o,aa(fun(fun(nat,rat),fun(nat,rat)),fun(fun(real,real),$o),bNF_rel_fun(fun(nat,rat),real,fun(nat,rat),real,pcr_real,pcr_real),aTP_Lamp_mu(fun(nat,rat),fun(nat,rat))),uminus_uminus(real)) ).

% uminus_real.transfer
tff(fact_6483_zero__int_Otransfer,axiom,
    aa(int,$o,aa(product_prod(nat,nat),fun(int,$o),pcr_int,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),zero_zero(nat))),zero_zero(int)) ).

% zero_int.transfer
tff(fact_6484_plus__real_Otransfer,axiom,
    aa(fun(real,fun(real,real)),$o,aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(fun(real,fun(real,real)),$o),bNF_rel_fun(fun(nat,rat),real,fun(fun(nat,rat),fun(nat,rat)),fun(real,real),pcr_real,bNF_rel_fun(fun(nat,rat),real,fun(nat,rat),real,pcr_real,pcr_real)),aTP_Lamp_mv(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),plus_plus(real)) ).

% plus_real.transfer
tff(fact_6485_times__real_Otransfer,axiom,
    aa(fun(real,fun(real,real)),$o,aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(fun(real,fun(real,real)),$o),bNF_rel_fun(fun(nat,rat),real,fun(fun(nat,rat),fun(nat,rat)),fun(real,real),pcr_real,bNF_rel_fun(fun(nat,rat),real,fun(nat,rat),real,pcr_real,pcr_real)),aTP_Lamp_ms(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),times_times(real)) ).

% times_real.transfer
tff(fact_6486_int__transfer,axiom,
    aa(fun(nat,int),$o,aa(fun(nat,product_prod(nat,nat)),fun(fun(nat,int),$o),bNF_rel_fun(nat,nat,product_prod(nat,nat),int,fequal(nat),pcr_int),aTP_Lamp_aak(nat,product_prod(nat,nat))),semiring_1_of_nat(int)) ).

% int_transfer
tff(fact_6487_uminus__int_Otransfer,axiom,
    aa(fun(int,int),$o,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(int,int),$o),bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_la(nat,fun(nat,product_prod(nat,nat))))),uminus_uminus(int)) ).

% uminus_int.transfer
tff(fact_6488_nat_Otransfer,axiom,
    aa(fun(int,nat),$o,aa(fun(product_prod(nat,nat),nat),fun(fun(int,nat),$o),bNF_rel_fun(product_prod(nat,nat),int,nat,nat,pcr_int,fequal(nat)),product_case_prod(nat,nat,nat,minus_minus(nat))),nat2) ).

% nat.transfer
tff(fact_6489_one__int_Otransfer,axiom,
    aa(int,$o,aa(product_prod(nat,nat),fun(int,$o),pcr_int,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,one_one(nat)),zero_zero(nat))),one_one(int)) ).

% one_int.transfer
tff(fact_6490_of__int_Otransfer,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => aa(fun(int,A),$o,aa(fun(product_prod(nat,nat),A),fun(fun(int,A),$o),bNF_rel_fun(product_prod(nat,nat),int,A,A,pcr_int,fequal(A)),product_case_prod(nat,nat,A,aTP_Lamp_lb(nat,fun(nat,A)))),ring_1_of_int(A)) ) ).

% of_int.transfer
tff(fact_6491_inverse__real_Otransfer,axiom,
    aa(fun(real,real),$o,aa(fun(fun(nat,rat),fun(nat,rat)),fun(fun(real,real),$o),bNF_rel_fun(fun(nat,rat),real,fun(nat,rat),real,pcr_real,pcr_real),aTP_Lamp_aah(fun(nat,rat),fun(nat,rat))),inverse_inverse(real)) ).

% inverse_real.transfer
tff(fact_6492_less__int_Otransfer,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),$o),fun(int,$o),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,$o,$o,pcr_int,fequal($o))),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_ld(nat,fun(nat,fun(product_prod(nat,nat),$o))))),ord_less(int)) ).

% less_int.transfer
tff(fact_6493_less__eq__int_Otransfer,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),$o),fun(int,$o),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,$o,$o,pcr_int,fequal($o))),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_lf(nat,fun(nat,fun(product_prod(nat,nat),$o))))),ord_less_eq(int)) ).

% less_eq_int.transfer
tff(fact_6494_plus__int_Otransfer,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lh(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),plus_plus(int)) ).

% plus_int.transfer
tff(fact_6495_minus__int_Otransfer,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lj(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),minus_minus(int)) ).

% minus_int.transfer
tff(fact_6496_num_Orec__transfer,axiom,
    ! [A: $tType,B: $tType,S3: fun(A,fun(B,$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))),S3,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,S3,S3)),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,S3,S3)),bNF_rel_fun(num,num,A,B,fequal(num),S3)))),rec_num(A)),rec_num(B)) ).

% num.rec_transfer
tff(fact_6497_num__of__integer__def,axiom,
    code_num_of_integer = aa(fun(int,num),fun(code_integer,num),map_fun(code_integer,int,num,num,code_int_of_integer,id(num)),aa(fun(int,nat),fun(int,num),comp(nat,num,int,num_of_nat),nat2)) ).

% num_of_integer_def
tff(fact_6498_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_6499_verit__eq__simplify_I20_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,fun(A,A)),F32: fun(num,fun(A,A)),X2: 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(X2)) = aa(A,A,aa(num,fun(A,A),F22,X2),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),X2)) ) ).

% verit_eq_simplify(20)
tff(fact_6500_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_6501_num_Ocase__transfer,axiom,
    ! [A: $tType,B: $tType,S3: fun(A,fun(B,$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))),S3,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),S3),bNF_rel_fun(fun(num,A),fun(num,B),fun(num,A),fun(num,B),bNF_rel_fun(num,num,A,B,fequal(num),S3),bNF_rel_fun(num,num,A,B,fequal(num),S3)))),case_num(A)),case_num(B)) ).

% num.case_transfer
tff(fact_6502_times__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_kz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_kz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int.rsp
tff(fact_6503_intrel__iff,axiom,
    ! [Xb: nat,Y: nat,U: nat,V2: nat] :
      ( aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xb),Y)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,U),V2))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xb),V2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),U),Y) ) ) ).

% intrel_iff
tff(fact_6504_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_6505_zero__int_Orsp,axiom,
    aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),zero_zero(nat))) ).

% zero_int.rsp
tff(fact_6506_verit__eq__simplify_I17_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X2: 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(X2)) = aa(num,A,F22,X2) ) ).

% verit_eq_simplify(17)
tff(fact_6507_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_6508_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_aal(fun(B,A),fun(fun(num,B),fun(num,A)),H),F22)),aa(fun(num,B),fun(num,A),aTP_Lamp_aal(fun(B,A),fun(fun(num,B),fun(num,A)),H),F32)),Num) ) ).

% num.case_distrib
tff(fact_6509_int_Oabs__eq__iff,axiom,
    ! [Xb: product_prod(nat,nat),Y: product_prod(nat,nat)] :
      ( ( aa(product_prod(nat,nat),int,abs_Integ,Xb) = aa(product_prod(nat,nat),int,abs_Integ,Y) )
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,Xb),Y) ) ).

% int.abs_eq_iff
tff(fact_6510_uminus__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),product_prod(nat,nat)),$o,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(nat,nat),product_prod(nat,nat)),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_la(nat,fun(nat,product_prod(nat,nat))))),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_la(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int.rsp
tff(fact_6511_nat_Orsp,axiom,
    aa(fun(product_prod(nat,nat),nat),$o,aa(fun(product_prod(nat,nat),nat),fun(fun(product_prod(nat,nat),nat),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),nat,nat,intrel,fequal(nat)),product_case_prod(nat,nat,nat,minus_minus(nat))),product_case_prod(nat,nat,nat,minus_minus(nat))) ).

% nat.rsp
tff(fact_6512_one__int_Orsp,axiom,
    aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,one_one(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,one_one(nat)),zero_zero(nat))) ).

% one_int.rsp
tff(fact_6513_of__int_Orsp,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => aa(fun(product_prod(nat,nat),A),$o,aa(fun(product_prod(nat,nat),A),fun(fun(product_prod(nat,nat),A),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),A,A,intrel,fequal(A)),product_case_prod(nat,nat,A,aTP_Lamp_lb(nat,fun(nat,A)))),product_case_prod(nat,nat,A,aTP_Lamp_lb(nat,fun(nat,A)))) ) ).

% of_int.rsp
tff(fact_6514_intrel__def,axiom,
    intrel = product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_aan(nat,fun(nat,fun(product_prod(nat,nat),$o)))) ).

% intrel_def
tff(fact_6515_less__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(product_prod(nat,nat),$o),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),$o,$o,intrel,fequal($o))),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_ld(nat,fun(nat,fun(product_prod(nat,nat),$o))))),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_ld(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_int.rsp
tff(fact_6516_less__eq__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(product_prod(nat,nat),$o),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),$o,$o,intrel,fequal($o))),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_lf(nat,fun(nat,fun(product_prod(nat,nat),$o))))),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_lf(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_eq_int.rsp
tff(fact_6517_int_Orel__eq__transfer,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),$o),fun(int,$o),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,$o,$o,pcr_int,fequal($o))),intrel),fequal(int)) ).

% int.rel_eq_transfer
tff(fact_6518_minus__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lj(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lj(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int.rsp
tff(fact_6519_plus__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lh(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lh(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int.rsp
tff(fact_6520_finite__enumerate__initial__segment,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Nb: nat,S: A] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),set_ord_lessThan(A,S))))
           => ( infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),set_ord_lessThan(A,S)),Nb) = infini527867602293511546merate(A,S3,Nb) ) ) ) ) ).

% finite_enumerate_initial_segment
tff(fact_6521_finite__enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Ma: nat,Nb: nat] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(set(A),nat,finite_card(A),S3))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(set(A),nat,finite_card(A),S3))
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S3,Ma)),infini527867602293511546merate(A,S3,Nb))
              <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_6522_enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Ma: nat,Nb: nat] :
          ( ~ aa(set(A),$o,finite_finite(A),S3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S3,Ma)),infini527867602293511546merate(A,S3,Nb))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ) ) ).

% enumerate_mono_iff
tff(fact_6523_enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Nb: nat] :
          ( ~ aa(set(A),$o,finite_finite(A),S3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S3,Nb)),infini527867602293511546merate(A,S3,aa(nat,nat,suc,Nb))) ) ) ).

% enumerate_step
tff(fact_6524_enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Ma: nat,Nb: nat,S3: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
         => ( ~ aa(set(A),$o,finite_finite(A),S3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S3,Ma)),infini527867602293511546merate(A,S3,Nb)) ) ) ) ).

% enumerate_mono
tff(fact_6525_finite__enumerate__in__set,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Nb: nat] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(set(A),nat,finite_card(A),S3))
           => member(A,infini527867602293511546merate(A,S3,Nb),S3) ) ) ) ).

% finite_enumerate_in_set
tff(fact_6526_finite__enumerate__Ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),S: A] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( member(A,S,S3)
           => ? [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(set(A),nat,finite_card(A),S3))
                & ( infini527867602293511546merate(A,S3,N) = S ) ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_6527_finite__enum__ext,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X6: set(A),Y6: set(A)] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(set(A),nat,finite_card(A),X6))
             => ( infini527867602293511546merate(A,X6,I2) = infini527867602293511546merate(A,Y6,I2) ) )
         => ( aa(set(A),$o,finite_finite(A),X6)
           => ( aa(set(A),$o,finite_finite(A),Y6)
             => ( ( aa(set(A),nat,finite_card(A),X6) = aa(set(A),nat,finite_card(A),Y6) )
               => ( X6 = Y6 ) ) ) ) ) ) ).

% finite_enum_ext
tff(fact_6528_finite__enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Ma: nat,Nb: nat,S3: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
         => ( aa(set(A),$o,finite_finite(A),S3)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(set(A),nat,finite_card(A),S3))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S3,Ma)),infini527867602293511546merate(A,S3,Nb)) ) ) ) ) ).

% finite_enumerate_mono
tff(fact_6529_finite__le__enumerate,axiom,
    ! [S3: set(nat),Nb: nat] :
      ( aa(set(nat),$o,finite_finite(nat),S3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(set(nat),nat,finite_card(nat),S3))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),infini527867602293511546merate(nat,S3,Nb)) ) ) ).

% finite_le_enumerate
tff(fact_6530_finite__enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Nb: nat] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),aa(set(A),nat,finite_card(A),S3))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,S3,Nb)),infini527867602293511546merate(A,S3,aa(nat,nat,suc,Nb))) ) ) ) ).

% finite_enumerate_step
tff(fact_6531_enumerate__Suc_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Nb: nat] : ( infini527867602293511546merate(A,S3,aa(nat,nat,suc,Nb)) = infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),infini527867602293511546merate(A,S3,zero_zero(nat))),bot_bot(set(A)))),Nb) ) ) ).

% enumerate_Suc'
tff(fact_6532_finite__enum__subset,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X6: set(A),Y6: set(A)] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(set(A),nat,finite_card(A),X6))
             => ( infini527867602293511546merate(A,X6,I2) = infini527867602293511546merate(A,Y6,I2) ) )
         => ( aa(set(A),$o,finite_finite(A),X6)
           => ( aa(set(A),$o,finite_finite(A),Y6)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),X6)),aa(set(A),nat,finite_card(A),Y6))
               => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),Y6) ) ) ) ) ) ).

% finite_enum_subset
tff(fact_6533_finite__enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Nb: nat] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),aa(set(A),nat,finite_card(A),S3))
           => ( infini527867602293511546merate(A,S3,aa(nat,nat,suc,Nb)) = ord_Least(A,aa(nat,fun(A,$o),aTP_Lamp_aao(set(A),fun(nat,fun(A,$o)),S3),Nb)) ) ) ) ) ).

% finite_enumerate_Suc''
tff(fact_6534_enumerate__Suc,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Nb: nat] : ( infini527867602293511546merate(A,S3,aa(nat,nat,suc,Nb)) = infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),ord_Least(A,aTP_Lamp_aap(set(A),fun(A,$o),S3))),bot_bot(set(A)))),Nb) ) ) ).

% enumerate_Suc
tff(fact_6535_Least__eq__0,axiom,
    ! [P: fun(nat,$o)] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ord_Least(nat,P) = zero_zero(nat) ) ) ).

% Least_eq_0
tff(fact_6536_Least__Suc2,axiom,
    ! [P: fun(nat,$o),Nb: nat,Q: fun(nat,$o),Ma: nat] :
      ( aa(nat,$o,P,Nb)
     => ( aa(nat,$o,Q,Ma)
       => ( ~ aa(nat,$o,P,zero_zero(nat))
         => ( ! [K2: nat] :
                ( aa(nat,$o,P,aa(nat,nat,suc,K2))
              <=> aa(nat,$o,Q,K2) )
           => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_6537_not__less__Least,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [K: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),ord_Least(A,P))
         => ~ aa(A,$o,P,K) ) ) ).

% not_less_Least
tff(fact_6538_Least__Suc,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_wn(fun(nat,$o),fun(nat,$o),P))) ) ) ) ).

% Least_Suc
tff(fact_6539_enumerate__0,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A)] : ( infini527867602293511546merate(A,S3,zero_zero(nat)) = ord_Least(A,aTP_Lamp_aap(set(A),fun(A,$o),S3)) ) ) ).

% enumerate_0
tff(fact_6540_Sup__real__def,axiom,
    ! [X6: set(real)] : ( aa(set(real),real,complete_Sup_Sup(real),X6) = ord_Least(real,aTP_Lamp_aaq(set(real),fun(real,$o),X6)) ) ).

% Sup_real_def
tff(fact_6541_enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),Nb: nat] :
          ( ~ aa(set(A),$o,finite_finite(A),S3)
         => ( infini527867602293511546merate(A,S3,aa(nat,nat,suc,Nb)) = ord_Least(A,aa(nat,fun(A,$o),aTP_Lamp_aao(set(A),fun(nat,fun(A,$o)),S3),Nb)) ) ) ) ).

% enumerate_Suc''
tff(fact_6542_rat__sgn__code,axiom,
    ! [P3: rat] : ( quotient_of(aa(rat,rat,sgn_sgn(rat),P3)) = aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,sgn_sgn(int),aa(product_prod(int,int),int,product_fst(int,int),quotient_of(P3)))),one_one(int)) ) ).

% rat_sgn_code
tff(fact_6543_independentD,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A),Ta: set(A),U: fun(A,real),V2: A] :
          ( ~ real_V358717886546972837endent(A,S)
         => ( aa(set(A),$o,finite_finite(A),Ta)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),S)
             => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aar(fun(A,real),fun(A,A),U)),Ta) = zero_zero(A) )
               => ( member(A,V2,Ta)
                 => ( aa(A,real,U,V2) = zero_zero(real) ) ) ) ) ) ) ) ).

% independentD
tff(fact_6544_dependent__single,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] :
          ( real_V358717886546972837endent(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))
        <=> ( Xb = zero_zero(A) ) ) ) ).

% dependent_single
tff(fact_6545_one__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),Nb)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,one2,Nb)) ) ) ).

% one_div_numeral
tff(fact_6546_dependent__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A)] :
          ( member(A,zero_zero(A),A3)
         => real_V358717886546972837endent(A,A3) ) ) ).

% dependent_zero
tff(fact_6547_dependent__finite,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( real_V358717886546972837endent(A,S3)
          <=> ? [U3: fun(A,real)] :
                ( ? [X4: A] :
                    ( member(A,X4,S3)
                    & ( aa(A,real,U3,X4) != zero_zero(real) ) )
                & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aar(fun(A,real),fun(A,A),U3)),S3) = zero_zero(A) ) ) ) ) ) ).

% dependent_finite
tff(fact_6548_independent__if__scalars__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ! [F3: fun(A,real),X: A] :
                ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aar(fun(A,real),fun(A,A),F3)),A3) = zero_zero(A) )
               => ( member(A,X,A3)
                 => ( aa(A,real,F3,X) = zero_zero(real) ) ) )
           => ~ real_V358717886546972837endent(A,A3) ) ) ) ).

% independent_if_scalars_zero
tff(fact_6549_independentD__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A),X6: fun(A,real),Xb: A] :
          ( ~ real_V358717886546972837endent(A,B3)
         => ( aa(set(A),$o,finite_finite(A),collect(A,aTP_Lamp_aas(fun(A,real),fun(A,$o),X6)))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aTP_Lamp_aas(fun(A,real),fun(A,$o),X6))),B3)
             => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aar(fun(A,real),fun(A,A),X6)),collect(A,aTP_Lamp_aas(fun(A,real),fun(A,$o),X6))) = zero_zero(A) )
               => ( aa(A,real,X6,Xb) = zero_zero(real) ) ) ) ) ) ) ).

% independentD_alt
tff(fact_6550_independent__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A)] :
          ( ~ real_V358717886546972837endent(A,B3)
        <=> ! [X7: fun(A,real)] :
              ( aa(set(A),$o,finite_finite(A),collect(A,aTP_Lamp_aas(fun(A,real),fun(A,$o),X7)))
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aTP_Lamp_aas(fun(A,real),fun(A,$o),X7))),B3)
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aar(fun(A,real),fun(A,A),X7)),collect(A,aTP_Lamp_aas(fun(A,real),fun(A,$o),X7))) = zero_zero(A) )
                 => ! [X4: A] : ( aa(A,real,X7,X4) = zero_zero(real) ) ) ) ) ) ) ).

% independent_alt
tff(fact_6551_dependent__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A)] :
          ( real_V358717886546972837endent(A,B3)
        <=> ? [X7: fun(A,real)] :
              ( aa(set(A),$o,finite_finite(A),collect(A,aTP_Lamp_aas(fun(A,real),fun(A,$o),X7)))
              & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aTP_Lamp_aas(fun(A,real),fun(A,$o),X7))),B3)
              & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aar(fun(A,real),fun(A,A),X7)),collect(A,aTP_Lamp_aas(fun(A,real),fun(A,$o),X7))) = zero_zero(A) )
              & ? [X4: A] : ( aa(A,real,X7,X4) != zero_zero(real) ) ) ) ) ).

% dependent_alt
tff(fact_6552_independent__explicit__finite__subsets,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A)] :
          ( ~ real_V358717886546972837endent(A,A3)
        <=> ! [S8: set(A)] :
              ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S8),A3)
             => ( aa(set(A),$o,finite_finite(A),S8)
               => ! [U3: fun(A,real)] :
                    ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aar(fun(A,real),fun(A,A),U3)),S8) = zero_zero(A) )
                   => ! [X4: A] :
                        ( member(A,X4,S8)
                       => ( aa(A,real,U3,X4) = zero_zero(real) ) ) ) ) ) ) ) ).

% independent_explicit_finite_subsets
tff(fact_6553_independent__explicit__module,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] :
          ( ~ real_V358717886546972837endent(A,S)
        <=> ! [T6: set(A),U3: fun(A,real),V5: A] :
              ( aa(set(A),$o,finite_finite(A),T6)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T6),S)
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aar(fun(A,real),fun(A,A),U3)),T6) = zero_zero(A) )
                 => ( member(A,V5,T6)
                   => ( aa(A,real,U3,V5) = zero_zero(real) ) ) ) ) ) ) ) ).

% independent_explicit_module
tff(fact_6554_dependent__explicit,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] :
          ( real_V358717886546972837endent(A,S)
        <=> ? [T6: set(A)] :
              ( aa(set(A),$o,finite_finite(A),T6)
              & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T6),S)
              & ? [U3: fun(A,real)] :
                  ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aar(fun(A,real),fun(A,A),U3)),T6) = zero_zero(A) )
                  & ? [X4: A] :
                      ( member(A,X4,T6)
                      & ( aa(A,real,U3,X4) != zero_zero(real) ) ) ) ) ) ) ).

% dependent_explicit
tff(fact_6555_eq__key__imp__eq__value,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),K: A,V1: B,V22: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,K),V1),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs))
       => ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,K),V22),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs))
         => ( V1 = V22 ) ) ) ) ).

% eq_key_imp_eq_value
tff(fact_6556_bezw_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod(int,int)] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Xb,Xaa)))))) ) ) ).

% bezw.elims
tff(fact_6557_map__fst__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : ( aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,Nb,Xs)) = upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% map_fst_enumerate
tff(fact_6558_Suc__0__div__numeral,axiom,
    ! [K: num] : ( divide_divide(nat,aa(nat,nat,suc,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_fst(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ) ).

% Suc_0_div_numeral
tff(fact_6559_one__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : ( modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),Nb)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,one2,Nb)) ) ) ).

% one_mod_numeral
tff(fact_6560_divides__aux__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Qr: product_prod(A,A)] :
          ( unique5940410009612947441es_aux(A,Qr)
        <=> ( aa(product_prod(A,A),A,product_snd(A,A),Qr) = zero_zero(A) ) ) ) ).

% divides_aux_def
tff(fact_6561_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),quotient_of(R2))) ).

% quotient_of_denom_pos'
tff(fact_6562_Eps__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : ( fChoice(product_prod(A,B),product_case_prod(A,B,$o,P)) = fChoice(product_prod(A,B),aTP_Lamp_aat(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P)) ) ).

% Eps_case_prod
tff(fact_6563_in__set__zip,axiom,
    ! [A: $tType,B: $tType,P3: product_prod(A,B),Xs: list(A),Ys2: list(B)] :
      ( member(product_prod(A,B),P3,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)))
    <=> ? [N4: nat] :
          ( ( aa(nat,A,nth(A,Xs),N4) = aa(product_prod(A,B),A,product_fst(A,B),P3) )
          & ( aa(nat,B,nth(B,Ys2),N4) = aa(product_prod(A,B),B,product_snd(A,B),P3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(B),nat,size_size(list(B)),Ys2)) ) ) ).

% in_set_zip
tff(fact_6564_bezw__non__0,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Y)
     => ( bezw(Xb,Y) = aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Xb,Y))))) ) ) ).

% bezw_non_0
tff(fact_6565_bezw_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      ( bezw(Xb,Y) = $ite(Y = zero_zero(nat),aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Xb,Y)))))) ) ).

% bezw.simps
tff(fact_6566_one__mod__minus__numeral,axiom,
    ! [Nb: num] : ( modulo_modulo(int,one_one(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),Nb),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,Nb)))) ) ).

% one_mod_minus_numeral
tff(fact_6567_minus__one__mod__numeral,axiom,
    ! [Nb: num] : ( modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),aa(num,int,numeral_numeral(int),Nb)) = adjust_mod(aa(num,int,numeral_numeral(int),Nb),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,Nb))) ) ).

% minus_one_mod_numeral
tff(fact_6568_Suc__0__mod__numeral,axiom,
    ! [K: num] : ( modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_snd(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ) ).

% Suc_0_mod_numeral
tff(fact_6569_minus__numeral__mod__numeral,axiom,
    ! [Ma: num,Nb: num] : ( modulo_modulo(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma)),aa(num,int,numeral_numeral(int),Nb)) = adjust_mod(aa(num,int,numeral_numeral(int),Nb),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,Ma,Nb))) ) ).

% minus_numeral_mod_numeral
tff(fact_6570_numeral__mod__minus__numeral,axiom,
    ! [Ma: num,Nb: num] : ( modulo_modulo(int,aa(num,int,numeral_numeral(int),Ma),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),Nb),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,Ma,Nb)))) ) ).

% numeral_mod_minus_numeral
tff(fact_6571_nths__shift__lemma__Suc,axiom,
    ! [A: $tType,P: fun(nat,$o),Xs: list(A),Is: list(nat)] : ( aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_aau(fun(nat,$o),fun(product_prod(A,nat),$o),P),zip(A,nat,Xs,Is))) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_aav(fun(nat,$o),fun(product_prod(A,nat),$o),P),zip(A,nat,Xs,aa(list(nat),list(nat),map(nat,nat,suc),Is)))) ) ).

% nths_shift_lemma_Suc
tff(fact_6572_nths__shift__lemma,axiom,
    ! [A: $tType,A3: set(nat),Xs: list(A),I: nat] : ( aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_aaw(set(nat),fun(product_prod(A,nat),$o),A3),zip(A,nat,Xs,upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))))) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter2(product_prod(A,nat),aa(nat,fun(product_prod(A,nat),$o),aTP_Lamp_aax(set(nat),fun(nat,fun(product_prod(A,nat),$o)),A3),I),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ) ).

% nths_shift_lemma
tff(fact_6573_nths__def,axiom,
    ! [A: $tType,Xs: list(A),A3: set(nat)] : ( nths(A,Xs,A3) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_aaw(set(nat),fun(product_prod(A,nat),$o),A3),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ) ).

% nths_def
tff(fact_6574_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R2: int] :
      ( adjust_mod(L,R2) = $ite(R2 = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),L),R2)) ) ).

% Divides.adjust_mod_def
tff(fact_6575_in__set__enumerate__eq,axiom,
    ! [A: $tType,P3: product_prod(nat,A),Nb: nat,Xs: list(A)] :
      ( member(product_prod(nat,A),P3,aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,Nb,Xs)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(product_prod(nat,A),nat,product_fst(nat,A),P3))
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb))
        & ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P3)),Nb)) = aa(product_prod(nat,A),A,product_snd(nat,A),P3) ) ) ) ).

% in_set_enumerate_eq
tff(fact_6576_size__prod__simp,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,nat),G: fun(B,nat),P3: product_prod(A,B)] : ( basic_BNF_size_prod(A,B,F2,G,P3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,aa(product_prod(A,B),A,product_fst(A,B),P3))),aa(B,nat,G,aa(product_prod(A,B),B,product_snd(A,B),P3)))),aa(nat,nat,suc,zero_zero(nat))) ) ).

% size_prod_simp
tff(fact_6577_bezw_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod(int,int)] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Xb,Xaa)))))) )
           => ~ accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xb),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_6578_prod__decode__aux_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod(nat,nat)] :
      ( ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(Xb),Xaa) = Y )
     => ( accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xb),Xaa))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xaa),Xb),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xaa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xb),Xaa)),aa(nat,product_prod(nat,nat),nat_prod_decode_aux(aa(nat,nat,suc,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xaa),aa(nat,nat,suc,Xb)))) )
           => ~ accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xb),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_6579_Rat_Opositive_Orep__eq,axiom,
    ! [Xb: rat] :
      ( aa(rat,$o,positive,Xb)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),aa(rat,product_prod(int,int),rep_Rat,Xb))),aa(product_prod(int,int),int,product_snd(int,int),aa(rat,product_prod(int,int),rep_Rat,Xb)))) ) ).

% Rat.positive.rep_eq
tff(fact_6580_Rat_Opositive__def,axiom,
    positive = aa(fun(product_prod(int,int),$o),fun(rat,$o),map_fun(rat,product_prod(int,int),$o,$o,rep_Rat,id($o)),aTP_Lamp_aay(product_prod(int,int),$o)) ).

% Rat.positive_def
tff(fact_6581_normalize__def,axiom,
    ! [P3: product_prod(int,int)] :
      ( normalize(P3) = $ite(
          aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P3)),
          $let(
            a2: int,
            a2:= aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3)),
            aa(int,product_prod(int,int),product_Pair(int,int,divide_divide(int,aa(product_prod(int,int),int,product_fst(int,int),P3),a2)),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P3),a2)) ),
          $ite(
            aa(product_prod(int,int),int,product_snd(int,int),P3) = zero_zero(int),
            aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)),
            $let(
              a2: int,
              a2:= aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3))),
              aa(int,product_prod(int,int),product_Pair(int,int,divide_divide(int,aa(product_prod(int,int),int,product_fst(int,int),P3),a2)),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P3),a2)) ) ) ) ) ).

% normalize_def
tff(fact_6582_gcd__right__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ).

% gcd_right_idem
tff(fact_6583_gcd__left__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ).

% gcd_left_idem
tff(fact_6584_gcd__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% gcd_eq_0_iff
tff(fact_6585_gcd_Obottom__left__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),one_one(A)),A2) = one_one(A) ) ) ).

% gcd.bottom_left_bottom
tff(fact_6586_gcd_Obottom__right__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),one_one(A)) = one_one(A) ) ) ).

% gcd.bottom_right_bottom
tff(fact_6587_gcd__add2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Ma: A,Nb: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Ma),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ma),Nb)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Ma),Nb) ) ) ).

% gcd_add2
tff(fact_6588_gcd__add1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Ma: A,Nb: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ma),Nb)),Nb) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Ma),Nb) ) ) ).

% gcd_add1
tff(fact_6589_gcd__neg1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ).

% gcd_neg1
tff(fact_6590_gcd__neg2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ).

% gcd_neg2
tff(fact_6591_gcd__exp,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A2: A,Nb: nat,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),Nb) ) ) ).

% gcd_exp
tff(fact_6592_gcd__dvd1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),A2) ) ).

% gcd_dvd1
tff(fact_6593_gcd__dvd2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),B2) ) ).

% gcd_dvd2
tff(fact_6594_gcd__greatest__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ) ).

% gcd_greatest_iff
tff(fact_6595_gcd__1__int,axiom,
    ! [Ma: int] : ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Ma),one_one(int)) = one_one(int) ) ).

% gcd_1_int
tff(fact_6596_gcd__neg2__int,axiom,
    ! [Xb: int,Y: int] : ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),aa(int,int,uminus_uminus(int),Y)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y) ) ).

% gcd_neg2_int
tff(fact_6597_gcd__neg1__int,axiom,
    ! [Xb: int,Y: int] : ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),Xb)),Y) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y) ) ).

% gcd_neg1_int
tff(fact_6598_abs__gcd__int,axiom,
    ! [Xb: int,Y: int] : ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y) ) ).

% abs_gcd_int
tff(fact_6599_gcd__abs1__int,axiom,
    ! [Xb: int,Y: int] : ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,abs_abs(int),Xb)),Y) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y) ) ).

% gcd_abs1_int
tff(fact_6600_gcd__abs2__int,axiom,
    ! [Xb: int,Y: int] : ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),aa(int,int,abs_abs(int),Y)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y) ) ).

% gcd_abs2_int
tff(fact_6601_gcd__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Nb: num,A2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))),A2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(num,A,numeral_numeral(A),Nb)),A2) ) ) ).

% gcd_neg_numeral_1
tff(fact_6602_gcd__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A,Nb: num] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(num,A,numeral_numeral(A),Nb)) ) ) ).

% gcd_neg_numeral_2
tff(fact_6603_is__unit__gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),one_one(A))
        <=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) = one_one(A) ) ) ) ).

% is_unit_gcd_iff
tff(fact_6604_gcd__pos__int,axiom,
    ! [Ma: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Ma),Nb))
    <=> ( ( Ma != zero_zero(int) )
        | ( Nb != zero_zero(int) ) ) ) ).

% gcd_pos_int
tff(fact_6605_Gcd__insert,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A3: set(A)] : ( gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),gcd_Gcd(A,A3)) ) ) ).

% Gcd_insert
tff(fact_6606_gcd__neg__numeral__1__int,axiom,
    ! [Nb: num,Xb: int] : ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))),Xb) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(num,int,numeral_numeral(int),Nb)),Xb) ) ).

% gcd_neg_numeral_1_int
tff(fact_6607_gcd__neg__numeral__2__int,axiom,
    ! [Xb: int,Nb: num] : ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),aa(num,int,numeral_numeral(int),Nb)) ) ).

% gcd_neg_numeral_2_int
tff(fact_6608_gcd__0__int,axiom,
    ! [Xb: int] : ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),zero_zero(int)) = aa(int,int,abs_abs(int),Xb) ) ).

% gcd_0_int
tff(fact_6609_gcd__0__left__int,axiom,
    ! [Xb: int] : ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),zero_zero(int)),Xb) = aa(int,int,abs_abs(int),Xb) ) ).

% gcd_0_left_int
tff(fact_6610_gcd__proj1__if__dvd__int,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Xb),Y)
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y) = aa(int,int,abs_abs(int),Xb) ) ) ).

% gcd_proj1_if_dvd_int
tff(fact_6611_gcd__proj2__if__dvd__int,axiom,
    ! [Y: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Y),Xb)
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y) = aa(int,int,abs_abs(int),Y) ) ) ).

% gcd_proj2_if_dvd_int
tff(fact_6612_Gcd__2,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,B2: A] : ( gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ).

% Gcd_2
tff(fact_6613_gcd__ge__0__int,axiom,
    ! [Xb: int,Y: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y)) ).

% gcd_ge_0_int
tff(fact_6614_gcd__dvd__prod,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,K: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2)) ) ).

% gcd_dvd_prod
tff(fact_6615_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Ma: A,K: A,Nb: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Ma),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),Ma)),Nb)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Ma),Nb) ) ) ).

% gcd_add_mult
tff(fact_6616_gcd__idem__int,axiom,
    ! [Xb: int] : ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Xb) = aa(int,int,abs_abs(int),Xb) ) ).

% gcd_idem_int
tff(fact_6617_gcd__mono,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),D3)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),C2),D3)) ) ) ) ).

% gcd_mono
tff(fact_6618_dvd__gcdD1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2))
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% dvd_gcdD1
tff(fact_6619_dvd__gcdD2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2))
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ).

% dvd_gcdD2
tff(fact_6620_gcd__dvdI1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),C2) ) ) ).

% gcd_dvdI1
tff(fact_6621_gcd__dvdI2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),C2) ) ) ).

% gcd_dvdI2
tff(fact_6622_gcd__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) ) ) ) ).

% gcd_greatest
tff(fact_6623_gcd__dvd__antisym,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),C2),D3))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),C2),D3)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2))
           => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),C2),D3) ) ) ) ) ).

% gcd_dvd_antisym
tff(fact_6624_gcd_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,A2: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2)) ) ) ).

% gcd.left_commute
tff(fact_6625_gcd_Ocommute,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),A2) ) ) ).

% gcd.commute
tff(fact_6626_gcd_Oassoc,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2)) ) ) ).

% gcd.assoc
tff(fact_6627_gcd__red__int,axiom,
    ! [Xb: int,Y: int] : ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Y),modulo_modulo(int,Xb,Y)) ) ).

% gcd_red_int
tff(fact_6628_gcd__diff2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Nb: A,Ma: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Nb),Ma)),Nb) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Ma),Nb) ) ) ).

% gcd_diff2
tff(fact_6629_gcd__diff1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Ma: A,Nb: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Ma),Nb)),Nb) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Ma),Nb) ) ) ).

% gcd_diff1
tff(fact_6630_bezout__int,axiom,
    ! [Xb: int,Y: int] :
    ? [U2: int,V4: int] : ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),U2),Xb)),aa(int,int,aa(int,fun(int,int),times_times(int),V4),Y)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y) ) ).

% bezout_int
tff(fact_6631_gcd__mult__distrib__int,axiom,
    ! [K: int,Ma: int,Nb: int] : ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),K)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Ma),Nb)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),Ma)),aa(int,int,aa(int,fun(int,int),times_times(int),K),Nb)) ) ).

% gcd_mult_distrib_int
tff(fact_6632_gcd__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))),gcd_gcd(int)),gcd_gcd(int)) ).

% gcd_integer.rsp
tff(fact_6633_gcd__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_mult_unit2
tff(fact_6634_gcd__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_mult_unit1
tff(fact_6635_gcd__div__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),divide_divide(A,C2,A2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_div_unit2
tff(fact_6636_gcd__div__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),divide_divide(A,B2,A2)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_div_unit1
tff(fact_6637_gcd__le2__int,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2)),B2) ) ).

% gcd_le2_int
tff(fact_6638_gcd__le1__int,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2)),A2) ) ).

% gcd_le1_int
tff(fact_6639_gcd__cases__int,axiom,
    ! [Xb: int,Y: int,P: fun(int,$o)] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y)) ) )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),zero_zero(int))
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),aa(int,int,uminus_uminus(int),Y))) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),zero_zero(int))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),Xb)),Y)) ) )
         => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),zero_zero(int))
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),zero_zero(int))
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),Xb)),aa(int,int,uminus_uminus(int),Y))) ) )
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_6640_gcd__unique__int,axiom,
    ! [D3: int,A2: int,B2: int] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),D3)
        & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),A2)
        & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),B2)
        & ! [E3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),E3),A2)
              & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),E3),B2) )
           => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),E3),D3) ) )
    <=> ( D3 = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2) ) ) ).

% gcd_unique_int
tff(fact_6641_gcd__non__0__int,axiom,
    ! [Y: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Y)
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Y),modulo_modulo(int,Xb,Y)) ) ) ).

% gcd_non_0_int
tff(fact_6642_gcd__code__int,axiom,
    ! [K: int,L: int] :
      ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),K),L) = aa(int,int,abs_abs(int),
          $ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),L),modulo_modulo(int,aa(int,int,abs_abs(int),K),aa(int,int,abs_abs(int),L))))) ) ).

% gcd_code_int
tff(fact_6643_Gcd__int__set__eq__fold,axiom,
    ! [Xs: list(int)] : ( gcd_Gcd(int,aa(list(int),set(int),set2(int),Xs)) = aa(int,int,fold(int,int,gcd_gcd(int),Xs),zero_zero(int)) ) ).

% Gcd_int_set_eq_fold
tff(fact_6644_Gcd__set__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Xs: list(A)] : ( gcd_Gcd(A,aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,gcd_gcd(A),Xs),zero_zero(A)) ) ) ).

% Gcd_set_eq_fold
tff(fact_6645_gcd__is__Max__divisors__int,axiom,
    ! [Nb: int,Ma: int] :
      ( ( Nb != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Ma),Nb) = lattic643756798349783984er_Max(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_aaz(int,fun(int,fun(int,$o)),Nb),Ma))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_6646_plus__rat_Otransfer,axiom,
    aa(fun(rat,fun(rat,rat)),$o,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),$o),bNF_rel_fun(product_prod(int,int),rat,fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),pcr_rat,bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat)),aTP_Lamp_aba(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),plus_plus(rat)) ).

% plus_rat.transfer
tff(fact_6647_inverse__rat_Otransfer,axiom,
    aa(fun(rat,rat),$o,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),$o),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_abb(product_prod(int,int),product_prod(int,int))),inverse_inverse(rat)) ).

% inverse_rat.transfer
tff(fact_6648_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = zero_zero(nat) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_6649_gcd__nat_Oleft__neutral,axiom,
    ! [A2: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),A2) = A2 ) ).

% gcd_nat.left_neutral
tff(fact_6650_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_6651_gcd__nat_Oright__neutral,axiom,
    ! [A2: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),zero_zero(nat)) = A2 ) ).

% gcd_nat.right_neutral
tff(fact_6652_gcd__0__nat,axiom,
    ! [Xb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),zero_zero(nat)) = Xb ) ).

% gcd_0_nat
tff(fact_6653_gcd__0__left__nat,axiom,
    ! [Xb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),Xb) = Xb ) ).

% gcd_0_left_nat
tff(fact_6654_gcd__1__nat,axiom,
    ! [Ma: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ma),one_one(nat)) = one_one(nat) ) ).

% gcd_1_nat
tff(fact_6655_gcd__proj2__if__dvd__nat,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Y),Xb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Y) = Y ) ) ).

% gcd_proj2_if_dvd_nat
tff(fact_6656_gcd__proj1__if__dvd__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Xb),Y)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Y) = Xb ) ) ).

% gcd_proj1_if_dvd_nat
tff(fact_6657_gcd__nat_Obounded__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2) ) ) ).

% gcd_nat.bounded_iff
tff(fact_6658_gcd__nat_Oabsorb2,axiom,
    ! [B2: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),A2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = B2 ) ) ).

% gcd_nat.absorb2
tff(fact_6659_gcd__nat_Oabsorb1,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = A2 ) ) ).

% gcd_nat.absorb1
tff(fact_6660_gcd__Suc__0,axiom,
    ! [Ma: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ma),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,suc,zero_zero(nat)) ) ).

% gcd_Suc_0
tff(fact_6661_gcd__pos__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ma),Nb))
    <=> ( ( Ma != zero_zero(nat) )
        | ( Nb != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_6662_gcd__int__int__eq,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(nat,int,semiring_1_of_nat(int),Ma)),aa(nat,int,semiring_1_of_nat(int),Nb)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ma),Nb)) ) ).

% gcd_int_int_eq
tff(fact_6663_gcd__nat__abs__right__eq,axiom,
    ! [Nb: nat,K: int] : ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Nb),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(nat,int,semiring_1_of_nat(int),Nb)),K)) ) ).

% gcd_nat_abs_right_eq
tff(fact_6664_gcd__nat__abs__left__eq,axiom,
    ! [K: int,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),Nb) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),K),aa(nat,int,semiring_1_of_nat(int),Nb))) ) ).

% gcd_nat_abs_left_eq
tff(fact_6665_gcd__le1__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),A2) ) ).

% gcd_le1_nat
tff(fact_6666_gcd__le2__nat,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),B2) ) ).

% gcd_le2_nat
tff(fact_6667_gcd__diff1__nat,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ma)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Nb)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ma),Nb) ) ) ).

% gcd_diff1_nat
tff(fact_6668_gcd__diff2__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ma),Nb) ) ) ).

% gcd_diff2_nat
tff(fact_6669_gcd__code__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),K),L) = aa(code_integer,code_integer,abs_abs(code_integer),
          $ite(L = zero_zero(code_integer),K,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),L),modulo_modulo(code_integer,aa(code_integer,code_integer,abs_abs(code_integer),K),aa(code_integer,code_integer,abs_abs(code_integer),L))))) ) ).

% gcd_code_integer
tff(fact_6670_gcd__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : ( aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),Xb),Xaa)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(code_integer,int,code_int_of_integer,Xb)),aa(code_integer,int,code_int_of_integer,Xaa)) ) ).

% gcd_integer.rep_eq
tff(fact_6671_gcd__red__nat,axiom,
    ! [Xb: nat,Y: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Y) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),modulo_modulo(nat,Xb,Y)) ) ).

% gcd_red_nat
tff(fact_6672_gcd__unique__nat,axiom,
    ! [D3: nat,A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),A2)
        & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),B2)
        & ! [E3: nat] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),E3),A2)
              & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),E3),B2) )
           => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),E3),D3) ) )
    <=> ( D3 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ).

% gcd_unique_nat
tff(fact_6673_gcd__nat_Ostrict__coboundedI2,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),C2)
        & ( B2 != C2 ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),C2)
        & ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) != C2 ) ) ) ).

% gcd_nat.strict_coboundedI2
tff(fact_6674_gcd__nat_Ostrict__coboundedI1,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2)
        & ( A2 != C2 ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),C2)
        & ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) != C2 ) ) ) ).

% gcd_nat.strict_coboundedI1
tff(fact_6675_gcd__nat_Ostrict__order__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
    <=> ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) )
        & ( A2 != B2 ) ) ) ).

% gcd_nat.strict_order_iff
tff(fact_6676_gcd__nat_Ostrict__boundedE,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2))
        & ( A2 != aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2) ) )
     => ~ ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
            & ( A2 != B2 ) )
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2)
              & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_boundedE
tff(fact_6677_gcd__nat_OcoboundedI2,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),C2)
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),C2) ) ).

% gcd_nat.coboundedI2
tff(fact_6678_gcd__nat_OcoboundedI1,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2)
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),C2) ) ).

% gcd_nat.coboundedI1
tff(fact_6679_gcd__nat_Oabsorb__iff2,axiom,
    ! [B2: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),A2)
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = B2 ) ) ).

% gcd_nat.absorb_iff2
tff(fact_6680_gcd__nat_Oabsorb__iff1,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = A2 ) ) ).

% gcd_nat.absorb_iff1
tff(fact_6681_gcd__nat_Ocobounded2,axiom,
    ! [A2: nat,B2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),B2) ).

% gcd_nat.cobounded2
tff(fact_6682_gcd__nat_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),A2) ).

% gcd_nat.cobounded1
tff(fact_6683_gcd__nat_Oorder__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
    <=> ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ).

% gcd_nat.order_iff
tff(fact_6684_gcd__nat_OboundedI,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2)) ) ) ).

% gcd_nat.boundedI
tff(fact_6685_gcd__nat_OboundedE,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2))
     => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
         => ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2) ) ) ).

% gcd_nat.boundedE
tff(fact_6686_gcd__nat_Oabsorb4,axiom,
    ! [B2: nat,A2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),A2)
        & ( B2 != A2 ) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = B2 ) ) ).

% gcd_nat.absorb4
tff(fact_6687_gcd__nat_Oabsorb3,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = A2 ) ) ).

% gcd_nat.absorb3
tff(fact_6688_gcd__nat_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) )
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2) ) ).

% gcd_nat.orderI
tff(fact_6689_gcd__nat_OorderE,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
     => ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ).

% gcd_nat.orderE
tff(fact_6690_gcd__nat_Omono,axiom,
    ! [A2: nat,C2: nat,B2: nat,D3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),D3)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),C2),D3)) ) ) ).

% gcd_nat.mono
tff(fact_6691_gcd__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),aa(int,code_integer,code_integer_of_int,Xaa)),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xaa),Xb)) ) ).

% gcd_integer.abs_eq
tff(fact_6692_gcd__mult__distrib__nat,axiom,
    ! [K: nat,Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ma),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) ) ).

% gcd_mult_distrib_nat
tff(fact_6693_gcd__nat_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero(nat),Xb,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xaa),modulo_modulo(nat,Xb,Xaa))) ) ) ).

% gcd_nat.elims
tff(fact_6694_gcd__nat_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Y) = $ite(Y = zero_zero(nat),Xb,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),modulo_modulo(nat,Xb,Y))) ) ).

% gcd_nat.simps
tff(fact_6695_gcd__non__0__nat,axiom,
    ! [Y: nat,Xb: nat] :
      ( ( Y != zero_zero(nat) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Y) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),modulo_modulo(nat,Xb,Y)) ) ) ).

% gcd_non_0_nat
tff(fact_6696_Gcd__in,axiom,
    ! [A3: set(nat)] :
      ( ! [A4: nat,B4: nat] :
          ( member(nat,A4,A3)
         => ( member(nat,B4,A3)
           => member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A4),B4),A3) ) )
     => ( ( A3 != bot_bot(set(nat)) )
       => member(nat,gcd_Gcd(nat,A3),A3) ) ) ).

% Gcd_in
tff(fact_6697_bezout__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [X: nat,Y3: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)) ) ) ).

% bezout_nat
tff(fact_6698_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A2: nat] :
    ? [X: nat,Y3: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) )
      | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_6699_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list(nat)] : ( gcd_Gcd(nat,aa(list(nat),set(nat),set2(nat),Xs)) = aa(nat,nat,fold(nat,nat,gcd_gcd(nat),Xs),zero_zero(nat)) ) ).

% Gcd_nat_set_eq_fold
tff(fact_6700_gcd__int__def,axiom,
    ! [Xb: int,Y: int] : ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),Xb))),aa(int,nat,nat2,aa(int,int,abs_abs(int),Y)))) ) ).

% gcd_int_def
tff(fact_6701_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,gcd_gcd(nat),zero_zero(nat),dvd_dvd(nat),aTP_Lamp_abc(nat,fun(nat,$o))) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_6702_one__rat_Otransfer,axiom,
    aa(rat,$o,aa(product_prod(int,int),fun(rat,$o),pcr_rat,aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int))),one_one(rat)) ).

% one_rat.transfer
tff(fact_6703_gcd__is__Max__divisors__nat,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ma),Nb) = lattic643756798349783984er_Max(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_abd(nat,fun(nat,fun(nat,$o)),Nb),Ma))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_6704_zero__rat_Otransfer,axiom,
    aa(rat,$o,aa(product_prod(int,int),fun(rat,$o),pcr_rat,aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int))),zero_zero(rat)) ).

% zero_rat.transfer
tff(fact_6705_Fract_Otransfer,axiom,
    aa(fun(int,fun(int,rat)),$o,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,rat)),$o),bNF_rel_fun(int,int,fun(int,product_prod(int,int)),fun(int,rat),fequal(int),bNF_rel_fun(int,int,product_prod(int,int),rat,fequal(int),pcr_rat)),aTP_Lamp_abe(int,fun(int,product_prod(int,int)))),fract) ).

% Fract.transfer
tff(fact_6706_bezw__aux,axiom,
    ! [Xb: nat,Y: nat] : ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xb,Y))),aa(nat,int,semiring_1_of_nat(int),Xb))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xb,Y))),aa(nat,int,semiring_1_of_nat(int),Y))) ) ).

% bezw_aux
tff(fact_6707_uminus__rat_Otransfer,axiom,
    aa(fun(rat,rat),$o,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),$o),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_abf(product_prod(int,int),product_prod(int,int))),uminus_uminus(rat)) ).

% uminus_rat.transfer
tff(fact_6708_Rat_Opositive_Otransfer,axiom,
    aa(fun(rat,$o),$o,aa(fun(product_prod(int,int),$o),fun(fun(rat,$o),$o),bNF_rel_fun(product_prod(int,int),rat,$o,$o,pcr_rat,fequal($o)),aTP_Lamp_aay(product_prod(int,int),$o)),positive) ).

% Rat.positive.transfer
tff(fact_6709_gcd__nat_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Xaa) = Y )
     => ( accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero(nat),Xb,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xaa),modulo_modulo(nat,Xb,Xaa))) )
           => ~ accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xb),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_6710_of__rat__def,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( field_char_0_of_rat(A) = aa(fun(product_prod(int,int),A),fun(rat,A),map_fun(rat,product_prod(int,int),A,A,rep_Rat,id(A)),aTP_Lamp_abg(product_prod(int,int),A)) ) ) ).

% of_rat_def
tff(fact_6711_gcd__idem__nat,axiom,
    ! [Xb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Xb) = Xb ) ).

% gcd_idem_nat
tff(fact_6712_gcd__nat_Oright__idem,axiom,
    ! [A2: nat,B2: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),B2) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ).

% gcd_nat.right_idem
tff(fact_6713_gcd__nat_Oleft__idem,axiom,
    ! [A2: nat,B2: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ).

% gcd_nat.left_idem
tff(fact_6714_gcd__nat_Oidem,axiom,
    ! [A2: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),A2) = A2 ) ).

% gcd_nat.idem
tff(fact_6715_of__rat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: rat] : ( archimedean_ceiling(A,aa(rat,A,field_char_0_of_rat(A),R2)) = archimedean_ceiling(rat,R2) ) ) ).

% of_rat_ceiling
tff(fact_6716_zero__eq__of__rat__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat] :
          ( ( zero_zero(A) = aa(rat,A,field_char_0_of_rat(A),A2) )
        <=> ( zero_zero(rat) = A2 ) ) ) ).

% zero_eq_of_rat_iff
tff(fact_6717_of__rat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat] :
          ( ( aa(rat,A,field_char_0_of_rat(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(rat) ) ) ) ).

% of_rat_eq_0_iff
tff(fact_6718_of__rat__0,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( aa(rat,A,field_char_0_of_rat(A),zero_zero(rat)) = zero_zero(A) ) ) ).

% of_rat_0
tff(fact_6719_of__rat__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( aa(rat,A,field_char_0_of_rat(A),one_one(rat)) = one_one(A) ) ) ).

% of_rat_1
tff(fact_6720_of__rat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat] :
          ( ( aa(rat,A,field_char_0_of_rat(A),A2) = one_one(A) )
        <=> ( A2 = one_one(rat) ) ) ) ).

% of_rat_eq_1_iff
tff(fact_6721_one__eq__of__rat__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat] :
          ( ( one_one(A) = aa(rat,A,field_char_0_of_rat(A),A2) )
        <=> ( one_one(rat) = A2 ) ) ) ).

% one_eq_of_rat_iff
tff(fact_6722_of__rat__of__int__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: int] : ( aa(rat,A,field_char_0_of_rat(A),aa(int,rat,ring_1_of_int(rat),Z)) = aa(int,A,ring_1_of_int(A),Z) ) ) ).

% of_rat_of_int_eq
tff(fact_6723_of__rat__of__nat__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Nb: nat] : ( aa(rat,A,field_char_0_of_rat(A),aa(nat,rat,semiring_1_of_nat(rat),Nb)) = aa(nat,A,semiring_1_of_nat(A),Nb) ) ) ).

% of_rat_of_nat_eq
tff(fact_6724_zero__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(rat,A,field_char_0_of_rat(A),R2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R2) ) ) ).

% zero_less_of_rat_iff
tff(fact_6725_of__rat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R2)),zero_zero(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R2),zero_zero(rat)) ) ) ).

% of_rat_less_0_iff
tff(fact_6726_one__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),one_one(rat)),R2) ) ) ).

% one_less_of_rat_iff
tff(fact_6727_of__rat__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R2)),one_one(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R2),one_one(rat)) ) ) ).

% of_rat_less_1_iff
tff(fact_6728_of__rat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R2)),zero_zero(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R2),zero_zero(rat)) ) ) ).

% of_rat_le_0_iff
tff(fact_6729_zero__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(rat,A,field_char_0_of_rat(A),R2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),zero_zero(rat)),R2) ) ) ).

% zero_le_of_rat_iff
tff(fact_6730_of__rat__neg__one,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% of_rat_neg_one
tff(fact_6731_of__rat__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R2)),one_one(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R2),one_one(rat)) ) ) ).

% of_rat_le_1_iff
tff(fact_6732_one__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),one_one(rat)),R2) ) ) ).

% one_le_of_rat_iff
tff(fact_6733_of__rat__neg__numeral__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [W: num] : ( aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),W))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) ) ) ).

% of_rat_neg_numeral_eq
tff(fact_6734_of__rat__Real,axiom,
    ! [Xb: rat] : ( aa(rat,real,field_char_0_of_rat(real),Xb) = aa(fun(nat,rat),real,real2,aTP_Lamp_mt(rat,fun(nat,rat),Xb)) ) ).

% of_rat_Real
tff(fact_6735_of__rat__inverse,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat] : ( aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,inverse_inverse(rat),A2)) = aa(A,A,inverse_inverse(A),aa(rat,A,field_char_0_of_rat(A),A2)) ) ) ).

% of_rat_inverse
tff(fact_6736_of__rat__diff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat,B2: rat] : ( aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),A2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(rat,A,field_char_0_of_rat(A),A2)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ) ).

% of_rat_diff
tff(fact_6737_of__rat__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat] : ( aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,uminus_uminus(rat),A2)) = aa(A,A,uminus_uminus(A),aa(rat,A,field_char_0_of_rat(A),A2)) ) ) ).

% of_rat_minus
tff(fact_6738_of__rat__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat,S: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R2)),aa(rat,A,field_char_0_of_rat(A),S))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R2),S) ) ) ).

% of_rat_less
tff(fact_6739_of__rat__dense,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
     => ? [Q2: rat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(rat,real,field_char_0_of_rat(real),Q2))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(rat,real,field_char_0_of_rat(real),Q2)),Y) ) ) ).

% of_rat_dense
tff(fact_6740_of__rat__add,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat,B2: rat] : ( aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(rat,A,field_char_0_of_rat(A),A2)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ) ).

% of_rat_add
tff(fact_6741_less__RealD,axiom,
    ! [Y6: fun(nat,rat),Xb: real] :
      ( cauchy(Y6)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(fun(nat,rat),real,real2,Y6))
       => ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(rat,real,field_char_0_of_rat(real),aa(nat,rat,Y6,N))) ) ) ).

% less_RealD
tff(fact_6742_le__RealI,axiom,
    ! [Y6: fun(nat,rat),Xb: real] :
      ( cauchy(Y6)
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(rat,real,field_char_0_of_rat(real),aa(nat,rat,Y6,N)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(fun(nat,rat),real,real2,Y6)) ) ) ).

% le_RealI
tff(fact_6743_Real__leI,axiom,
    ! [X6: fun(nat,rat),Y: real] :
      ( cauchy(X6)
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(rat,real,field_char_0_of_rat(real),aa(nat,rat,X6,N))),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(fun(nat,rat),real,real2,X6)),Y) ) ) ).

% Real_leI
tff(fact_6744_nonzero__of__rat__inverse,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat] :
          ( ( A2 != zero_zero(rat) )
         => ( aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,inverse_inverse(rat),A2)) = aa(A,A,inverse_inverse(A),aa(rat,A,field_char_0_of_rat(A),A2)) ) ) ) ).

% nonzero_of_rat_inverse
tff(fact_6745_of__rat__rat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [B2: int,A2: int] :
          ( ( B2 != zero_zero(int) )
         => ( aa(rat,A,field_char_0_of_rat(A),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)) = divide_divide(A,aa(int,A,ring_1_of_int(A),A2),aa(int,A,ring_1_of_int(A),B2)) ) ) ) ).

% of_rat_rat
tff(fact_6746_of__rat_Orep__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Xb: rat] : ( aa(rat,A,field_char_0_of_rat(A),Xb) = divide_divide(A,aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_fst(int,int),aa(rat,product_prod(int,int),rep_Rat,Xb))),aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_snd(int,int),aa(rat,product_prod(int,int),rep_Rat,Xb)))) ) ) ).

% of_rat.rep_eq
tff(fact_6747_of__rat_Otransfer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => aa(fun(rat,A),$o,aa(fun(product_prod(int,int),A),fun(fun(rat,A),$o),bNF_rel_fun(product_prod(int,int),rat,A,A,pcr_rat,fequal(A)),aTP_Lamp_abg(product_prod(int,int),A)),field_char_0_of_rat(A)) ) ).

% of_rat.transfer
tff(fact_6748_plus__rat__def,axiom,
    plus_plus(rat) = aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(rat,fun(rat,rat)),map_fun(rat,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),rep_Rat,map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat)),aTP_Lamp_aba(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% plus_rat_def
tff(fact_6749_inverse__rat__def,axiom,
    inverse_inverse(rat) = aa(fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat),aTP_Lamp_abb(product_prod(int,int),product_prod(int,int))) ).

% inverse_rat_def
tff(fact_6750_one__rat__def,axiom,
    one_one(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int))) ).

% one_rat_def
tff(fact_6751_Fract_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      ( aa(int,rat,aa(int,fun(int,rat),fract,Xaa),Xb) = aa(product_prod(int,int),rat,abs_Rat,
          $ite(Xb = zero_zero(int),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),product_Pair(int,int,Xaa),Xb))) ) ).

% Fract.abs_eq
tff(fact_6752_zero__rat__def,axiom,
    zero_zero(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int))) ).

% zero_rat_def
tff(fact_6753_uminus__rat__def,axiom,
    uminus_uminus(rat) = aa(fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat),aTP_Lamp_abf(product_prod(int,int),product_prod(int,int))) ).

% uminus_rat_def
tff(fact_6754_map__of__is__SomeI,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(A,B)),Xb: A,Y: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Xb),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys))
       => ( aa(A,option(B),map_of(A,B,Xys),Xb) = aa(B,option(B),some(B),Y) ) ) ) ).

% map_of_is_SomeI
tff(fact_6755_Some__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Y: B,Xb: A] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( ( aa(B,option(B),some(B),Y) = aa(A,option(B),map_of(A,B,Xys),Xb) )
      <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Xb),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)) ) ) ).

% Some_eq_map_of_iff
tff(fact_6756_map__of__zip__is__None,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys2)),Xb) = none(B) )
      <=> ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ) ).

% map_of_zip_is_None
tff(fact_6757_map__of__eq__Some__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Xb: A,Y: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( ( aa(A,option(B),map_of(A,B,Xys),Xb) = aa(B,option(B),some(B),Y) )
      <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Xb),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)) ) ) ).

% map_of_eq_Some_iff
tff(fact_6758_map__of__SomeD,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(B,A)),K: B,Y: A] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),Y) )
     => member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,K),Y),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs)) ) ).

% map_of_SomeD
tff(fact_6759_weak__map__of__SomeI,axiom,
    ! [A: $tType,B: $tType,K: A,Xb: B,L: list(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,K),Xb),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L))
     => ? [X: B] : ( aa(A,option(B),map_of(A,B,L),K) = aa(B,option(B),some(B),X) ) ) ).

% weak_map_of_SomeI
tff(fact_6760_map__of__eqI,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys2: list(product_prod(A,B))] :
      ( ( aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)) = aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys2)) )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
           => ( aa(A,option(B),map_of(A,B,Xs),X) = aa(A,option(B),map_of(A,B,Ys2),X) ) )
       => ( map_of(A,B,Xs) = map_of(A,B,Ys2) ) ) ) ).

% map_of_eqI
tff(fact_6761_map__of__eq__dom,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys2: list(product_prod(A,B))] :
      ( ( map_of(A,B,Xs) = map_of(A,B,Ys2) )
     => ( image(product_prod(A,B),A,product_fst(A,B),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)) = image(product_prod(A,B),A,product_fst(A,B),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Ys2)) ) ) ).

% map_of_eq_dom
tff(fact_6762_map__of__zip__is__Some,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
      <=> ? [Y5: B] : ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys2)),Xb) = aa(B,option(B),some(B),Y5) ) ) ) ).

% map_of_zip_is_Some
tff(fact_6763_map__of__eq__None__iff,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(B,A)),Xb: B] :
      ( ( aa(B,option(A),map_of(B,A,Xys),Xb) = none(A) )
    <=> ~ member(B,Xb,image(product_prod(B,A),B,product_fst(B,A),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xys))) ) ).

% map_of_eq_None_iff
tff(fact_6764_map__of__inject__set,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys2: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys2))
       => ( ( map_of(A,B,Xs) = map_of(A,B,Ys2) )
        <=> ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Ys2) ) ) ) ) ).

% map_of_inject_set
tff(fact_6765_map__of__zip__map,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),F2: fun(A,B),X3: A] :
      ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,aa(list(A),list(B),map(A,B,F2),Xs))),X3) = $ite(member(A,X3,aa(list(A),set(A),set2(A),Xs)),aa(B,option(B),some(B),aa(A,B,F2,X3)),none(B)) ) ).

% map_of_zip_map
tff(fact_6766_map__of__zip__nth,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),I: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( distinct(A,Xs)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys2))
         => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys2)),aa(nat,A,nth(A,Xs),I)) = aa(B,option(B),some(B),aa(nat,B,nth(B,Ys2),I)) ) ) ) ) ).

% map_of_zip_nth
tff(fact_6767_set__map__of__compr,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs) = collect(product_prod(A,B),product_case_prod(A,B,$o,aTP_Lamp_abh(list(product_prod(A,B)),fun(A,fun(B,$o)),Xs))) ) ) ).

% set_map_of_compr
tff(fact_6768_ran__distinct,axiom,
    ! [B: $tType,A: $tType,Al: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Al))
     => ( ran(A,B,map_of(A,B,Al)) = image(product_prod(A,B),B,product_snd(A,B),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Al)) ) ) ).

% ran_distinct
tff(fact_6769_plus__rat_Oabs__eq,axiom,
    ! [Xaa: product_prod(int,int),Xb: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Xaa),Xaa)
     => ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Xb),Xb)
       => ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(product_prod(int,int),rat,abs_Rat,Xaa)),aa(product_prod(int,int),rat,abs_Rat,Xb)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xaa)),aa(product_prod(int,int),int,product_snd(int,int),Xb))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xb)),aa(product_prod(int,int),int,product_snd(int,int),Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xaa)),aa(product_prod(int,int),int,product_snd(int,int),Xb)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_6770_ratrel__iff,axiom,
    ! [Xb: product_prod(int,int),Y: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Xb),Y)
    <=> ( ( aa(product_prod(int,int),int,product_snd(int,int),Xb) != zero_zero(int) )
        & ( aa(product_prod(int,int),int,product_snd(int,int),Y) != zero_zero(int) )
        & ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xb)),aa(product_prod(int,int),int,product_snd(int,int),Y)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Y)),aa(product_prod(int,int),int,product_snd(int,int),Xb)) ) ) ) ).

% ratrel_iff
tff(fact_6771_one__rat_Orsp,axiom,
    aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int))),aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int))) ).

% one_rat.rsp
tff(fact_6772_zero__rat_Orsp,axiom,
    aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int))),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int))) ).

% zero_rat.rsp
tff(fact_6773_Fract_Orsp,axiom,
    aa(fun(int,fun(int,product_prod(int,int))),$o,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,product_prod(int,int))),$o),bNF_rel_fun(int,int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int)),fequal(int),bNF_rel_fun(int,int,product_prod(int,int),product_prod(int,int),fequal(int),ratrel)),aTP_Lamp_abe(int,fun(int,product_prod(int,int)))),aTP_Lamp_abe(int,fun(int,product_prod(int,int)))) ).

% Fract.rsp
tff(fact_6774_of__rat_Orsp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => aa(fun(product_prod(int,int),A),$o,aa(fun(product_prod(int,int),A),fun(fun(product_prod(int,int),A),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),A,A,ratrel,fequal(A)),aTP_Lamp_abg(product_prod(int,int),A)),aTP_Lamp_abg(product_prod(int,int),A)) ) ).

% of_rat.rsp
tff(fact_6775_ratrel__def,axiom,
    ! [X3: product_prod(int,int),Xa: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X3),Xa)
    <=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X3) != zero_zero(int) )
        & ( aa(product_prod(int,int),int,product_snd(int,int),Xa) != zero_zero(int) )
        & ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X3)),aa(product_prod(int,int),int,product_snd(int,int),Xa)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X3)) ) ) ) ).

% ratrel_def
tff(fact_6776_uminus__rat_Orsp,axiom,
    aa(fun(product_prod(int,int),product_prod(int,int)),$o,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_abf(product_prod(int,int),product_prod(int,int))),aTP_Lamp_abf(product_prod(int,int),product_prod(int,int))) ).

% uminus_rat.rsp
tff(fact_6777_Rat_Opositive_Orsp,axiom,
    aa(fun(product_prod(int,int),$o),$o,aa(fun(product_prod(int,int),$o),fun(fun(product_prod(int,int),$o),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),$o,$o,ratrel,fequal($o)),aTP_Lamp_aay(product_prod(int,int),$o)),aTP_Lamp_aay(product_prod(int,int),$o)) ).

% Rat.positive.rsp
tff(fact_6778_inverse__rat_Orsp,axiom,
    aa(fun(product_prod(int,int),product_prod(int,int)),$o,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_abb(product_prod(int,int),product_prod(int,int))),aTP_Lamp_abb(product_prod(int,int),product_prod(int,int))) ).

% inverse_rat.rsp
tff(fact_6779_plus__rat_Orsp,axiom,
    aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),$o,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(product_prod(int,int),product_prod(int,int)),ratrel,bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel)),aTP_Lamp_aba(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_aba(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% plus_rat.rsp
tff(fact_6780_of__rat_Oabs__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Xb: product_prod(int,int)] :
          ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Xb),Xb)
         => ( aa(rat,A,field_char_0_of_rat(A),aa(product_prod(int,int),rat,abs_Rat,Xb)) = divide_divide(A,aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_fst(int,int),Xb)),aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_snd(int,int),Xb))) ) ) ) ).

% of_rat.abs_eq
tff(fact_6781_uminus__rat_Oabs__eq,axiom,
    ! [Xb: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Xb),Xb)
     => ( aa(rat,rat,uminus_uminus(rat),aa(product_prod(int,int),rat,abs_Rat,Xb)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,uminus_uminus(int),aa(product_prod(int,int),int,product_fst(int,int),Xb))),aa(product_prod(int,int),int,product_snd(int,int),Xb))) ) ) ).

% uminus_rat.abs_eq
tff(fact_6782_ran__map__of__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( distinct(A,Xs)
       => ( ran(A,B,map_of(A,B,zip(A,B,Xs,Ys2))) = aa(list(B),set(B),set2(B),Ys2) ) ) ) ).

% ran_map_of_zip
tff(fact_6783_Rat_Opositive_Oabs__eq,axiom,
    ! [Xb: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Xb),Xb)
     => ( aa(rat,$o,positive,aa(product_prod(int,int),rat,abs_Rat,Xb))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xb)),aa(product_prod(int,int),int,product_snd(int,int),Xb))) ) ) ).

% Rat.positive.abs_eq
tff(fact_6784_inverse__rat_Oabs__eq,axiom,
    ! [Xb: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Xb),Xb)
     => ( aa(rat,rat,inverse_inverse(rat),aa(product_prod(int,int),rat,abs_Rat,Xb)) = aa(product_prod(int,int),rat,abs_Rat,
            $ite(aa(product_prod(int,int),int,product_fst(int,int),Xb) = zero_zero(int),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),Xb)),aa(product_prod(int,int),int,product_fst(int,int),Xb)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_6785_set__relcomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,Xys: list(product_prod(A,C)),Yzs: list(product_prod(C,B))] : ( relcomp(A,C,B,aa(list(product_prod(A,C)),set(product_prod(A,C)),set2(product_prod(A,C)),Xys),aa(list(product_prod(C,B)),set(product_prod(C,B)),set2(product_prod(C,B)),Yzs)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(product_prod(A,C)),list(list(product_prod(A,B))),map(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_abj(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs)),Xys))) ) ).

% set_relcomp
tff(fact_6786_graph__map__of__if__distinct__dom,axiom,
    ! [B: $tType,A: $tType,Al: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Al))
     => ( graph(A,B,map_of(A,B,Al)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Al) ) ) ).

% graph_map_of_if_distinct_dom
tff(fact_6787_map__of__map__restrict,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Ks: list(A)] : ( map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_abk(fun(A,B),fun(A,product_prod(A,B)),F2)),Ks)) = restrict_map(A,B,aa(fun(A,B),fun(A,option(B)),comp(B,option(B),A,some(B)),F2),aa(list(A),set(A),set2(A),Ks)) ) ).

% map_of_map_restrict
tff(fact_6788_quotient__of__def,axiom,
    ! [Xb: rat] : ( quotient_of(Xb) = the(product_prod(int,int),aTP_Lamp_abl(rat,fun(product_prod(int,int),$o),Xb)) ) ).

% quotient_of_def
tff(fact_6789_coprime__mult__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,C2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( algebr8660921524188924756oprime(A,C2,A2)
            & algebr8660921524188924756oprime(A,C2,B2) ) ) ) ).

% coprime_mult_right_iff
tff(fact_6790_coprime__mult__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2)
        <=> ( algebr8660921524188924756oprime(A,A2,C2)
            & algebr8660921524188924756oprime(A,B2,C2) ) ) ) ).

% coprime_mult_left_iff
tff(fact_6791_coprime__minus__left__iff,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,uminus_uminus(A),A2),B2)
        <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% coprime_minus_left_iff
tff(fact_6792_coprime__minus__right__iff,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,aa(A,A,uminus_uminus(A),B2))
        <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% coprime_minus_right_iff
tff(fact_6793_coprime__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( algebr8660921524188924756oprime(A,A2,A2)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A)) ) ) ).

% coprime_self
tff(fact_6794_coprime__mod__right__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( algebr8660921524188924756oprime(A,A2,modulo_modulo(A,B2,A2))
          <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ) ).

% coprime_mod_right_iff
tff(fact_6795_coprime__mod__left__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( algebr8660921524188924756oprime(A,modulo_modulo(A,A2,B2),B2)
          <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ) ).

% coprime_mod_left_iff
tff(fact_6796_coprime__power__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb),B2)
        <=> ( algebr8660921524188924756oprime(A,A2,B2)
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% coprime_power_left_iff
tff(fact_6797_coprime__power__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( algebr8660921524188924756oprime(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
        <=> ( algebr8660921524188924756oprime(A,A2,B2)
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% coprime_power_right_iff
tff(fact_6798_coprime__imp__gcd__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,B2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) = one_one(A) ) ) ) ).

% coprime_imp_gcd_eq_1
tff(fact_6799_coprime__0__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( algebr8660921524188924756oprime(A,zero_zero(A),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A)) ) ) ).

% coprime_0_left_iff
tff(fact_6800_coprime__0__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( algebr8660921524188924756oprime(A,A2,zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A)) ) ) ).

% coprime_0_right_iff
tff(fact_6801_coprime__mult__self__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),one_one(A))
            & algebr8660921524188924756oprime(A,A2,B2) ) ) ) ).

% coprime_mult_self_left_iff
tff(fact_6802_coprime__mult__self__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),one_one(A))
            & algebr8660921524188924756oprime(A,A2,B2) ) ) ) ).

% coprime_mult_self_right_iff
tff(fact_6803_is__unit__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),one_one(A))
        <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% is_unit_gcd
tff(fact_6804_normalize__stable,axiom,
    ! [Q4: int,P3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q4)
     => ( algebr8660921524188924756oprime(int,P3,Q4)
       => ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),Q4)) = aa(int,product_prod(int,int),product_Pair(int,int,P3),Q4) ) ) ) ).

% normalize_stable
tff(fact_6805_coprime__iff__gcd__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,B2)
        <=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) = one_one(A) ) ) ) ).

% coprime_iff_gcd_eq_1
tff(fact_6806_gcd__eq__1__imp__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) = one_one(A) )
         => algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% gcd_eq_1_imp_coprime
tff(fact_6807_gcd__mult__left__left__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C2: A,A2: A] :
          ( algebr8660921524188924756oprime(A,B2,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ) ).

% gcd_mult_left_left_cancel
tff(fact_6808_gcd__mult__left__right__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C2: A,A2: A] :
          ( algebr8660921524188924756oprime(A,B2,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ) ).

% gcd_mult_left_right_cancel
tff(fact_6809_gcd__mult__right__left__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ) ).

% gcd_mult_right_left_cancel
tff(fact_6810_gcd__mult__right__right__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,C2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ) ).

% gcd_mult_right_right_cancel
tff(fact_6811_mult__mod__cancel__right,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [A2: A,Nb: A,Ma: A,B2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),Nb),Ma) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),Nb),Ma) )
         => ( algebr8660921524188924756oprime(A,Ma,Nb)
           => ( modulo_modulo(A,A2,Ma) = modulo_modulo(A,B2,Ma) ) ) ) ) ).

% mult_mod_cancel_right
tff(fact_6812_mult__mod__cancel__left,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [Nb: A,A2: A,Ma: A,B2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),Nb),A2),Ma) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),Nb),B2),Ma) )
         => ( algebr8660921524188924756oprime(A,Ma,Nb)
           => ( modulo_modulo(A,A2,Ma) = modulo_modulo(A,B2,Ma) ) ) ) ) ).

% mult_mod_cancel_left
tff(fact_6813_prod__list__coprime__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A),A2: A] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xs))
             => algebr8660921524188924756oprime(A,X,A2) )
         => algebr8660921524188924756oprime(A,aa(list(A),A,groups5270119922927024881d_list(A),Xs),A2) ) ) ).

% prod_list_coprime_left
tff(fact_6814_prod__list__coprime__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A),A2: A] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xs))
             => algebr8660921524188924756oprime(A,A2,X) )
         => algebr8660921524188924756oprime(A,A2,aa(list(A),A,groups5270119922927024881d_list(A),Xs)) ) ) ).

% prod_list_coprime_right
tff(fact_6815_coprime__crossproduct__int,axiom,
    ! [A2: int,D3: int,B2: int,C2: int] :
      ( algebr8660921524188924756oprime(int,A2,D3)
     => ( algebr8660921524188924756oprime(int,B2,C2)
       => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),A2)),aa(int,int,abs_abs(int),C2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),B2)),aa(int,int,abs_abs(int),D3)) )
        <=> ( ( aa(int,int,abs_abs(int),A2) = aa(int,int,abs_abs(int),B2) )
            & ( aa(int,int,abs_abs(int),C2) = aa(int,int,abs_abs(int),D3) ) ) ) ) ) ).

% coprime_crossproduct_int
tff(fact_6816_is__unit__right__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% is_unit_right_imp_coprime
tff(fact_6817_is__unit__left__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% is_unit_left_imp_coprime
tff(fact_6818_coprime__common__divisor,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( algebr8660921524188924756oprime(A,A2,B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),one_one(A)) ) ) ) ) ).

% coprime_common_divisor
tff(fact_6819_coprime__absorb__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Y),Xb)
         => ( algebr8660921524188924756oprime(A,Xb,Y)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Y),one_one(A)) ) ) ) ).

% coprime_absorb_right
tff(fact_6820_coprime__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,D3: A,A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,C2,D3)
         => ( ! [E: A] :
                ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E),one_one(A))
               => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E),A2)
                 => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E),B2)
                   => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E),C2) ) ) )
           => ( ! [E: A] :
                  ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E),one_one(A))
                 => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E),A2)
                   => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E),B2)
                     => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E),D3) ) ) )
             => algebr8660921524188924756oprime(A,A2,B2) ) ) ) ) ).

% coprime_imp_coprime
tff(fact_6821_coprime__absorb__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),Y)
         => ( algebr8660921524188924756oprime(A,Xb,Y)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),one_one(A)) ) ) ) ).

% coprime_absorb_left
tff(fact_6822_not__coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
           => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),one_one(A))
             => ~ algebr8660921524188924756oprime(A,A2,B2) ) ) ) ) ).

% not_coprimeI
tff(fact_6823_not__coprimeE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ~ algebr8660921524188924756oprime(A,A2,B2)
         => ~ ! [C3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),A2)
               => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B2)
                 => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),one_one(A)) ) ) ) ) ).

% not_coprimeE
tff(fact_6824_coprime__def,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,B2)
        <=> ! [C4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C4),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C4),B2)
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C4),one_one(A)) ) ) ) ) ).

% coprime_def
tff(fact_6825_coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ! [C3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B2)
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),one_one(A)) ) )
         => algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% coprimeI
tff(fact_6826_coprime__add__one__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : algebr8660921524188924756oprime(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))) ) ).

% coprime_add_one_right
tff(fact_6827_coprime__add__one__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)),A2) ) ).

% coprime_add_one_left
tff(fact_6828_coprime__doff__one__right,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A] : algebr8660921524188924756oprime(A,A2,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))) ) ).

% coprime_doff_one_right
tff(fact_6829_coprime__diff__one__left,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A)),A2) ) ).

% coprime_diff_one_left
tff(fact_6830_coprime__dvd__mult__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,C2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% coprime_dvd_mult_right_iff
tff(fact_6831_coprime__dvd__mult__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,C2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% coprime_dvd_mult_left_iff
tff(fact_6832_divides__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2)
           => ( algebr8660921524188924756oprime(A,A2,B2)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ) ) ) ).

% divides_mult
tff(fact_6833_coprime__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( algebr8660921524188924756oprime(A,B2,A2)
        <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% coprime_commute
tff(fact_6834_coprime__divisors,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),D3)
           => ( algebr8660921524188924756oprime(A,C2,D3)
             => algebr8660921524188924756oprime(A,A2,B2) ) ) ) ) ).

% coprime_divisors
tff(fact_6835_coprime__1__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] : algebr8660921524188924756oprime(A,one_one(A),A2) ) ).

% coprime_1_left
tff(fact_6836_coprime__1__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] : algebr8660921524188924756oprime(A,A2,one_one(A)) ) ).

% coprime_1_right
tff(fact_6837_prod__coprime__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_gcd(B)
     => ! [A3: set(A),F2: fun(A,B),A2: B] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => algebr8660921524188924756oprime(B,aa(A,B,F2,I2),A2) )
         => algebr8660921524188924756oprime(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3),A2) ) ) ).

% prod_coprime_left
tff(fact_6838_prod__coprime__right,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_gcd(B)
     => ! [A3: set(A),A2: B,F2: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => algebr8660921524188924756oprime(B,A2,aa(A,B,F2,I2)) )
         => algebr8660921524188924756oprime(B,A2,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ).

% prod_coprime_right
tff(fact_6839_invertible__coprime,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) = one_one(A) )
         => algebr8660921524188924756oprime(A,A2,C2) ) ) ).

% invertible_coprime
tff(fact_6840_gcd__coprime__exists,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) != zero_zero(A) )
         => ? [A10: A,B7: A] :
              ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),A10),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) )
              & ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B7),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) )
              & algebr8660921524188924756oprime(A,A10,B7) ) ) ) ).

% gcd_coprime_exists
tff(fact_6841_gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,A6: A,B6: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) != zero_zero(A) )
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),A6),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) )
           => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B6),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) )
             => algebr8660921524188924756oprime(A,A6,B6) ) ) ) ) ).

% gcd_coprime
tff(fact_6842_div__gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( ( A2 != zero_zero(A) )
            | ( B2 != zero_zero(A) ) )
         => algebr8660921524188924756oprime(A,divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),divide_divide(A,B2,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2))) ) ) ).

% div_gcd_coprime
tff(fact_6843_Rat__cases,axiom,
    ! [Q4: rat] :
      ~ ! [A4: int,B4: int] :
          ( ( Q4 = aa(int,rat,aa(int,fun(int,rat),fract,A4),B4) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B4)
           => ~ algebr8660921524188924756oprime(int,A4,B4) ) ) ).

% Rat_cases
tff(fact_6844_Rat__induct,axiom,
    ! [P: fun(rat,$o),Q4: rat] :
      ( ! [A4: int,B4: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B4)
         => ( algebr8660921524188924756oprime(int,A4,B4)
           => aa(rat,$o,P,aa(int,rat,aa(int,fun(int,rat),fract,A4),B4)) ) )
     => aa(rat,$o,P,Q4) ) ).

% Rat_induct
tff(fact_6845_coprime__common__divisor__int,axiom,
    ! [A2: int,B2: int,Xb: int] :
      ( algebr8660921524188924756oprime(int,A2,B2)
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Xb),A2)
       => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Xb),B2)
         => ( aa(int,int,abs_abs(int),Xb) = one_one(int) ) ) ) ) ).

% coprime_common_divisor_int
tff(fact_6846_Rat__cases__nonzero,axiom,
    ! [Q4: rat] :
      ( ! [A4: int,B4: int] :
          ( ( Q4 = aa(int,rat,aa(int,fun(int,rat),fract,A4),B4) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B4)
           => ( ( A4 != zero_zero(int) )
             => ~ algebr8660921524188924756oprime(int,A4,B4) ) ) )
     => ( Q4 = zero_zero(rat) ) ) ).

% Rat_cases_nonzero
tff(fact_6847_Rats__cases_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Xb: A] :
          ( member(A,Xb,field_char_0_Rats(A))
         => ~ ! [A4: int,B4: int] :
                ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B4)
               => ( algebr8660921524188924756oprime(int,A4,B4)
                 => ( Xb != divide_divide(A,aa(int,A,ring_1_of_int(A),A4),aa(int,A,ring_1_of_int(A),B4)) ) ) ) ) ) ).

% Rats_cases'
tff(fact_6848_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X: product_prod(int,int)] :
      ( ( R2 = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),X)) )
      & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),X))
      & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),X),aa(product_prod(int,int),int,product_snd(int,int),X))
      & ! [Y4: product_prod(int,int)] :
          ( ( ( R2 = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),Y4)),aa(product_prod(int,int),int,product_snd(int,int),Y4)) )
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),Y4))
            & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Y4),aa(product_prod(int,int),int,product_snd(int,int),Y4)) )
         => ( Y4 = X ) ) ) ).

% quotient_of_unique
tff(fact_6849_restrict__map__upds,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),D7: set(A),Ma: fun(A,option(B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),D7)
       => ( restrict_map(A,B,map_upds(A,B,Ma,Xs,Ys2),D7) = map_upds(A,B,restrict_map(A,B,Ma,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D7),aa(list(A),set(A),set2(A),Xs))),Xs,Ys2) ) ) ) ).

% restrict_map_upds
tff(fact_6850_possible__bit__def,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),Nb: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,Nb)
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) != zero_zero(A) ) ) ) ).

% possible_bit_def
tff(fact_6851_map__upds__apply__nontin,axiom,
    ! [B: $tType,A: $tType,Xb: A,Xs: list(A),F2: fun(A,option(B)),Ys2: list(B)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(A,option(B),map_upds(A,B,F2,Xs,Ys2),Xb) = aa(A,option(B),F2,Xb) ) ) ).

% map_upds_apply_nontin
tff(fact_6852_coprime__int__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( algebr8660921524188924756oprime(int,aa(nat,int,semiring_1_of_nat(int),Ma),aa(nat,int,semiring_1_of_nat(int),Nb))
    <=> algebr8660921524188924756oprime(nat,Ma,Nb) ) ).

% coprime_int_iff
tff(fact_6853_coprime__nat__abs__left__iff,axiom,
    ! [K: int,Nb: nat] :
      ( algebr8660921524188924756oprime(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),Nb)
    <=> algebr8660921524188924756oprime(int,K,aa(nat,int,semiring_1_of_nat(int),Nb)) ) ).

% coprime_nat_abs_left_iff
tff(fact_6854_coprime__nat__abs__right__iff,axiom,
    ! [Nb: nat,K: int] :
      ( algebr8660921524188924756oprime(nat,Nb,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)))
    <=> algebr8660921524188924756oprime(int,aa(nat,int,semiring_1_of_nat(int),Nb),K) ) ).

% coprime_nat_abs_right_iff
tff(fact_6855_coprime__crossproduct__nat,axiom,
    ! [A2: nat,D3: nat,B2: nat,C2: nat] :
      ( algebr8660921524188924756oprime(nat,A2,D3)
     => ( algebr8660921524188924756oprime(nat,B2,C2)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),C2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),D3) )
        <=> ( ( A2 = B2 )
            & ( C2 = D3 ) ) ) ) ) ).

% coprime_crossproduct_nat
tff(fact_6856_coprime__Suc__left__nat,axiom,
    ! [Nb: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,Nb),Nb) ).

% coprime_Suc_left_nat
tff(fact_6857_coprime__Suc__right__nat,axiom,
    ! [Nb: nat] : algebr8660921524188924756oprime(nat,Nb,aa(nat,nat,suc,Nb)) ).

% coprime_Suc_right_nat
tff(fact_6858_coprime__Suc__0__left,axiom,
    ! [Nb: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,zero_zero(nat)),Nb) ).

% coprime_Suc_0_left
tff(fact_6859_coprime__Suc__0__right,axiom,
    ! [Nb: nat] : algebr8660921524188924756oprime(nat,Nb,aa(nat,nat,suc,zero_zero(nat))) ).

% coprime_Suc_0_right
tff(fact_6860_coprime__common__divisor__nat,axiom,
    ! [A2: nat,B2: nat,Xb: nat] :
      ( algebr8660921524188924756oprime(nat,A2,B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Xb),A2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Xb),B2)
         => ( Xb = one_one(nat) ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_6861_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_6862_coprime__diff__one__right__nat,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => algebr8660921524188924756oprime(nat,Nb,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_6863_coprime__diff__one__left__nat,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => algebr8660921524188924756oprime(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Nb) ) ).

% coprime_diff_one_left_nat
tff(fact_6864_Rats__abs__nat__div__natE,axiom,
    ! [Xb: real] :
      ( member(real,Xb,field_char_0_Rats(real))
     => ~ ! [M: nat,N: nat] :
            ( ( N != zero_zero(nat) )
           => ( ( aa(real,real,abs_abs(real),Xb) = divide_divide(real,aa(nat,real,semiring_1_of_nat(real),M),aa(nat,real,semiring_1_of_nat(real),N)) )
             => ~ algebr8660921524188924756oprime(nat,M,N) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_6865_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,Nb: nat] :
          ( aa(A,A,bit_se4197421643247451524op_bit(A,Ma),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,
              aa(A,fun(A,A),times_times(A),
                aa($o,A,zero_neq_one_of_bool(A),
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
                  & bit_se6407376104438227557le_bit(A,type2(A),Nb) ))),
              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),Nb),Ma))) ) ) ).

% drop_bit_exp_eq
tff(fact_6866_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2)))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% bit_minus_2_iff
tff(fact_6867_CHAR__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) ) ) ).

% CHAR_eq_0
tff(fact_6868_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_6869_bit__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),one_one(A))),Nb)
        <=> bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ).

% bit_minus_1_iff
tff(fact_6870_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,bit_se2239418461657761734s_mask(A,Ma)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma) ) ) ) ).

% bit_mask_iff
tff(fact_6871_of__nat__eq__0__iff__char__dvd,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Nb) = zero_zero(A) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),semiri4206861660011772517g_char(A,type2(A))),Nb) ) ) ).

% of_nat_eq_0_iff_char_dvd
tff(fact_6872_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,aa(nat,fun(nat,$o),dvd_dvd(nat),C2),X) )
           => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ).

% CHAR_eqI
tff(fact_6873_bit__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(int,A,ring_1_of_int(A),K)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ) ) ).

% bit_of_int_iff
tff(fact_6874_bit__of__nat__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ma: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(nat,A,semiring_1_of_nat(A),Ma)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(nat,Ma),Nb) ) ) ) ).

% bit_of_nat_iff
tff(fact_6875_bit__minus__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),A2)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),Nb) ) ) ) ).

% bit_minus_iff
tff(fact_6876_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N4)
           => ( aa(nat,A,semiring_1_of_nat(A),N4) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_6877_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_6878_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A)))
      <=> ? [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N4)
            & ( aa(nat,A,semiring_1_of_nat(A),N4) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_6879_bit__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ma: nat,A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se4730199178511100633sh_bit(A,Ma),A2)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
            & bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma)) ) ) ) ).

% bit_push_bit_iff
tff(fact_6880_fold__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) = zero_zero(A) )
        <=> ~ bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ).

% fold_possible_bit
tff(fact_6881_bit__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),one_one(nat))
            & ( Nb = one_one(nat) ) ) ) ) ).

% bit_2_iff
tff(fact_6882_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Ma: nat,Nb: nat] :
          ( aa(nat,$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))),Ma))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb) ) ) ) ).

% bit_minus_exp_iff
tff(fact_6883_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ma: nat,Nb: nat] :
          ( aa(nat,$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))),Ma)),one_one(A))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma) ) ) ) ).

% bit_mask_sub_iff
tff(fact_6884_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$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)),Nb)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))
            & ( Nb != zero_zero(nat) )
            & bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ) ).

% bit_double_iff
tff(fact_6885_map__upds__append1,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),Ma: fun(A,option(B)),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys2))
     => ( map_upds(A,B,Ma,append(A,Xs,cons(A,Xb,nil(A))),Ys2) = fun_upd(A,option(B),map_upds(A,B,Ma,Xs,Ys2),Xb,aa(B,option(B),some(B),aa(nat,B,nth(B,Ys2),aa(list(A),nat,size_size(list(A)),Xs)))) ) ) ).

% map_upds_append1
tff(fact_6886_mult__inj__if__coprime__nat,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,nat),A3: set(A),G: fun(B,nat),B3: set(B)] :
      ( inj_on(A,nat,F2,A3)
     => ( inj_on(B,nat,G,B3)
       => ( ! [A4: A,B4: B] :
              ( member(A,A4,A3)
             => ( member(B,B4,B3)
               => algebr8660921524188924756oprime(nat,aa(A,nat,F2,A4),aa(B,nat,G,B4)) ) )
         => inj_on(product_prod(A,B),nat,product_case_prod(A,B,nat,aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_abm(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),F2),G)),product_Sigma(A,B,A3,aTP_Lamp_abn(set(B),fun(A,set(B)),B3))) ) ) ) ).

% mult_inj_if_coprime_nat
tff(fact_6887_map__fun__upd,axiom,
    ! [B: $tType,A: $tType,Y: A,Xs: list(A),F2: fun(A,B),V2: B] :
      ( ~ member(A,Y,aa(list(A),set(A),set2(A),Xs))
     => ( aa(list(A),list(B),map(A,B,fun_upd(A,B,F2,Y,V2)),Xs) = aa(list(A),list(B),map(A,B,F2),Xs) ) ) ).

% map_fun_upd
tff(fact_6888_map__upds__twist,axiom,
    ! [A: $tType,B: $tType,A2: A,As: list(A),Ma: fun(A,option(B)),B2: B,Bs: list(B)] :
      ( ~ member(A,A2,aa(list(A),set(A),set2(A),As))
     => ( map_upds(A,B,fun_upd(A,option(B),Ma,A2,aa(B,option(B),some(B),B2)),As,Bs) = fun_upd(A,option(B),map_upds(A,B,Ma,As,Bs),A2,aa(B,option(B),some(B),B2)) ) ) ).

% map_upds_twist
tff(fact_6889_set__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] : ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),product(A,B,Xs,Ys2)) = product_Sigma(A,B,aa(list(A),set(A),set2(A),Xs),aTP_Lamp_abo(list(B),fun(A,set(B)),Ys2)) ) ).

% set_product
tff(fact_6890_restrict__upd__same,axiom,
    ! [B: $tType,A: $tType,Ma: fun(A,option(B)),Xb: A,Y: B] : ( restrict_map(A,B,fun_upd(A,option(B),Ma,Xb,aa(B,option(B),some(B),Y)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))) = restrict_map(A,B,Ma,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))) ) ).

% restrict_upd_same
tff(fact_6891_Ex__inj__on__UNION__Sigma,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),I5: set(B)] :
    ? [F3: fun(A,product_prod(B,A))] :
      ( inj_on(A,product_prod(B,A),F3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),A3,I5)))
      & aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),image(A,product_prod(B,A),F3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),A3,I5)))),product_Sigma(B,A,I5,A3)) ) ).

% Ex_inj_on_UNION_Sigma
tff(fact_6892_product__atMost__eq__Un,axiom,
    ! [A3: set(nat),Ma: nat] : ( product_Sigma(nat,nat,A3,aTP_Lamp_abp(nat,fun(nat,set(nat)),Ma)) = aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),sup_sup(set(product_prod(nat,nat))),product_Sigma(nat,nat,A3,aTP_Lamp_abq(nat,fun(nat,set(nat)),Ma))),product_Sigma(nat,nat,A3,aTP_Lamp_abr(nat,fun(nat,set(nat)),Ma))) ) ).

% product_atMost_eq_Un
tff(fact_6893_map__of__zip__upd,axiom,
    ! [B: $tType,A: $tType,Ys2: list(A),Xs: list(B),Zs3: list(A),Xb: B,Y: A,Z: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Ys2) = aa(list(B),nat,size_size(list(B)),Xs) )
     => ( ( aa(list(A),nat,size_size(list(A)),Zs3) = aa(list(B),nat,size_size(list(B)),Xs) )
       => ( ~ member(B,Xb,aa(list(B),set(B),set2(B),Xs))
         => ( ( fun_upd(B,option(A),map_of(B,A,zip(B,A,Xs,Ys2)),Xb,aa(A,option(A),some(A),Y)) = fun_upd(B,option(A),map_of(B,A,zip(B,A,Xs,Zs3)),Xb,aa(A,option(A),some(A),Z)) )
           => ( map_of(B,A,zip(B,A,Xs,Ys2)) = map_of(B,A,zip(B,A,Xs,Zs3)) ) ) ) ) ) ).

% map_of_zip_upd
tff(fact_6894_restrict__complement__singleton__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),Xb: A] : ( restrict_map(A,B,F2,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))) = fun_upd(A,option(B),F2,Xb,none(B)) ) ).

% restrict_complement_singleton_eq
tff(fact_6895_map__upd__upds__conv__if,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),Xb: A,Y: B,Xs: list(A),Ys2: list(B)] :
      ( map_upds(A,B,fun_upd(A,option(B),F2,Xb,aa(B,option(B),some(B),Y)),Xs,Ys2) = $ite(member(A,Xb,aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys2),Xs))),map_upds(A,B,F2,Xs,Ys2),fun_upd(A,option(B),map_upds(A,B,F2,Xs,Ys2),Xb,aa(B,option(B),some(B),Y))) ) ).

% map_upd_upds_conv_if
tff(fact_6896_pairs__le__eq__Sigma,axiom,
    ! [Ma: nat] : ( collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_hp(nat,fun(nat,fun(nat,$o)),Ma))) = product_Sigma(nat,nat,set_ord_atMost(nat,Ma),aTP_Lamp_abq(nat,fun(nat,set(nat)),Ma)) ) ).

% pairs_le_eq_Sigma
tff(fact_6897_Compl__Times__UNIV1,axiom,
    ! [A: $tType,B: $tType,A3: set(B)] : ( aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_abn(set(B),fun(A,set(B)),A3))) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_abs(set(B),fun(A,set(B)),A3)) ) ).

% Compl_Times_UNIV1
tff(fact_6898_Compl__Times__UNIV2,axiom,
    ! [B: $tType,A: $tType,A3: set(A)] : ( aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_abt(A,set(B)))) = product_Sigma(A,B,aa(set(A),set(A),uminus_uminus(set(A)),A3),aTP_Lamp_abt(A,set(B))) ) ).

% Compl_Times_UNIV2
tff(fact_6899_list_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),A2: list(A),B2: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),A2),B2)
    <=> ? [Z5: list(product_prod(A,B))] :
          ( member(list(product_prod(A,B)),Z5,collect(list(product_prod(A,B)),aTP_Lamp_abu(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),R)))
          & ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Z5) = A2 )
          & ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Z5) = B2 ) ) ) ).

% list.in_rel
tff(fact_6900_dom__map__upds,axiom,
    ! [B: $tType,A: $tType,Ma: fun(A,option(B)),Xs: list(A),Ys2: list(B)] : ( dom(A,B,map_upds(A,B,Ma,Xs,Ys2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys2),Xs))),dom(A,B,Ma)) ) ).

% dom_map_upds
tff(fact_6901_dom__map__of__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( dom(A,B,map_of(A,B,zip(A,B,Xs,Ys2))) = aa(list(A),set(A),set2(A),Xs) ) ) ).

% dom_map_of_zip
tff(fact_6902_list__all2__same,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),list_all2(A,A,P),Xs),Xs)
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,aa(A,fun(A,$o),P,X4),X4) ) ) ).

% list_all2_same
tff(fact_6903_list_Orel__refl__strong,axiom,
    ! [A: $tType,Xb: list(A),Ra: fun(A,fun(A,$o))] :
      ( ! [Z2: A] :
          ( member(A,Z2,aa(list(A),set(A),set2(A),Xb))
         => aa(A,$o,aa(A,fun(A,$o),Ra,Z2),Z2) )
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),list_all2(A,A,Ra),Xb),Xb) ) ).

% list.rel_refl_strong
tff(fact_6904_list_Orel__mono__strong,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),Xb: list(A),Y: list(B),Ra: fun(A,fun(B,$o))] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),Xb),Y)
     => ( ! [Z2: A,Yb: B] :
            ( member(A,Z2,aa(list(A),set(A),set2(A),Xb))
           => ( member(B,Yb,aa(list(B),set(B),set2(B),Y))
             => ( aa(B,$o,aa(A,fun(B,$o),R,Z2),Yb)
               => aa(B,$o,aa(A,fun(B,$o),Ra,Z2),Yb) ) ) )
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,Ra),Xb),Y) ) ) ).

% list.rel_mono_strong
tff(fact_6905_list_Orel__cong,axiom,
    ! [A: $tType,B: $tType,Xb: list(A),Ya: list(A),Y: list(B),Xaa: list(B),R: fun(A,fun(B,$o)),Ra: fun(A,fun(B,$o))] :
      ( ( Xb = Ya )
     => ( ( Y = Xaa )
       => ( ! [Z2: A,Yb: B] :
              ( member(A,Z2,aa(list(A),set(A),set2(A),Ya))
             => ( member(B,Yb,aa(list(B),set(B),set2(B),Xaa))
               => ( aa(B,$o,aa(A,fun(B,$o),R,Z2),Yb)
                <=> aa(B,$o,aa(A,fun(B,$o),Ra,Z2),Yb) ) ) )
         => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),Xb),Y)
          <=> aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,Ra),Ya),Xaa) ) ) ) ) ).

% list.rel_cong
tff(fact_6906_list__all2__conv__all__nth,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xs: list(A),Ys2: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Xs),Ys2)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),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(B,$o,aa(A,fun(B,$o),P,aa(nat,A,nth(A,Xs),I3)),aa(nat,B,nth(B,Ys2),I3)) ) ) ) ).

% list_all2_conv_all_nth
tff(fact_6907_list__all2__all__nthI,axiom,
    ! [A: $tType,B: $tType,A2: list(A),B2: list(B),P: fun(A,fun(B,$o))] :
      ( ( aa(list(A),nat,size_size(list(A)),A2) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),A2))
           => aa(B,$o,aa(A,fun(B,$o),P,aa(nat,A,nth(A,A2),N)),aa(nat,B,nth(B,B2),N)) )
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),A2),B2) ) ) ).

% list_all2_all_nthI
tff(fact_6908_list__all2__nthD2,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xs: list(A),Ys2: list(B),P3: nat] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Xs),Ys2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P3),aa(list(B),nat,size_size(list(B)),Ys2))
       => aa(B,$o,aa(A,fun(B,$o),P,aa(nat,A,nth(A,Xs),P3)),aa(nat,B,nth(B,Ys2),P3)) ) ) ).

% list_all2_nthD2
tff(fact_6909_list__all2__nthD,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xs: list(A),Ys2: list(B),P3: nat] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Xs),Ys2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P3),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(B,$o,aa(A,fun(B,$o),P,aa(nat,A,nth(A,Xs),P3)),aa(nat,B,nth(B,Ys2),P3)) ) ) ).

% list_all2_nthD
tff(fact_6910_product__lists__set,axiom,
    ! [A: $tType,Xss: list(list(A))] : ( aa(list(list(A)),set(list(A)),set2(list(A)),product_lists(A,Xss)) = collect(list(A),aTP_Lamp_abw(list(list(A)),fun(list(A),$o),Xss)) ) ).

% product_lists_set
tff(fact_6911_list__all2I,axiom,
    ! [A: $tType,B: $tType,A2: list(A),B2: list(B),P: fun(A,fun(B,$o))] :
      ( ! [X: product_prod(A,B)] :
          ( member(product_prod(A,B),X,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,A2,B2)))
         => aa(product_prod(A,B),$o,product_case_prod(A,B,$o,P),X) )
     => ( ( aa(list(A),nat,size_size(list(A)),A2) = aa(list(B),nat,size_size(list(B)),B2) )
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),A2),B2) ) ) ).

% list_all2I
tff(fact_6912_dom__map__of__conv__image__fst,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] : ( dom(A,B,map_of(A,B,Xys)) = image(product_prod(A,B),A,product_fst(A,B),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)) ) ).

% dom_map_of_conv_image_fst
tff(fact_6913_sum__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & monoid_add(A) )
     => ! [A3: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),A3,zero_zero(A)),zero_zero(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),A3,bNF_rel_fun(A,B,A,B,A3,A3)),plus_plus(A)),plus_plus(B))
           => aa(fun(list(B),B),$o,aa(fun(list(A),A),fun(fun(list(B),B),$o),bNF_rel_fun(list(A),list(B),A,B,list_all2(A,B,A3),A3),groups8242544230860333062m_list(A)),groups8242544230860333062m_list(B)) ) ) ) ).

% sum_list_transfer
tff(fact_6914_prod__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_mult(B)
        & monoid_mult(A) )
     => ! [A3: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),A3,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),A3,bNF_rel_fun(A,B,A,B,A3,A3)),times_times(A)),times_times(B))
           => aa(fun(list(B),B),$o,aa(fun(list(A),A),fun(fun(list(B),B),$o),bNF_rel_fun(list(A),list(B),A,B,list_all2(A,B,A3),A3),groups5270119922927024881d_list(A)),groups5270119922927024881d_list(B)) ) ) ) ).

% prod_list_transfer
tff(fact_6915_list__all2__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Xs: list(A),Ys2: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Xs),Ys2)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [X4: product_prod(A,B)] :
            ( member(product_prod(A,B),X4,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)))
           => aa(product_prod(A,B),$o,product_case_prod(A,B,$o,P),X4) ) ) ) ).

% list_all2_iff
tff(fact_6916_horner__sum__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( comm_semiring_0(B)
        & comm_semiring_0(A) )
     => ! [A3: fun(A,fun(B,$o)),B3: fun(C,fun(D,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),A3,zero_zero(A)),zero_zero(B))
         => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A3,bNF_rel_fun(A,B,A,B,A3,A3)),plus_plus(A)),plus_plus(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A3,bNF_rel_fun(A,B,A,B,A3,A3)),times_times(A)),times_times(B))
             => aa(fun(fun(D,B),fun(B,fun(list(D),B))),$o,aa(fun(fun(C,A),fun(A,fun(list(C),A))),fun(fun(fun(D,B),fun(B,fun(list(D),B))),$o),bNF_rel_fun(fun(C,A),fun(D,B),fun(A,fun(list(C),A)),fun(B,fun(list(D),B)),bNF_rel_fun(C,D,A,B,B3,A3),bNF_rel_fun(A,B,fun(list(C),A),fun(list(D),B),A3,bNF_rel_fun(list(C),list(D),A,B,list_all2(C,D,B3),A3))),groups4207007520872428315er_sum(C,A)),groups4207007520872428315er_sum(D,B)) ) ) ) ) ).

% horner_sum_transfer
tff(fact_6917_map__of__map__keys,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ma: fun(A,option(B))] :
      ( ( aa(list(A),set(A),set2(A),Xs) = dom(A,B,Ma) )
     => ( map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_abx(fun(A,option(B)),fun(A,product_prod(A,B)),Ma)),Xs)) = Ma ) ) ).

% map_of_map_keys
tff(fact_6918_numeral__xor__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Ma: num,Nb: num] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Ma)),aa(num,A,numeral_numeral(A),Nb)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_un2480387367778600638or_num(Ma,Nb)) ) ) ).

% numeral_xor_num
tff(fact_6919_xor__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Ma: num,Nb: num] :
          ( ( bit_un2480387367778600638or_num(Ma,Nb) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Ma)),aa(num,A,numeral_numeral(A),Nb)) = zero_zero(A) ) ) ) ).

% xor_num_eq_None_iff
tff(fact_6920_Code__Target__Nat_ONat__def,axiom,
    code_Target_Nat = aa(fun(int,nat),fun(code_integer,nat),map_fun(code_integer,int,nat,nat,code_int_of_integer,id(nat)),nat2) ).

% Code_Target_Nat.Nat_def
tff(fact_6921_set__rec,axiom,
    ! [A: $tType,Xs: list(A)] : ( aa(list(A),set(A),set2(A),Xs) = rec_list(set(A),A,bot_bot(set(A)),aTP_Lamp_aby(A,fun(list(A),fun(set(A),set(A)))),Xs) ) ).

% set_rec
tff(fact_6922_nat__of__integer__def,axiom,
    code_nat_of_integer = aa(fun(int,nat),fun(code_integer,nat),map_fun(code_integer,int,nat,nat,code_int_of_integer,id(nat)),nat2) ).

% nat_of_integer_def
tff(fact_6923_numeral__and__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Ma: num,Nb: num] : ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Ma)),aa(num,A,numeral_numeral(A),Nb)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_un7362597486090784418nd_num(Ma,Nb)) ) ) ).

% numeral_and_num
tff(fact_6924_Gcd__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A3) = zero_zero(A) )
        <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A))))
            & aa(set(A),$o,finite_finite(A),A3) ) ) ) ).

% Gcd_fin_0_iff
tff(fact_6925_Gcd__fin_Oempty,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_fin.empty
tff(fact_6926_Gcd__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( ~ aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A3) = one_one(A) ) ) ) ).

% Gcd_fin.infinite
tff(fact_6927_is__unit__Gcd__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A3)),one_one(A))
        <=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A3) = one_one(A) ) ) ) ).

% is_unit_Gcd_fin_iff
tff(fact_6928_Gcd__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A3: set(A)] : ( aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(set(A),A,semiring_gcd_Gcd_fin(A),A3)) ) ) ).

% Gcd_fin.insert
tff(fact_6929_Gcd__fin__eq__Gcd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A3) = gcd_Gcd(A,A3) ) ) ) ).

% Gcd_fin_eq_Gcd
tff(fact_6930_Gcd__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( member(A,A2,A3)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(set(A),A,semiring_gcd_Gcd_fin(A),A3)) = aa(set(A),A,semiring_gcd_Gcd_fin(A),A3) ) ) ) ).

% Gcd_fin.in_idem
tff(fact_6931_gcd__list__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Bs: list(A),A2: A] :
          ( ! [B4: A] :
              ( member(A,B4,aa(list(A),set(A),set2(A),Bs))
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B4) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(list(A),set(A),set2(A),Bs))) ) ) ).

% gcd_list_greatest
tff(fact_6932_dvd__gcd__list__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,Xs: list(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(list(A),set(A),set2(A),Xs)))
        <=> ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),X4) ) ) ) ).

% dvd_gcd_list_iff
tff(fact_6933_Gcd__fin__dvd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( member(A,A2,A3)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A3)),A2) ) ) ).

% Gcd_fin_dvd
tff(fact_6934_Gcd__fin__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ! [B4: A] :
                ( member(A,B4,A3)
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B4) )
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(set(A),A,semiring_gcd_Gcd_fin(A),A3)) ) ) ) ).

% Gcd_fin_greatest
tff(fact_6935_dvd__Gcd__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A),B2: A] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),aa(set(A),A,semiring_gcd_Gcd_fin(A),A3))
          <=> ! [X4: A] :
                ( member(A,X4,A3)
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),X4) ) ) ) ) ).

% dvd_Gcd_fin_iff
tff(fact_6936_Gcd__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A),B3: set(A)] : ( aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A3)),aa(set(A),A,semiring_gcd_Gcd_fin(A),B3)) ) ) ).

% Gcd_fin.union
tff(fact_6937_Gcd__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B3: set(A),A3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),B3)),aa(set(A),A,semiring_gcd_Gcd_fin(A),A3)) = aa(set(A),A,semiring_gcd_Gcd_fin(A),A3) ) ) ) ).

% Gcd_fin.subset
tff(fact_6938_Gcd__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( member(A,A2,A3)
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A3) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))) ) ) ) ).

% Gcd_fin.remove
tff(fact_6939_Gcd__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A3: set(A)] : ( aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))) ) ) ).

% Gcd_fin.insert_remove
tff(fact_6940_Gcd__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A)] : ( aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,gcd_gcd(A),Xs),zero_zero(A)) ) ) ).

% Gcd_fin.set_eq_fold
tff(fact_6941_and__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Ma: num,Nb: num] :
          ( ( bit_un7362597486090784418nd_num(Ma,Nb) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Ma)),aa(num,A,numeral_numeral(A),Nb)) = zero_zero(A) ) ) ) ).

% and_num_eq_None_iff
tff(fact_6942_continuous__on__arcosh,axiom,
    ! [A3: set(real)] :
      ( aa(set(real),$o,aa(set(real),fun(set(real),$o),ord_less_eq(set(real)),A3),set_ord_atLeast(real,one_one(real)))
     => topolo81223032696312382ous_on(real,real,A3,arcosh(real)) ) ).

% continuous_on_arcosh
tff(fact_6943_last__list__update,axiom,
    ! [A: $tType,Xs: list(A),K: nat,Xb: A] :
      ( ( Xs != nil(A) )
     => ( last(A,list_update(A,Xs,K,Xb)) = $ite(K = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),Xb,last(A,Xs)) ) ) ).

% last_list_update
tff(fact_6944_image__add__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A] : ( image(A,A,aa(A,fun(A,A),plus_plus(A),K),set_ord_atLeast(A,I)) = set_ord_atLeast(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),K),I)) ) ) ).

% image_add_atLeast
tff(fact_6945_last__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( Nb != zero_zero(nat) )
     => ( last(A,replicate(A,Nb,Xb)) = Xb ) ) ).

% last_replicate
tff(fact_6946_Compl__atLeast,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : ( aa(set(A),set(A),uminus_uminus(set(A)),set_ord_atLeast(A,K)) = set_ord_lessThan(A,K) ) ) ).

% Compl_atLeast
tff(fact_6947_Compl__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : ( aa(set(A),set(A),uminus_uminus(set(A)),set_ord_lessThan(A,K)) = set_ord_atLeast(A,K) ) ) ).

% Compl_lessThan
tff(fact_6948_image__minus__const__AtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] : ( image(A,A,aa(A,fun(A,A),minus_minus(A),C2),set_ord_atMost(A,B2)) = set_ord_atLeast(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)) ) ) ).

% image_minus_const_AtMost
tff(fact_6949_image__minus__const__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] : ( image(A,A,aa(A,fun(A,A),minus_minus(A),C2),set_ord_atLeast(A,A2)) = set_ord_atMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)) ) ) ).

% image_minus_const_atLeast
tff(fact_6950_image__uminus__atLeast,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A] : ( image(A,A,uminus_uminus(A),set_ord_atLeast(A,Xb)) = set_ord_atMost(A,aa(A,A,uminus_uminus(A),Xb)) ) ) ).

% image_uminus_atLeast
tff(fact_6951_image__uminus__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A] : ( image(A,A,uminus_uminus(A),set_ord_atMost(A,Xb)) = set_ord_atLeast(A,aa(A,A,uminus_uminus(A),Xb)) ) ) ).

% image_uminus_atMost
tff(fact_6952_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( last(nat,upt(I,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),one_one(nat)) ) ) ).

% last_upt
tff(fact_6953_last__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( last(A,drop(A,Nb,Xs)) = last(A,Xs) ) ) ).

% last_drop
tff(fact_6954_dropWhile__last,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P,Xb)
       => ( last(A,dropWhile(A,P,Xs)) = last(A,Xs) ) ) ) ).

% dropWhile_last
tff(fact_6955_last__in__set,axiom,
    ! [A: $tType,As: list(A)] :
      ( ( As != nil(A) )
     => member(A,last(A,As),aa(list(A),set(A),set2(A),As)) ) ).

% last_in_set
tff(fact_6956_Ici__subset__Ioi__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atLeast(A,A2)),set_ord_greaterThan(A,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_6957_ivl__disj__un__one_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),set_ord_atLeast(A,U)) = set_ord_greaterThan(A,L) ) ) ) ).

% ivl_disj_un_one(6)
tff(fact_6958_last__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( last(A,Xs) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))) ) ) ).

% last_conv_nth
tff(fact_6959_minus__coset__filter,axiom,
    ! [A: $tType,A3: set(A),Xs: list(A)] : ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),coset(A,Xs)) = aa(list(A),set(A),set2(A),filter2(A,aTP_Lamp_a(set(A),fun(A,$o),A3),Xs)) ) ).

% minus_coset_filter
tff(fact_6960_insort__insert__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xb: B,Xs: list(B)] :
          ( ~ member(A,aa(B,A,F2,Xb),image(B,A,F2,aa(list(B),set(B),set2(B),Xs)))
         => ( linord329482645794927042rt_key(B,A,F2,Xb,Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Xb),Xs) ) ) ) ).

% insort_insert_insort_key
tff(fact_6961_atLeast__0,axiom,
    set_ord_atLeast(nat,zero_zero(nat)) = top_top(set(nat)) ).

% atLeast_0
tff(fact_6962_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : ( set_ord_atLeast(nat,aa(nat,nat,suc,K)) = set_ord_greaterThan(nat,K) ) ).

% atLeast_Suc_greaterThan
tff(fact_6963_insort__insert__triv,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Xs: list(A)] :
          ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
         => ( linord329482645794927042rt_key(A,A,aTP_Lamp_os(A,A),Xb,Xs) = Xs ) ) ) ).

% insort_insert_triv
tff(fact_6964_subset__code_I2_J,axiom,
    ! [A: $tType,A3: set(A),Ys2: list(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),coset(A,Ys2))
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Ys2))
         => ~ member(A,X4,A3) ) ) ).

% subset_code(2)
tff(fact_6965_coset__def,axiom,
    ! [A: $tType,Xs: list(A)] : ( coset(A,Xs) = aa(set(A),set(A),uminus_uminus(set(A)),aa(list(A),set(A),set2(A),Xs)) ) ).

% coset_def
tff(fact_6966_compl__coset,axiom,
    ! [A: $tType,Xs: list(A)] : ( aa(set(A),set(A),uminus_uminus(set(A)),coset(A,Xs)) = aa(list(A),set(A),set2(A),Xs) ) ).

% compl_coset
tff(fact_6967_insort__insert__key__triv,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xb: B,Xs: list(B)] :
          ( member(A,aa(B,A,F2,Xb),image(B,A,F2,aa(list(B),set(B),set2(B),Xs)))
         => ( linord329482645794927042rt_key(B,A,F2,Xb,Xs) = Xs ) ) ) ).

% insort_insert_key_triv
tff(fact_6968_set__insort__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Xs: list(A)] : ( aa(list(A),set(A),set2(A),linord329482645794927042rt_key(A,A,aTP_Lamp_os(A,A),Xb,Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),aa(list(A),set(A),set2(A),Xs)) ) ) ).

% set_insort_insert
tff(fact_6969_insort__insert__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Xs: list(A)] :
          ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
         => ( linord329482645794927042rt_key(A,A,aTP_Lamp_os(A,A),Xb,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_os(A,A)),Xb),Xs) ) ) ) ).

% insort_insert_insort
tff(fact_6970_subset__code_I3_J,axiom,
    ! [A: $tType] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),coset(A,nil(A))),aa(list(A),set(A),set2(A),nil(A))) ).

% subset_code(3)
tff(fact_6971_atLeast__Suc,axiom,
    ! [K: nat] : ( set_ord_atLeast(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_atLeast(nat,K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),K),bot_bot(set(nat)))) ) ).

% atLeast_Suc
tff(fact_6972_insort__insert__key__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xb: A,Xs: list(A)] :
          ( linord329482645794927042rt_key(A,B,F2,Xb,Xs) = $ite(member(B,aa(A,B,F2,Xb),image(A,B,F2,aa(list(A),set(A),set2(A),Xs))),Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xb),Xs)) ) ) ).

% insort_insert_key_def
tff(fact_6973_differentiable__cmult__right__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Q4: fun(A,B),C2: B,Ta: A] :
          ( differentiable(A,B,aa(B,fun(A,B),aTP_Lamp_abz(fun(A,B),fun(B,fun(A,B)),Q4),C2),topolo174197925503356063within(A,Ta,top_top(set(A))))
        <=> ( ( C2 = zero_zero(B) )
            | differentiable(A,B,Q4,topolo174197925503356063within(A,Ta,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_right_iff
tff(fact_6974_differentiable__cmult__left__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [C2: B,Q4: fun(A,B),Ta: A] :
          ( differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aca(B,fun(fun(A,B),fun(A,B)),C2),Q4),topolo174197925503356063within(A,Ta,top_top(set(A))))
        <=> ( ( C2 = zero_zero(B) )
            | differentiable(A,B,Q4,topolo174197925503356063within(A,Ta,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_left_iff
tff(fact_6975_differentiable__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( differentiable(A,B,F2,F4)
         => differentiable(A,B,aTP_Lamp_rl(fun(A,B),fun(A,B),F2),F4) ) ) ).

% differentiable_minus
tff(fact_6976_differentiable__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,F4)
         => ( differentiable(A,B,G,F4)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rn(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F4) ) ) ) ).

% differentiable_diff
tff(fact_6977_differentiable__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,F4)
         => ( differentiable(A,B,G,F4)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F4) ) ) ) ).

% differentiable_add
tff(fact_6978_differentiable__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xb: A,S: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xb,S))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,B,G,Xb) != zero_zero(B) )
             => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% differentiable_divide
tff(fact_6979_differentiable__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xb: A,S: set(A)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,B,F2,Xb) != zero_zero(B) )
           => differentiable(A,B,aTP_Lamp_acb(fun(A,B),fun(A,B),F2),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% differentiable_inverse
tff(fact_6980_differentiable__power__int,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xb: A,S: set(A),Nb: int] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,B,F2,Xb) != zero_zero(B) )
           => differentiable(A,B,aa(int,fun(A,B),aTP_Lamp_acc(fun(A,B),fun(int,fun(A,B)),F2),Nb),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% differentiable_power_int
tff(fact_6981_sort__key__conv__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xs: list(A)] :
          ( inj_on(A,B,F2,aa(list(A),set(A),set2(A),Xs))
         => ( linorder_sort_key(A,B,F2,Xs) = aa(list(A),list(A),fold(A,list(A),linorder_insort_key(A,B,F2),Xs),nil(A)) ) ) ) ).

% sort_key_conv_fold
tff(fact_6982_in__measures_I2_J,axiom,
    ! [A: $tType,Xb: A,Y: A,F2: fun(A,nat),Fs: list(fun(A,nat))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),measures(A,cons(fun(A,nat),F2,Fs)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y))
        | ( ( aa(A,nat,F2,Xb) = aa(A,nat,F2,Y) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),measures(A,Fs)) ) ) ) ).

% in_measures(2)
tff(fact_6983_set__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xs: list(A)] : ( aa(list(A),set(A),set2(A),linorder_sort_key(A,B,F2,Xs)) = aa(list(A),set(A),set2(A),Xs) ) ) ).

% set_sort
tff(fact_6984_measures__less,axiom,
    ! [A: $tType,F2: fun(A,nat),Xb: A,Y: A,Fs: list(fun(A,nat))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y))
     => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),measures(A,cons(fun(A,nat),F2,Fs))) ) ).

% measures_less
tff(fact_6985_sorted__list__of__set__sort__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : ( linord4507533701916653071of_set(A,aa(list(A),set(A),set2(A),Xs)) = linorder_sort_key(A,A,aTP_Lamp_os(A,A),remdups(A,Xs)) ) ) ).

% sorted_list_of_set_sort_remdups
tff(fact_6986_VEBT_Osimps_I7_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),F22: fun($o,fun($o,A)),X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : ( aa(vEBT_VEBT,A,aa(fun($o,fun($o,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,A)),vEBT_rec_VEBT(A),F1),F22),vEBT_Node(X11,X12,X13,X14)) = aa(A,A,aa(vEBT_VEBT,fun(A,A),aa(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A)),aa(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A)))),F1,X11),X12),aa(list(vEBT_VEBT),list(product_prod(vEBT_VEBT,A)),map(vEBT_VEBT,product_prod(vEBT_VEBT,A),aa(fun($o,fun($o,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A)),aTP_Lamp_acd(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A))),F1),F22)),X13)),X14),aa(vEBT_VEBT,A,aa(fun($o,fun($o,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,A)),vEBT_rec_VEBT(A),F1),F22),X14)) ) ).

% VEBT.simps(7)
tff(fact_6987_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F2: fun(nat,real),M9: nat] :
      ( bfun(nat,real,aa(nat,fun(nat,real),aTP_Lamp_ace(fun(nat,real),fun(nat,fun(nat,real)),F2),M9),at_top(nat))
     => ( ! [M: nat,N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N)),aa(nat,real,F2,M)) ) )
       => topolo6863149650580417670ergent(real,F2) ) ) ).

% Bseq_monoseq_convergent'_dec
tff(fact_6988_convergent__minus__iff,axiom,
    ! [A: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [X6: fun(nat,A)] :
          ( topolo6863149650580417670ergent(A,X6)
        <=> topolo6863149650580417670ergent(A,aTP_Lamp_acf(fun(nat,A),fun(nat,A),X6)) ) ) ).

% convergent_minus_iff
tff(fact_6989_VEBT_Osimps_I8_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),F22: fun($o,fun($o,A)),X21: $o,X22: $o] : ( aa(vEBT_VEBT,A,aa(fun($o,fun($o,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,A)),vEBT_rec_VEBT(A),F1),F22),vEBT_Leaf((X21),(X22))) = aa($o,A,aa($o,fun($o,A),F22,(X21)),(X22)) ) ).

% VEBT.simps(8)
tff(fact_6990_convergent__mult__const__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_uz(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
          <=> topolo6863149650580417670ergent(A,F2) ) ) ) ).

% convergent_mult_const_right_iff
tff(fact_6991_convergent__mult__const__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_uy(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
          <=> topolo6863149650580417670ergent(A,F2) ) ) ) ).

% convergent_mult_const_iff
tff(fact_6992_convergent__Suc__iff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A)] :
          ( topolo6863149650580417670ergent(A,aTP_Lamp_vb(fun(nat,A),fun(nat,A),F2))
        <=> topolo6863149650580417670ergent(A,F2) ) ) ).

% convergent_Suc_iff
tff(fact_6993_convergent__add__const__right__iff,axiom,
    ! [A: $tType] :
      ( topolo1287966508704411220up_add(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( topolo6863149650580417670ergent(A,aa(A,fun(nat,A),aTP_Lamp_acg(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> topolo6863149650580417670ergent(A,F2) ) ) ).

% convergent_add_const_right_iff
tff(fact_6994_convergent__add__const__iff,axiom,
    ! [A: $tType] :
      ( topolo1287966508704411220up_add(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ach(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
        <=> topolo6863149650580417670ergent(A,F2) ) ) ).

% convergent_add_const_iff
tff(fact_6995_convergent__add,axiom,
    ! [A: $tType] :
      ( topolo6943815403480290642id_add(A)
     => ! [X6: fun(nat,A),Y6: fun(nat,A)] :
          ( topolo6863149650580417670ergent(A,X6)
         => ( topolo6863149650580417670ergent(A,Y6)
           => topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aci(fun(nat,A),fun(fun(nat,A),fun(nat,A)),X6),Y6)) ) ) ) ).

% convergent_add
tff(fact_6996_convergent__diff__const__right__iff,axiom,
    ! [A: $tType] :
      ( topolo1287966508704411220up_add(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( topolo6863149650580417670ergent(A,aa(A,fun(nat,A),aTP_Lamp_acj(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> topolo6863149650580417670ergent(A,F2) ) ) ).

% convergent_diff_const_right_iff
tff(fact_6997_convergent__diff,axiom,
    ! [A: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [X6: fun(nat,A),Y6: fun(nat,A)] :
          ( topolo6863149650580417670ergent(A,X6)
         => ( topolo6863149650580417670ergent(A,Y6)
           => topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ack(fun(nat,A),fun(fun(nat,A),fun(nat,A)),X6),Y6)) ) ) ) ).

% convergent_diff
tff(fact_6998_convergent__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),Ma: nat] :
          ( topolo6863149650580417670ergent(A,aa(nat,fun(nat,A),aTP_Lamp_vc(fun(nat,A),fun(nat,fun(nat,A)),F2),Ma))
        <=> topolo6863149650580417670ergent(A,F2) ) ) ).

% convergent_ignore_initial_segment
tff(fact_6999_convergent__realpow,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => topolo6863149650580417670ergent(real,aa(real,fun(nat,real),power_power(real),Xb)) ) ) ).

% convergent_realpow
tff(fact_7000_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F2: fun(nat,real),M9: nat] :
      ( bfun(nat,real,aa(nat,fun(nat,real),aTP_Lamp_ace(fun(nat,real),fun(nat,fun(nat,real)),F2),M9),at_top(nat))
     => ( ! [M: nat,N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,M)),aa(nat,real,F2,N)) ) )
       => topolo6863149650580417670ergent(real,F2) ) ) ).

% Bseq_monoseq_convergent'_inc
tff(fact_7001_pair__lessI2,axiom,
    ! [A2: nat,B2: nat,S: nat,Ta: 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),S),Ta)
       => member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A2),S)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,B2),Ta)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_7002_pair__less__iff1,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xb),Y)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xb),Z)),fun_pair_less)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Z) ) ).

% pair_less_iff1
tff(fact_7003_pair__lessI1,axiom,
    ! [A2: nat,B2: nat,S: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A2),S)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,B2),Ta)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_7004_pair__leqI1,axiom,
    ! [A2: nat,B2: nat,S: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A2),S)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,B2),Ta)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_7005_bot_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ordering_top(A,aTP_Lamp_acl(A,fun(A,$o)),aTP_Lamp_acm(A,fun(A,$o)),bot_bot(A)) ) ).

% bot.ordering_top_axioms
tff(fact_7006_gcd__nat_Oordering__top__axioms,axiom,
    ordering_top(nat,dvd_dvd(nat),aTP_Lamp_abc(nat,fun(nat,$o)),zero_zero(nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_7007_top_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ordering_top(A,ord_less_eq(A),ord_less(A),top_top(A)) ) ).

% top.ordering_top_axioms
tff(fact_7008_bot__nat__0_Oordering__top__axioms,axiom,
    ordering_top(nat,aTP_Lamp_ab(nat,fun(nat,$o)),aTP_Lamp_aa(nat,fun(nat,$o)),zero_zero(nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_7009_set__encode__vimage__Suc,axiom,
    ! [A3: set(nat)] : ( aa(set(nat),nat,nat_set_encode,vimage(nat,nat,suc,A3)) = divide_divide(nat,aa(set(nat),nat,nat_set_encode,A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% set_encode_vimage_Suc
tff(fact_7010_euclidean__size__times__nonunit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),B2)),aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ) ) ).

% euclidean_size_times_nonunit
tff(fact_7011_euclidean__size__of__nat,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : ( aa(A,nat,euclid6346220572633701492n_size(A),aa(nat,A,semiring_1_of_nat(A),Nb)) = Nb ) ) ).

% euclidean_size_of_nat
tff(fact_7012_size__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ( aa(A,nat,euclid6346220572633701492n_size(A),zero_zero(A)) = zero_zero(nat) ) ) ).

% size_0
tff(fact_7013_euclidean__size__eq__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A] :
          ( ( aa(A,nat,euclid6346220572633701492n_size(A),B2) = zero_zero(nat) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% euclidean_size_eq_0_iff
tff(fact_7014_euclidean__size__1,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( aa(A,nat,euclid6346220572633701492n_size(A),one_one(A)) = one_one(nat) ) ) ).

% euclidean_size_1
tff(fact_7015_euclidean__size__greater__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,euclid6346220572633701492n_size(A),B2))
        <=> ( B2 != zero_zero(A) ) ) ) ).

% euclidean_size_greater_0_iff
tff(fact_7016_vimage__if,axiom,
    ! [B: $tType,A: $tType,B3: set(A),C2: B,D3: B,A3: set(B)] :
      ( vimage(A,B,aa(B,fun(A,B),aa(B,fun(B,fun(A,B)),aTP_Lamp_acn(set(A),fun(B,fun(B,fun(A,B))),B3),C2),D3),A3) = $ite(
          member(B,C2,A3),
          $ite(member(B,D3,A3),top_top(set(A)),B3),
          $ite(member(B,D3,A3),aa(set(A),set(A),uminus_uminus(set(A)),B3),bot_bot(set(A))) ) ) ).

% vimage_if
tff(fact_7017_dvd__euclidean__size__eq__imp__dvd,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( aa(A,nat,euclid6346220572633701492n_size(A),A2) = aa(A,nat,euclid6346220572633701492n_size(A),B2) )
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ) ).

% dvd_euclidean_size_eq_imp_dvd
tff(fact_7018_vimage__Suc__insert__Suc,axiom,
    ! [Nb: nat,A3: set(nat)] : ( vimage(nat,nat,suc,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(nat,nat,suc,Nb)),A3)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),Nb),vimage(nat,nat,suc,A3)) ) ).

% vimage_Suc_insert_Suc
tff(fact_7019_finite__vimage__Suc__iff,axiom,
    ! [F4: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),vimage(nat,nat,suc,F4))
    <=> aa(set(nat),$o,finite_finite(nat),F4) ) ).

% finite_vimage_Suc_iff
tff(fact_7020_euclidean__size__unit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,nat,euclid6346220572633701492n_size(A),A2) = aa(A,nat,euclid6346220572633701492n_size(A),one_one(A)) ) ) ) ).

% euclidean_size_unit
tff(fact_7021_vimage__Compl,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] : ( vimage(A,B,F2,aa(set(B),set(B),uminus_uminus(set(B)),A3)) = aa(set(A),set(A),uminus_uminus(set(A)),vimage(A,B,F2,A3)) ) ).

% vimage_Compl
tff(fact_7022_vimage__Suc__insert__0,axiom,
    ! [A3: set(nat)] : ( vimage(nat,nat,suc,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),A3)) = vimage(nat,nat,suc,A3) ) ).

% vimage_Suc_insert_0
tff(fact_7023_unit__iff__euclidean__size,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
        <=> ( ( aa(A,nat,euclid6346220572633701492n_size(A),A2) = aa(A,nat,euclid6346220572633701492n_size(A),one_one(A)) )
            & ( A2 != zero_zero(A) ) ) ) ) ).

% unit_iff_euclidean_size
tff(fact_7024_size__mult__mono_H,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,euclid6346220572633701492n_size(A),A2)),aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2))) ) ) ).

% size_mult_mono'
tff(fact_7025_size__mult__mono,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,euclid6346220572633701492n_size(A),A2)),aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ).

% size_mult_mono
tff(fact_7026_euclidean__size__times__unit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,nat,euclid6346220572633701492n_size(A),B2) ) ) ) ).

% euclidean_size_times_unit
tff(fact_7027_dvd__proper__imp__size__less,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
           => ( ( B2 != zero_zero(A) )
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),A2)),aa(A,nat,euclid6346220572633701492n_size(A),B2)) ) ) ) ) ).

% dvd_proper_imp_size_less
tff(fact_7028_dvd__imp__size__le,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( ( B2 != zero_zero(A) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,euclid6346220572633701492n_size(A),A2)),aa(A,nat,euclid6346220572633701492n_size(A),B2)) ) ) ) ).

% dvd_imp_size_le
tff(fact_7029_mod__size__less,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),modulo_modulo(A,A2,B2))),aa(A,nat,euclid6346220572633701492n_size(A),B2)) ) ) ).

% mod_size_less
tff(fact_7030_euclidean__size__int__def,axiom,
    euclid6346220572633701492n_size(int) = aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int)) ).

% euclidean_size_int_def
tff(fact_7031_set__decode__div__2,axiom,
    ! [Xb: nat] : ( nat_set_decode(divide_divide(nat,Xb,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = vimage(nat,nat,suc,nat_set_decode(Xb)) ) ).

% set_decode_div_2
tff(fact_7032_divmod__cases,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,A2: A] :
          ( ( ( B2 != zero_zero(A) )
           => ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
             => ( A2 != aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),B2) ) ) )
         => ( ( ( B2 != zero_zero(A) )
             => ! [Q2: A,R4: A] :
                  ( ( euclid7384307370059645450egment(A,R4) = euclid7384307370059645450egment(A,B2) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R4)),aa(A,nat,euclid6346220572633701492n_size(A),B2))
                   => ( ( R4 != zero_zero(A) )
                     => ( ( divide_divide(A,A2,B2) = Q2 )
                       => ( ( modulo_modulo(A,A2,B2) = R4 )
                         => ( A2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q2),B2)),R4) ) ) ) ) ) ) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% divmod_cases
tff(fact_7033_mod__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R2: A,Q4: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B2) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R2)),aa(A,nat,euclid6346220572633701492n_size(A),B2))
             => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q4),B2)),R2) = A2 )
               => ( modulo_modulo(A,A2,B2) = R2 ) ) ) ) ) ) ).

% mod_eqI
tff(fact_7034_abs__division__segment,axiom,
    ! [K: int] : ( aa(int,int,abs_abs(int),euclid7384307370059645450egment(int,K)) = one_one(int) ) ).

% abs_division_segment
tff(fact_7035_division__segment__1,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( euclid7384307370059645450egment(A,one_one(A)) = one_one(A) ) ) ).

% division_segment_1
tff(fact_7036_division__segment__numeral,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [K: num] : ( euclid7384307370059645450egment(A,aa(num,A,numeral_numeral(A),K)) = one_one(A) ) ) ).

% division_segment_numeral
tff(fact_7037_division__segment__of__nat,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : ( euclid7384307370059645450egment(A,aa(nat,A,semiring_1_of_nat(A),Nb)) = one_one(A) ) ) ).

% division_segment_of_nat
tff(fact_7038_division__segment__euclidean__size,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),euclid7384307370059645450egment(A,A2)),aa(nat,A,semiring_1_of_nat(A),aa(A,nat,euclid6346220572633701492n_size(A),A2))) = A2 ) ) ).

% division_segment_euclidean_size
tff(fact_7039_is__unit__division__segment,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),euclid7384307370059645450egment(A,A2)),one_one(A)) ) ).

% is_unit_division_segment
tff(fact_7040_division__segment__eq__sgn,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( euclid7384307370059645450egment(int,K) = aa(int,int,sgn_sgn(int),K) ) ) ).

% division_segment_eq_sgn
tff(fact_7041_division__segment__not__0,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A] : ( euclid7384307370059645450egment(A,A2) != zero_zero(A) ) ) ).

% division_segment_not_0
tff(fact_7042_division__segment__nat__def,axiom,
    ! [Nb: nat] : ( euclid7384307370059645450egment(nat,Nb) = one_one(nat) ) ).

% division_segment_nat_def
tff(fact_7043_division__segment__mult,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( euclid7384307370059645450egment(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),euclid7384307370059645450egment(A,A2)),euclid7384307370059645450egment(A,B2)) ) ) ) ) ).

% division_segment_mult
tff(fact_7044_division__segment__mod,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
           => ( euclid7384307370059645450egment(A,modulo_modulo(A,A2,B2)) = euclid7384307370059645450egment(A,B2) ) ) ) ) ).

% division_segment_mod
tff(fact_7045_of__nat__euclidean__size,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] : ( aa(nat,A,semiring_1_of_nat(A),aa(A,nat,euclid6346220572633701492n_size(A),A2)) = divide_divide(A,A2,euclid7384307370059645450egment(A,A2)) ) ) ).

% of_nat_euclidean_size
tff(fact_7046_division__segment__int__def,axiom,
    ! [K: int] :
      ( euclid7384307370059645450egment(int,K) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ) ).

% division_segment_int_def
tff(fact_7047_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A] :
          ( ( euclid7384307370059645450egment(A,A2) = euclid7384307370059645450egment(A,B2) )
         => ( ( divide_divide(A,A2,B2) = zero_zero(A) )
          <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),A2)),aa(A,nat,euclid6346220572633701492n_size(A),B2))
              | ( B2 = zero_zero(A) ) ) ) ) ) ).

% unique_euclidean_semiring_class.div_eq_0_iff
tff(fact_7048_div__bounded,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R2: A,Q4: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B2) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R2)),aa(A,nat,euclid6346220572633701492n_size(A),B2))
             => ( 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),Q4),B2)),R2),B2) = Q4 ) ) ) ) ) ).

% div_bounded
tff(fact_7049_div__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R2: A,Q4: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B2) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R2)),aa(A,nat,euclid6346220572633701492n_size(A),B2))
             => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q4),B2)),R2) = A2 )
               => ( divide_divide(A,A2,B2) = Q4 ) ) ) ) ) ) ).

% div_eqI
tff(fact_7050_inv__image__partition,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Ys2: list(A)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X) )
     => ( ! [Y3: A] :
            ( member(A,Y3,aa(list(A),set(A),set2(A),Ys2))
           => ~ aa(A,$o,P,Y3) )
       => ( vimage(list(A),product_prod(list(A),list(A)),partition(A,P),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A)))),insert2(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2)),bot_bot(set(product_prod(list(A),list(A)))))) = shuffles(A,Xs,Ys2) ) ) ) ).

% inv_image_partition
tff(fact_7051_subseqs__powset,axiom,
    ! [A: $tType,Xs: list(A)] : ( image(list(A),set(A),set2(A),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))) = pow2(A,aa(list(A),set(A),set2(A),Xs)) ) ).

% subseqs_powset
tff(fact_7052_Pow__Compl,axiom,
    ! [A: $tType,A3: set(A)] : ( pow2(A,aa(set(A),set(A),uminus_uminus(set(A)),A3)) = collect(set(A),aTP_Lamp_aco(set(A),fun(set(A),$o),A3)) ) ).

% Pow_Compl
tff(fact_7053_partition__P,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Yes: list(A),No4: list(A)] :
      ( ( aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs) = aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Yes),No4) )
     => ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Yes))
           => aa(A,$o,P,X3) )
        & ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),No4))
           => ~ aa(A,$o,P,X3) ) ) ) ).

% partition_P
tff(fact_7054_Pow__set_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( pow2(A,aa(list(A),set(A),set2(A),cons(A,Xb,Xs))) = $let(
          a3: set(set(A)),
          a3:= pow2(A,aa(list(A),set(A),set2(A),Xs)),
          aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),a3),image(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),a3)) ) ) ).

% Pow_set(2)
tff(fact_7055_Pow__set_I1_J,axiom,
    ! [A: $tType] : ( pow2(A,aa(list(A),set(A),set2(A),nil(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ) ).

% Pow_set(1)
tff(fact_7056_binomial__def,axiom,
    ! [Nb: nat,K: nat] : ( aa(nat,nat,binomial(Nb),K) = aa(set(set(nat)),nat,finite_card(set(nat)),collect(set(nat),aa(nat,fun(set(nat),$o),aTP_Lamp_acp(nat,fun(nat,fun(set(nat),$o)),Nb),K))) ) ).

% binomial_def
tff(fact_7057_partition__set,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Yes: list(A),No4: list(A)] :
      ( ( aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs) = aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Yes),No4) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Yes)),aa(list(A),set(A),set2(A),No4)) = aa(list(A),set(A),set2(A),Xs) ) ) ).

% partition_set
tff(fact_7058_lexn_Osimps_I1_J,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : ( lexn(A,R2,zero_zero(nat)) = bot_bot(set(product_prod(list(A),list(A)))) ) ).

% lexn.simps(1)
tff(fact_7059_lists__length__Suc__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] : ( collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_acq(set(A),fun(nat,fun(list(A),$o)),A3),Nb)) = image(product_prod(list(A),A),list(A),product_case_prod(list(A),A,list(A),aTP_Lamp_ol(list(A),fun(A,list(A)))),product_Sigma(list(A),A,collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_in(set(A),fun(nat,fun(list(A),$o)),A3),Nb)),aTP_Lamp_acr(set(A),fun(list(A),set(A)),A3))) ) ).

% lists_length_Suc_eq
tff(fact_7060_sorted__wrt__iff__nth__Suc__transp,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A)] :
      ( transp(A,P)
     => ( sorted_wrt(A,P,Xs)
      <=> ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I3))) ) ) ) ).

% sorted_wrt_iff_nth_Suc_transp
tff(fact_7061_mlex__eq,axiom,
    ! [A: $tType,F2: fun(A,nat),R: set(product_prod(A,A))] : ( mlex_prod(A,F2,R) = collect(product_prod(A,A),product_case_prod(A,A,$o,aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_acs(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),F2),R))) ) ).

% mlex_eq
tff(fact_7062_transp__realrel,axiom,
    transp(fun(nat,rat),realrel) ).

% transp_realrel
tff(fact_7063_mlex__less,axiom,
    ! [A: $tType,F2: fun(A,nat),Xb: A,Y: A,R: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y))
     => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),mlex_prod(A,F2,R)) ) ).

% mlex_less
tff(fact_7064_mlex__iff,axiom,
    ! [A: $tType,Xb: A,Y: A,F2: fun(A,nat),R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),mlex_prod(A,F2,R))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y))
        | ( ( aa(A,nat,F2,Xb) = aa(A,nat,F2,Y) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),R) ) ) ) ).

% mlex_iff
tff(fact_7065_in__measure,axiom,
    ! [A: $tType,Xb: A,Y: A,F2: fun(A,nat)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),measure(A,F2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y)) ) ).

% in_measure
tff(fact_7066_transp__gr,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,aTP_Lamp_act(A,fun(A,$o))) ) ).

% transp_gr
tff(fact_7067_transp__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,ord_less(A)) ) ).

% transp_less
tff(fact_7068_pred__nat__def,axiom,
    pred_nat = collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_acu(nat,fun(nat,$o)))) ).

% pred_nat_def
tff(fact_7069_lexordp__conv__lexord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys2)
        <=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2),lexord(A,collect(product_prod(A,A),product_case_prod(A,A,$o,ord_less(A))))) ) ) ).

% lexordp_conv_lexord
tff(fact_7070_lexordp__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Xs: list(A),Y: A,Ys2: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),cons(A,Xb,Xs)),cons(A,Y,Ys2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
            | ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
              & aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys2) ) ) ) ) ).

% lexordp_simps(3)
tff(fact_7071_lexordp_OCons__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A,Xs: list(A),Ys2: list(A)] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
           => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys2)
             => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),cons(A,Xb,Xs)),cons(A,Y,Ys2)) ) ) ) ) ).

% lexordp.Cons_eq
tff(fact_7072_lexordp_OCons,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A,Xs: list(A),Ys2: list(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),cons(A,Xb,Xs)),cons(A,Y,Ys2)) ) ) ).

% lexordp.Cons
tff(fact_7073_lexordp__append__leftD,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A),Us3: list(A),Vs3: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),append(A,Xs,Us3)),append(A,Xs,Vs3))
         => ( ! [A4: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),A4)
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Us3),Vs3) ) ) ) ).

% lexordp_append_leftD
tff(fact_7074_lexordp__irreflexive,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A)] :
          ( ! [X: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X)
         => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Xs) ) ) ).

% lexordp_irreflexive
tff(fact_7075_lexordp_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),A1),A22)
         => ( ( ( A1 = nil(A) )
             => ! [Y3: A,Ys3: list(A)] : ( A22 != cons(A,Y3,Ys3) ) )
           => ( ! [X: A] :
                  ( ? [Xs2: list(A)] : ( A1 = cons(A,X,Xs2) )
                 => ! [Y3: A] :
                      ( ? [Ys3: list(A)] : ( A22 = cons(A,Y3,Ys3) )
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3) ) )
             => ~ ! [X: A,Y3: A,Xs2: list(A)] :
                    ( ( A1 = cons(A,X,Xs2) )
                   => ! [Ys3: list(A)] :
                        ( ( A22 = cons(A,Y3,Ys3) )
                       => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
                         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X)
                           => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs2),Ys3) ) ) ) ) ) ) ) ) ).

% lexordp.cases
tff(fact_7076_lexordp_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),A1),A22)
        <=> ( ? [Y5: A,Ys4: list(A)] :
                ( ( A1 = nil(A) )
                & ( A22 = cons(A,Y5,Ys4) ) )
            | ? [X4: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A1 = cons(A,X4,Xs3) )
                & ( A22 = cons(A,Y5,Ys4) )
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y5) )
            | ? [X4: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A1 = cons(A,X4,Xs3) )
                & ( A22 = cons(A,Y5,Ys4) )
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y5)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X4)
                & aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs3),Ys4) ) ) ) ) ).

% lexordp.simps
tff(fact_7077_lexordp__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys2)
         => ( ( ( Xs = nil(A) )
             => ! [Y3: A,Ys6: list(A)] : ( Ys2 != cons(A,Y3,Ys6) ) )
           => ( ! [X: A] :
                  ( ? [Xs5: list(A)] : ( Xs = cons(A,X,Xs5) )
                 => ! [Y3: A] :
                      ( ? [Ys6: list(A)] : ( Ys2 = cons(A,Y3,Ys6) )
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3) ) )
             => ~ ! [X: A,Xs5: list(A)] :
                    ( ( Xs = cons(A,X,Xs5) )
                   => ! [Ys6: list(A)] :
                        ( ( Ys2 = cons(A,X,Ys6) )
                       => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs5),Ys6) ) ) ) ) ) ) ).

% lexordp_cases
tff(fact_7078_lexordp__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A),P: fun(list(A),fun(list(A),$o))] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys2)
         => ( ! [Y3: A,Ys3: list(A)] : aa(list(A),$o,aa(list(A),fun(list(A),$o),P,nil(A)),cons(A,Y3,Ys3))
           => ( ! [X: A,Xs2: list(A),Y3: A,Ys3: list(A)] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
                 => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,cons(A,X,Xs2)),cons(A,Y3,Ys3)) )
             => ( ! [X: A,Xs2: list(A),Ys3: list(A)] :
                    ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs2),Ys3)
                   => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Xs2),Ys3)
                     => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,cons(A,X,Xs2)),cons(A,X,Ys3)) ) )
               => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Xs),Ys2) ) ) ) ) ) ).

% lexordp_induct
tff(fact_7079_lexordp__append__left__rightI,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A,Us3: list(A),Xs: list(A),Ys2: list(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),append(A,Us3,cons(A,Xb,Xs))),append(A,Us3,cons(A,Y,Ys2))) ) ) ).

% lexordp_append_left_rightI
tff(fact_7080_lexordp__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys2)
        <=> ( ? [X4: A,Vs2: list(A)] : ( Ys2 = append(A,Xs,cons(A,X4,Vs2)) )
            | ? [Us2: list(A),A5: A,B5: A,Vs2: list(A),Ws: list(A)] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A5),B5)
                & ( Xs = append(A,Us2,cons(A,A5,Vs2)) )
                & ( Ys2 = append(A,Us2,cons(A,B5,Ws)) ) ) ) ) ) ).

% lexordp_iff
tff(fact_7081_ord__class_Olexordp__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( ord_lexordp(A) = complete_lattice_lfp(fun(list(A),fun(list(A),$o)),aTP_Lamp_pd(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)))) ) ) ).

% ord_class.lexordp_def
tff(fact_7082_less__enat__def,axiom,
    ! [Ma: extended_enat,Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Ma),Nb)
    <=> extended_case_enat($o,aTP_Lamp_acv(extended_enat,fun(nat,$o),Nb),$false,Ma) ) ).

% less_enat_def
tff(fact_7083_lfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),Nb: nat] :
          ( order_mono(A,A,F2)
         => ( complete_lattice_lfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Nb)),F2)) = complete_lattice_lfp(A,F2) ) ) ) ).

% lfp_funpow
tff(fact_7084_lfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),K: nat] :
          ( order_mono(A,A,F2)
         => ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,K)),F2),bot_bot(A)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),bot_bot(A)) )
           => ( complete_lattice_lfp(A,F2) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),bot_bot(A)) ) ) ) ) ).

% lfp_Kleene_iter
tff(fact_7085_bounded__bilinear__def,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C))] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
        <=> ( ! [A5: A,A11: A,B5: B] : ( aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(A,fun(A,A),plus_plus(A),A5),A11)),B5) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A5),B5)),aa(B,C,aa(A,fun(B,C),Prod,A11),B5)) )
            & ! [A5: A,B5: B,B11: B] : ( aa(B,C,aa(A,fun(B,C),Prod,A5),aa(B,B,aa(B,fun(B,B),plus_plus(B),B5),B11)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A5),B5)),aa(B,C,aa(A,fun(B,C),Prod,A5),B11)) )
            & ! [R5: real,A5: A,B5: B] : ( aa(B,C,aa(A,fun(B,C),Prod,real_V8093663219630862766scaleR(A,R5,A5)),B5) = real_V8093663219630862766scaleR(C,R5,aa(B,C,aa(A,fun(B,C),Prod,A5),B5)) )
            & ! [A5: A,R5: real,B5: B] : ( aa(B,C,aa(A,fun(B,C),Prod,A5),real_V8093663219630862766scaleR(B,R5,B5)) = real_V8093663219630862766scaleR(C,R5,aa(B,C,aa(A,fun(B,C),Prod,A5),B5)) )
            & ? [K5: real] :
              ! [A5: A,B5: B] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(B,C,aa(A,fun(B,C),Prod,A5),B5))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A5)),real_V7770717601297561774m_norm(B,B5))),K5)) ) ) ) ).

% bounded_bilinear_def
tff(fact_7086_bounded__bilinear_Ointro,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C))] :
          ( ! [A4: A,A10: A,B4: B] : ( aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),A10)),B4) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A4),B4)),aa(B,C,aa(A,fun(B,C),Prod,A10),B4)) )
         => ( ! [A4: A,B4: B,B7: B] : ( aa(B,C,aa(A,fun(B,C),Prod,A4),aa(B,B,aa(B,fun(B,B),plus_plus(B),B4),B7)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A4),B4)),aa(B,C,aa(A,fun(B,C),Prod,A4),B7)) )
           => ( ! [R4: real,A4: A,B4: B] : ( aa(B,C,aa(A,fun(B,C),Prod,real_V8093663219630862766scaleR(A,R4,A4)),B4) = real_V8093663219630862766scaleR(C,R4,aa(B,C,aa(A,fun(B,C),Prod,A4),B4)) )
             => ( ! [A4: A,R4: real,B4: B] : ( aa(B,C,aa(A,fun(B,C),Prod,A4),real_V8093663219630862766scaleR(B,R4,B4)) = real_V8093663219630862766scaleR(C,R4,aa(B,C,aa(A,fun(B,C),Prod,A4),B4)) )
               => ( ? [K8: real] :
                    ! [A4: A,B4: B] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(B,C,aa(A,fun(B,C),Prod,A4),B4))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A4)),real_V7770717601297561774m_norm(B,B4))),K8))
                 => real_V2442710119149674383linear(A,B,C,Prod) ) ) ) ) ) ) ).

% bounded_bilinear.intro
tff(fact_7087_bounded__bilinear_Ominus__left,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Prod: fun(A,fun(B,C)),A2: A,B2: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,uminus_uminus(A),A2)),B2) = aa(C,C,uminus_uminus(C),aa(B,C,aa(A,fun(B,C),Prod,A2),B2)) ) ) ) ).

% bounded_bilinear.minus_left
tff(fact_7088_bounded__bilinear_Ominus__right,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Prod: fun(A,fun(B,C)),A2: A,B2: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,A2),aa(B,B,uminus_uminus(B),B2)) = aa(C,C,uminus_uminus(C),aa(B,C,aa(A,fun(B,C),Prod,A2),B2)) ) ) ) ).

% bounded_bilinear.minus_right
tff(fact_7089_bounded__bilinear_Ozero__left,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Prod: fun(A,fun(B,C)),B2: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,zero_zero(A)),B2) = zero_zero(C) ) ) ) ).

% bounded_bilinear.zero_left
tff(fact_7090_bounded__bilinear_Ozero__right,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Prod: fun(A,fun(B,C)),A2: A] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,A2),zero_zero(B)) = zero_zero(C) ) ) ) ).

% bounded_bilinear.zero_right
tff(fact_7091_bounded__bilinear_Odiff__left,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Prod: fun(A,fun(B,C)),A2: A,A6: A,B2: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),A6)),B2) = aa(C,C,aa(C,fun(C,C),minus_minus(C),aa(B,C,aa(A,fun(B,C),Prod,A2),B2)),aa(B,C,aa(A,fun(B,C),Prod,A6),B2)) ) ) ) ).

% bounded_bilinear.diff_left
tff(fact_7092_bounded__bilinear_Odiff__right,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Prod: fun(A,fun(B,C)),A2: A,B2: B,B6: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,A2),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),B6)) = aa(C,C,aa(C,fun(C,C),minus_minus(C),aa(B,C,aa(A,fun(B,C),Prod,A2),B2)),aa(B,C,aa(A,fun(B,C),Prod,A2),B6)) ) ) ) ).

% bounded_bilinear.diff_right
tff(fact_7093_bounded__bilinear_Oadd__right,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Prod: fun(A,fun(B,C)),A2: A,B2: B,B6: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),B6)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A2),B2)),aa(B,C,aa(A,fun(B,C),Prod,A2),B6)) ) ) ) ).

% bounded_bilinear.add_right
tff(fact_7094_bounded__bilinear_Oadd__left,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Prod: fun(A,fun(B,C)),A2: A,A6: A,B2: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A6)),B2) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A2),B2)),aa(B,C,aa(A,fun(B,C),Prod,A6),B2)) ) ) ) ).

% bounded_bilinear.add_left
tff(fact_7095_bounded__bilinear_Oprod__diff__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Prod: fun(A,fun(B,C)),Xb: A,Y: B,A2: A,B2: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(C,C,aa(C,fun(C,C),minus_minus(C),aa(B,C,aa(A,fun(B,C),Prod,Xb),Y)),aa(B,C,aa(A,fun(B,C),Prod,A2),B2)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A2)),aa(B,B,aa(B,fun(B,B),minus_minus(B),Y),B2))),aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A2)),B2))),aa(B,C,aa(A,fun(B,C),Prod,A2),aa(B,B,aa(B,fun(B,B),minus_minus(B),Y),B2))) ) ) ) ).

% bounded_bilinear.prod_diff_prod
tff(fact_7096_bounded__bilinear_Otendsto__zero,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C)),F2: fun(D,A),F4: filter(D),G: fun(D,B)] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( filterlim(D,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => ( filterlim(D,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
             => filterlim(D,C,aa(fun(D,B),fun(D,C),aa(fun(D,A),fun(fun(D,B),fun(D,C)),aTP_Lamp_acw(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,B),fun(D,C))),Prod),F2),G),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ) ).

% bounded_bilinear.tendsto_zero
tff(fact_7097_bounded__bilinear_Otendsto__left__zero,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C)),F2: fun(D,A),F4: filter(D),C2: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( filterlim(D,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(D,C,aa(B,fun(D,C),aa(fun(D,A),fun(B,fun(D,C)),aTP_Lamp_acx(fun(A,fun(B,C)),fun(fun(D,A),fun(B,fun(D,C))),Prod),F2),C2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ).

% bounded_bilinear.tendsto_left_zero
tff(fact_7098_bounded__bilinear_Otendsto__right__zero,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C)),F2: fun(D,B),F4: filter(D),C2: A] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( filterlim(D,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(D,C,aa(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_acy(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Prod),F2),C2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ).

% bounded_bilinear.tendsto_right_zero
tff(fact_7099_bounded__bilinear_OFDERIV,axiom,
    ! [B: $tType,C: $tType,A: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [Prod: fun(A,fun(B,C)),F2: fun(D,A),F7: fun(D,A),Xb: D,S: set(D),G: fun(D,B),G2: fun(D,B)] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( has_derivative(D,A,F2,F7,topolo174197925503356063within(D,Xb,S))
           => ( has_derivative(D,B,G,G2,topolo174197925503356063within(D,Xb,S))
             => has_derivative(D,C,aa(fun(D,B),fun(D,C),aa(fun(D,A),fun(fun(D,B),fun(D,C)),aTP_Lamp_acz(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,B),fun(D,C))),Prod),F2),G),aa(fun(D,B),fun(D,C),aa(fun(D,B),fun(fun(D,B),fun(D,C)),aa(D,fun(fun(D,B),fun(fun(D,B),fun(D,C))),aa(fun(D,A),fun(D,fun(fun(D,B),fun(fun(D,B),fun(D,C)))),aa(fun(D,A),fun(fun(D,A),fun(D,fun(fun(D,B),fun(fun(D,B),fun(D,C))))),aTP_Lamp_ada(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,A),fun(D,fun(fun(D,B),fun(fun(D,B),fun(D,C)))))),Prod),F2),F7),Xb),G),G2),topolo174197925503356063within(D,Xb,S)) ) ) ) ) ).

% bounded_bilinear.FDERIV
tff(fact_7100_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_adb(int,fun(int,int))),Qr) ) ).

% Divides.adjust_div_def
tff(fact_7101_suminf__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,nat),F2: fun(nat,A)] :
          ( order_strict_mono(nat,nat,G)
         => ( ! [N: nat] :
                ( ~ member(nat,N,image(nat,nat,G,top_top(set(nat))))
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => ( suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_adc(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2)) = suminf(A,F2) ) ) ) ) ).

% suminf_mono_reindex
tff(fact_7102_strict__mono__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_strict_mono(nat,A,F2)
        <=> ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N4)),aa(nat,A,F2,aa(nat,nat,suc,N4))) ) ) ).

% strict_mono_Suc_iff
tff(fact_7103_strict__mono__add,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A] : order_strict_mono(A,A,aTP_Lamp_ls(A,fun(A,A),K)) ) ).

% strict_mono_add
tff(fact_7104_strict__mono__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_strict_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ) ).

% strict_mono_less
tff(fact_7105_strict__mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_strict_mono(A,B,F2)
        <=> ! [X4: A,Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,F2,Y5)) ) ) ) ).

% strict_mono_def
tff(fact_7106_strict__monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X: A,Y3: A] :
              ( aa(A,$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)) )
         => order_strict_mono(A,B,F2) ) ) ).

% strict_monoI
tff(fact_7107_strict__monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_strict_mono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)) ) ) ) ).

% strict_monoD
tff(fact_7108_strict__mono__imp__increasing,axiom,
    ! [F2: fun(nat,nat),Nb: nat] :
      ( order_strict_mono(nat,nat,F2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,F2,Nb)) ) ).

% strict_mono_imp_increasing
tff(fact_7109_summable__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [G: fun(nat,nat),F2: fun(nat,A)] :
          ( order_strict_mono(nat,nat,G)
         => ( ! [N: nat] :
                ( ~ member(nat,N,image(nat,nat,G,top_top(set(nat))))
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_add(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2))
            <=> summable(A,F2) ) ) ) ) ).

% summable_mono_reindex
tff(fact_7110_sums__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [G: fun(nat,nat),F2: fun(nat,A),C2: A] :
          ( order_strict_mono(nat,nat,G)
         => ( ! [N: nat] :
                ( ~ member(nat,N,image(nat,nat,G,top_top(set(nat))))
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_add(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2),C2)
            <=> sums(A,F2,C2) ) ) ) ) ).

% sums_mono_reindex
tff(fact_7111_prod__decode__triangle__add,axiom,
    ! [K: nat,Ma: nat] : ( aa(nat,product_prod(nat,nat),nat_prod_decode,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),Ma)) = aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K),Ma) ) ).

% prod_decode_triangle_add
tff(fact_7112_Max_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic4895041142388067077er_set(A,ord_max(A),aTP_Lamp_ade(A,fun(A,$o)),aTP_Lamp_adf(A,fun(A,$o))) ) ).

% Max.semilattice_order_set_axioms
tff(fact_7113_prod__decode__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa(nat,product_prod(nat,nat),nat_prod_decode,Xb) = aa(nat,product_prod(nat,nat),nat_prod_decode,Y) )
    <=> ( Xb = Y ) ) ).

% prod_decode_eq
tff(fact_7114_prod__decode__inverse,axiom,
    ! [Nb: nat] : ( aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),nat_prod_decode,Nb)) = Nb ) ).

% prod_decode_inverse
tff(fact_7115_prod__encode__inverse,axiom,
    ! [Xb: product_prod(nat,nat)] : ( aa(nat,product_prod(nat,nat),nat_prod_decode,aa(product_prod(nat,nat),nat,nat_prod_encode,Xb)) = Xb ) ).

% prod_encode_inverse
tff(fact_7116_inj__prod__decode,axiom,
    ! [A3: set(nat)] : inj_on(nat,product_prod(nat,nat),nat_prod_decode,A3) ).

% inj_prod_decode
tff(fact_7117_prod__decode__def,axiom,
    nat_prod_decode = nat_prod_decode_aux(zero_zero(nat)) ).

% prod_decode_def
tff(fact_7118_surj__prod__decode,axiom,
    image(nat,product_prod(nat,nat),nat_prod_decode,top_top(set(nat))) = top_top(set(product_prod(nat,nat))) ).

% surj_prod_decode
tff(fact_7119_bij__prod__decode,axiom,
    bij_betw(nat,product_prod(nat,nat),nat_prod_decode,top_top(set(nat)),top_top(set(product_prod(nat,nat)))) ).

% bij_prod_decode
tff(fact_7120_Inf__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => lattic4895041142388067077er_set(A,inf_inf(A),ord_less_eq(A),ord_less(A)) ) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_7121_Min_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic4895041142388067077er_set(A,ord_min(A),ord_less_eq(A),ord_less(A)) ) ).

% Min.semilattice_order_set_axioms
tff(fact_7122_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => lattic4895041142388067077er_set(A,sup_sup(A),aTP_Lamp_adg(A,fun(A,$o)),aTP_Lamp_adh(A,fun(A,$o))) ) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_7123_list__decode_Opinduct,axiom,
    ! [A0: nat,P: fun(nat,$o)] :
      ( accp(nat,nat_list_decode_rel,A0)
     => ( ( accp(nat,nat_list_decode_rel,zero_zero(nat))
         => aa(nat,$o,P,zero_zero(nat)) )
       => ( ! [N: nat] :
              ( accp(nat,nat_list_decode_rel,aa(nat,nat,suc,N))
             => ( ! [X3: nat,Y4: nat] :
                    ( ( aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X3),Y4) = aa(nat,product_prod(nat,nat),nat_prod_decode,N) )
                   => aa(nat,$o,P,Y4) )
               => aa(nat,$o,P,aa(nat,nat,suc,N)) ) )
         => aa(nat,$o,P,A0) ) ) ) ).

% list_decode.pinduct
tff(fact_7124_list__decode_Oelims,axiom,
    ! [Xb: nat,Y: list(nat)] :
      ( ( aa(nat,list(nat),nat_list_decode,Xb) = Y )
     => ( ( ( Xb = zero_zero(nat) )
         => ( Y != nil(nat) ) )
       => ~ ! [N: nat] :
              ( ( Xb = aa(nat,nat,suc,N) )
             => ( Y != aa(product_prod(nat,nat),list(nat),product_case_prod(nat,nat,list(nat),aTP_Lamp_adi(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) ) ) ) ) ).

% list_decode.elims
tff(fact_7125_list__decode__inverse,axiom,
    ! [Nb: nat] : ( aa(list(nat),nat,nat_list_encode,aa(nat,list(nat),nat_list_decode,Nb)) = Nb ) ).

% list_decode_inverse
tff(fact_7126_list__encode__inverse,axiom,
    ! [Xb: list(nat)] : ( aa(nat,list(nat),nat_list_decode,aa(list(nat),nat,nat_list_encode,Xb)) = Xb ) ).

% list_encode_inverse
tff(fact_7127_list__decode__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa(nat,list(nat),nat_list_decode,Xb) = aa(nat,list(nat),nat_list_decode,Y) )
    <=> ( Xb = Y ) ) ).

% list_decode_eq
tff(fact_7128_inj__list__decode,axiom,
    ! [A3: set(nat)] : inj_on(nat,list(nat),nat_list_decode,A3) ).

% inj_list_decode
tff(fact_7129_list__decode_Opsimps_I1_J,axiom,
    ( accp(nat,nat_list_decode_rel,zero_zero(nat))
   => ( aa(nat,list(nat),nat_list_decode,zero_zero(nat)) = nil(nat) ) ) ).

% list_decode.psimps(1)
tff(fact_7130_list__decode_Osimps_I1_J,axiom,
    aa(nat,list(nat),nat_list_decode,zero_zero(nat)) = nil(nat) ).

% list_decode.simps(1)
tff(fact_7131_list__decode_Opsimps_I2_J,axiom,
    ! [Nb: nat] :
      ( accp(nat,nat_list_decode_rel,aa(nat,nat,suc,Nb))
     => ( aa(nat,list(nat),nat_list_decode,aa(nat,nat,suc,Nb)) = aa(product_prod(nat,nat),list(nat),product_case_prod(nat,nat,list(nat),aTP_Lamp_adi(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,Nb)) ) ) ).

% list_decode.psimps(2)
tff(fact_7132_bij__list__decode,axiom,
    bij_betw(nat,list(nat),nat_list_decode,top_top(set(nat)),top_top(set(list(nat)))) ).

% bij_list_decode
tff(fact_7133_surj__list__decode,axiom,
    image(nat,list(nat),nat_list_decode,top_top(set(nat))) = top_top(set(list(nat))) ).

% surj_list_decode
tff(fact_7134_list__decode_Osimps_I2_J,axiom,
    ! [Nb: nat] : ( aa(nat,list(nat),nat_list_decode,aa(nat,nat,suc,Nb)) = aa(product_prod(nat,nat),list(nat),product_case_prod(nat,nat,list(nat),aTP_Lamp_adi(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,Nb)) ) ).

% list_decode.simps(2)
tff(fact_7135_list__decode_Opelims,axiom,
    ! [Xb: nat,Y: list(nat)] :
      ( ( aa(nat,list(nat),nat_list_decode,Xb) = Y )
     => ( accp(nat,nat_list_decode_rel,Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = nil(nat) )
             => ~ accp(nat,nat_list_decode_rel,zero_zero(nat)) ) )
         => ~ ! [N: nat] :
                ( ( Xb = aa(nat,nat,suc,N) )
               => ( ( Y = aa(product_prod(nat,nat),list(nat),product_case_prod(nat,nat,list(nat),aTP_Lamp_adi(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) )
                 => ~ accp(nat,nat_list_decode_rel,aa(nat,nat,suc,N)) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_7136_Gcd__fin__def,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( semiring_gcd_Gcd_fin(A) = bounde2362111253966948842tice_F(A,gcd_gcd(A),zero_zero(A),one_one(A)) ) ) ).

% Gcd_fin_def
tff(fact_7137_gen__length__def,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : ( aa(list(A),nat,gen_length(A,Nb),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% gen_length_def
tff(fact_7138_bounded__quasi__semilattice__set_OF_Ocong,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A] : ( bounde2362111253966948842tice_F(A,F2,Top,Bot) = bounde2362111253966948842tice_F(A,F2,Top,Bot) ) ).

% bounded_quasi_semilattice_set.F.cong
tff(fact_7139_gen__length__code_I2_J,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : ( aa(list(A),nat,gen_length(A,Nb),cons(A,Xb,Xs)) = aa(list(A),nat,gen_length(A,aa(nat,nat,suc,Nb)),Xs) ) ).

% gen_length_code(2)
tff(fact_7140_length__code,axiom,
    ! [A: $tType] : ( size_size(list(A)) = gen_length(A,zero_zero(nat)) ) ).

% length_code
tff(fact_7141_bounded__quasi__semilattice__set_Oinsert__remove,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A2: A,A3: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)) = aa(A,A,aa(A,fun(A,A),F2,A2),aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),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)),insert2(A),A2),bot_bot(set(A)))))) ) ) ).

% bounded_quasi_semilattice_set.insert_remove
tff(fact_7142_bounded__quasi__semilattice__set_Oremove,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A2: A,A3: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( member(A,A2,A3)
       => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A3) = aa(A,A,aa(A,fun(A,A),F2,A2),aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),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)),insert2(A),A2),bot_bot(set(A)))))) ) ) ) ).

% bounded_quasi_semilattice_set.remove
tff(fact_7143_bounded__quasi__semilattice__set_Onormalize,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( aa(A,A,Normalize,aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A3)) = aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A3) ) ) ).

% bounded_quasi_semilattice_set.normalize
tff(fact_7144_bounded__quasi__semilattice__set_Oin__idem,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A2: A,A3: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( member(A,A2,A3)
       => ( aa(A,A,aa(A,fun(A,A),F2,A2),aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A3)) = aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A3) ) ) ) ).

% bounded_quasi_semilattice_set.in_idem
tff(fact_7145_bounded__quasi__semilattice__set_Oempty,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),bot_bot(set(A))) = Top ) ) ).

% bounded_quasi_semilattice_set.empty
tff(fact_7146_bounded__quasi__semilattice__set_Oinfinite,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( ~ aa(set(A),$o,finite_finite(A),A3)
       => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A3) = Bot ) ) ) ).

% bounded_quasi_semilattice_set.infinite
tff(fact_7147_bounded__quasi__semilattice__set_Osubset,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),B3: set(A),A3: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
       => ( aa(A,A,aa(A,fun(A,A),F2,aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),B3)),aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A3)) = aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A3) ) ) ) ).

% bounded_quasi_semilattice_set.subset
tff(fact_7148_bounded__quasi__semilattice__set_Oinsert,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A2: A,A3: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)) = aa(A,A,aa(A,fun(A,A),F2,A2),aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A3)) ) ) ).

% bounded_quasi_semilattice_set.insert
tff(fact_7149_bounded__quasi__semilattice__set_Ounion,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: set(A),B3: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),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),F2,aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A3)),aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),B3)) ) ) ).

% bounded_quasi_semilattice_set.union
tff(fact_7150_bounded__quasi__semilattice__set_Oset__eq__fold,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),Xs: list(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,F2,Xs),Top) ) ) ).

% bounded_quasi_semilattice_set.set_eq_fold
tff(fact_7151_num__of__integer_Otransfer,axiom,
    aa(fun(code_integer,num),$o,aa(fun(int,num),fun(fun(code_integer,num),$o),bNF_rel_fun(int,code_integer,num,num,code_pcr_integer,fequal(num)),aa(fun(int,nat),fun(int,num),comp(nat,num,int,num_of_nat),nat2)),code_num_of_integer) ).

% num_of_integer.transfer
tff(fact_7152_compute__powr__real,axiom,
    ! [B2: real,I: real] :
      ( powr_real(B2,I) = $ite(
          aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),zero_zero(real)),
          abort(real,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$false,$true,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$true,$false,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($false,$false,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$true,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($false,$false,$false,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$false,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$true,$true,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$false,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($true,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,zero_zero(literal)))))))))))))))))))))))))))))))),aa(real,fun(product_unit,real),aTP_Lamp_adj(real,fun(real,fun(product_unit,real)),B2),I)),
          $ite(
            aa(int,real,ring_1_of_int(real),aa(real,int,archim6421214686448440834_floor(real),I)) = I,
            $ite(aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),I),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(int,nat,nat2,aa(real,int,archim6421214686448440834_floor(real),I))),divide_divide(real,one_one(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(int,nat,nat2,aa(real,int,archim6421214686448440834_floor(real),aa(real,real,uminus_uminus(real),I)))))),
            abort(real,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$false,$true,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$true,$false,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($false,$false,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$true,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($false,$false,$false,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$false,$true,$true,$false,$true,$false,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$true,$true,$false,$false,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$false,$false,$true,$true,$true,$true,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,zero_zero(literal)))))))))))))))))))))))))))))))))))),aa(real,fun(product_unit,real),aTP_Lamp_adj(real,fun(real,fun(product_unit,real)),B2),I)) ) ) ) ).

% compute_powr_real
tff(fact_7153_Code__Target__Nat_ONat_Otransfer,axiom,
    aa(fun(code_integer,nat),$o,aa(fun(int,nat),fun(fun(code_integer,nat),$o),bNF_rel_fun(int,code_integer,nat,nat,code_pcr_integer,fequal(nat)),nat2),code_Target_Nat) ).

% Code_Target_Nat.Nat.transfer
tff(fact_7154_gcd__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(code_integer,code_integer)),$o,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),$o),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),gcd_gcd(int)),gcd_gcd(code_integer)) ).

% gcd_integer.transfer
tff(fact_7155_nat__of__integer_Otransfer,axiom,
    aa(fun(code_integer,nat),$o,aa(fun(int,nat),fun(fun(code_integer,nat),$o),bNF_rel_fun(int,code_integer,nat,nat,code_pcr_integer,fequal(nat)),nat2),code_nat_of_integer) ).

% nat_of_integer.transfer
tff(fact_7156_plus__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(code_integer,code_integer)),$o,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),$o),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),plus_plus(int)),plus_plus(code_integer)) ).

% plus_integer.transfer
tff(fact_7157_minus__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(code_integer,code_integer)),$o,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),$o),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),minus_minus(int)),minus_minus(code_integer)) ).

% minus_integer.transfer
tff(fact_7158_less__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(code_integer,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(code_integer,fun(code_integer,$o)),$o),bNF_rel_fun(int,code_integer,fun(int,$o),fun(code_integer,$o),code_pcr_integer,bNF_rel_fun(int,code_integer,$o,$o,code_pcr_integer,fequal($o))),ord_less(int)),ord_less(code_integer)) ).

% less_integer.transfer
tff(fact_7159_uminus__integer_Otransfer,axiom,
    aa(fun(code_integer,code_integer),$o,aa(fun(int,int),fun(fun(code_integer,code_integer),$o),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),uminus_uminus(int)),uminus_uminus(code_integer)) ).

% uminus_integer.transfer
tff(fact_7160_one__integer_Otransfer,axiom,
    aa(code_integer,$o,aa(int,fun(code_integer,$o),code_pcr_integer,one_one(int)),one_one(code_integer)) ).

% one_integer.transfer
tff(fact_7161_zero__integer_Otransfer,axiom,
    aa(code_integer,$o,aa(int,fun(code_integer,$o),code_pcr_integer,zero_zero(int)),zero_zero(code_integer)) ).

% zero_integer.transfer
tff(fact_7162_int__of__integer__integer__of__nat,axiom,
    ! [Nb: nat] : ( aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,code_integer_of_nat,Nb)) = aa(nat,int,semiring_1_of_nat(int),Nb) ) ).

% int_of_integer_integer_of_nat
tff(fact_7163_integer__of__nat_Orep__eq,axiom,
    ! [Xb: nat] : ( aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,code_integer_of_nat,Xb)) = aa(nat,int,semiring_1_of_nat(int),Xb) ) ).

% integer_of_nat.rep_eq
tff(fact_7164_integer__of__nat__0,axiom,
    aa(nat,code_integer,code_integer_of_nat,zero_zero(nat)) = zero_zero(code_integer) ).

% integer_of_nat_0
tff(fact_7165_integer__of__nat_Oabs__eq,axiom,
    ! [Xb: nat] : ( aa(nat,code_integer,code_integer_of_nat,Xb) = aa(int,code_integer,code_integer_of_int,aa(nat,int,semiring_1_of_nat(int),Xb)) ) ).

% integer_of_nat.abs_eq
tff(fact_7166_integer__of__nat_Otransfer,axiom,
    aa(fun(nat,code_integer),$o,aa(fun(nat,int),fun(fun(nat,code_integer),$o),bNF_rel_fun(nat,nat,int,code_integer,fequal(nat),code_pcr_integer),semiring_1_of_nat(int)),code_integer_of_nat) ).

% integer_of_nat.transfer
tff(fact_7167_integer__of__nat__1,axiom,
    aa(nat,code_integer,code_integer_of_nat,one_one(nat)) = one_one(code_integer) ).

% integer_of_nat_1
tff(fact_7168_dup_Otransfer,axiom,
    aa(fun(code_integer,code_integer),$o,aa(fun(int,int),fun(fun(code_integer,code_integer),$o),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),aTP_Lamp_aai(int,int)),code_dup) ).

% dup.transfer
tff(fact_7169_integer__of__nat__def,axiom,
    code_integer_of_nat = aa(fun(nat,int),fun(nat,code_integer),map_fun(nat,nat,int,code_integer,id(nat),code_integer_of_int),semiring_1_of_nat(int)) ).

% integer_of_nat_def
tff(fact_7170_dup_Orep__eq,axiom,
    ! [Xb: code_integer] : ( aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,code_dup,Xb)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(code_integer,int,code_int_of_integer,Xb)),aa(code_integer,int,code_int_of_integer,Xb)) ) ).

% dup.rep_eq
tff(fact_7171_dup_Oabs__eq,axiom,
    ! [Xb: int] : ( aa(code_integer,code_integer,code_dup,aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xb)) ) ).

% dup.abs_eq
tff(fact_7172_Code__Numeral_Osub__code_I9_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,bit0(Ma)),aa(num,num,bit1,Nb)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,code_dup,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,Ma),Nb))),one_one(code_integer)) ) ).

% Code_Numeral.sub_code(9)
tff(fact_7173_Code__Numeral_Osub__code_I8_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,aa(num,num,bit1,Ma)),bit0(Nb)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,code_dup,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,Ma),Nb))),one_one(code_integer)) ) ).

% Code_Numeral.sub_code(8)
tff(fact_7174_sub_Orep__eq,axiom,
    ! [Xb: num,Xaa: num] : ( aa(code_integer,int,code_int_of_integer,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,Xb),Xaa)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Xb)),aa(num,int,numeral_numeral(int),Xaa)) ) ).

% sub.rep_eq
tff(fact_7175_sub_Oabs__eq,axiom,
    ! [Xaa: num,Xb: num] : ( aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,Xaa),Xb) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Xaa)),aa(num,int,numeral_numeral(int),Xb))) ) ).

% sub.abs_eq
tff(fact_7176_sub_Otransfer,axiom,
    aa(fun(num,fun(num,code_integer)),$o,aa(fun(num,fun(num,int)),fun(fun(num,fun(num,code_integer)),$o),bNF_rel_fun(num,num,fun(num,int),fun(num,code_integer),fequal(num),bNF_rel_fun(num,num,int,code_integer,fequal(num),code_pcr_integer)),aTP_Lamp_aaj(num,fun(num,int))),code_sub) ).

% sub.transfer
tff(fact_7177_card__def,axiom,
    ! [A: $tType] : ( finite_card(A) = finite_folding_F(A,nat,aTP_Lamp_adk(A,fun(nat,nat)),zero_zero(nat)) ) ).

% card_def
tff(fact_7178_Gcd__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A3) = $ite(aa(set(A),$o,finite_finite(A),A3),finite_fold(A,A,gcd_gcd(A),zero_zero(A),A3),one_one(A)) ) ) ).

% Gcd_fin.eq_fold
tff(fact_7179_card_Oeq__fold,axiom,
    ! [A: $tType,A3: set(A)] : ( aa(set(A),nat,finite_card(A),A3) = finite_fold(A,nat,aTP_Lamp_adk(A,fun(nat,nat)),zero_zero(nat),A3) ) ).

% card.eq_fold
tff(fact_7180_bounded__quasi__semilattice__set_Oeq__fold,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A3) = $ite(aa(set(A),$o,finite_finite(A),A3),finite_fold(A,A,F2,Top,A3),Bot) ) ) ).

% bounded_quasi_semilattice_set.eq_fold
tff(fact_7181_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,plus_plus(A)),G),zero_zero(A),A3) ) ) ).

% sum.eq_fold
tff(fact_7182_prod_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A3: set(B)] : ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,times_times(A)),G),one_one(A),A3) ) ) ).

% prod.eq_fold
tff(fact_7183_comp__fun__idem__on_Ofold__set__fold,axiom,
    ! [A: $tType,B: $tType,S3: set(A),F2: fun(A,fun(B,B)),Xs: list(A),Y: B] :
      ( finite673082921795544331dem_on(A,B,S3,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),S3)
       => ( finite_fold(A,B,F2,Y,aa(list(A),set(A),set2(A),Xs)) = aa(B,B,fold(A,B,F2,Xs),Y) ) ) ) ).

% comp_fun_idem_on.fold_set_fold
tff(fact_7184_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
    ! [A: $tType,B: $tType,S3: set(A),F2: fun(A,fun(B,B)),Xs: list(A),Y: B] :
      ( finite4664212375090638736ute_on(A,B,S3,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),S3)
       => ( finite_fold(A,B,F2,Y,aa(list(A),set(A),set2(A),Xs)) = aa(B,B,fold(A,B,F2,remdups(A,Xs)),Y) ) ) ) ).

% comp_fun_commute_on.fold_set_fold_remdups
tff(fact_7185_tendsto__add__Pair,axiom,
    ! [A: $tType] :
      ( topolo6943815403480290642id_add(A)
     => ! [A2: A,B2: A] : filterlim(product_prod(A,A),A,aTP_Lamp_adl(product_prod(A,A),A),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),prod_filter(A,A,topolo7230453075368039082e_nhds(A,A2),topolo7230453075368039082e_nhds(A,B2))) ) ).

% tendsto_add_Pair
tff(fact_7186_less__than__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xb),Y),less_than)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y) ) ).

% less_than_iff
tff(fact_7187_elimnum,axiom,
    ! [Info2: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info2,Dega,TreeLista,Summarya),Nb)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info2,Dega,TreeLista,Summarya),extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) = vEBT_Node(Info2,Dega,TreeLista,Summarya) ) ) ).

% elimnum
tff(fact_7188_at__top__to__right,axiom,
    at_top(real) = filtermap(real,real,inverse_inverse(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% at_top_to_right
tff(fact_7189_idiff__enat__0,axiom,
    ! [Nb: extended_enat] : ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),extended_enat2(zero_zero(nat))),Nb) = extended_enat2(zero_zero(nat)) ) ).

% idiff_enat_0
tff(fact_7190_idiff__enat__0__right,axiom,
    ! [Nb: extended_enat] : ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),Nb),extended_enat2(zero_zero(nat))) = Nb ) ).

% idiff_enat_0_right
tff(fact_7191_plus__enat__simps_I1_J,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),extended_enat2(Ma)),extended_enat2(Nb)) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)) ) ).

% plus_enat_simps(1)
tff(fact_7192_idiff__enat__enat,axiom,
    ! [A2: nat,B2: nat] : ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),extended_enat2(A2)),extended_enat2(B2)) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) ) ).

% idiff_enat_enat
tff(fact_7193_enat__ord__simps_I2_J,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extended_enat2(Ma)),extended_enat2(Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ).

% enat_ord_simps(2)
tff(fact_7194_numeral__less__enat__iff,axiom,
    ! [Ma: num,Nb: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),Ma)),extended_enat2(Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),Ma)),Nb) ) ).

% numeral_less_enat_iff
tff(fact_7195_zero__enat__def,axiom,
    zero_zero(extended_enat) = extended_enat2(zero_zero(nat)) ).

% zero_enat_def
tff(fact_7196_enat__0__iff_I1_J,axiom,
    ! [Xb: nat] :
      ( ( extended_enat2(Xb) = zero_zero(extended_enat) )
    <=> ( Xb = zero_zero(nat) ) ) ).

% enat_0_iff(1)
tff(fact_7197_enat__0__iff_I2_J,axiom,
    ! [Xb: nat] :
      ( ( zero_zero(extended_enat) = extended_enat2(Xb) )
    <=> ( Xb = zero_zero(nat) ) ) ).

% enat_0_iff(2)
tff(fact_7198_one__enat__def,axiom,
    one_one(extended_enat) = extended_enat2(one_one(nat)) ).

% one_enat_def
tff(fact_7199_enat__1__iff_I1_J,axiom,
    ! [Xb: nat] :
      ( ( extended_enat2(Xb) = one_one(extended_enat) )
    <=> ( Xb = one_one(nat) ) ) ).

% enat_1_iff(1)
tff(fact_7200_enat__1__iff_I2_J,axiom,
    ! [Xb: nat] :
      ( ( one_one(extended_enat) = extended_enat2(Xb) )
    <=> ( Xb = one_one(nat) ) ) ).

% enat_1_iff(2)
tff(fact_7201_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Uu: extended_enat] : ( vEBT_VEBT_elim_dead(vEBT_Leaf((A2),(B2)),Uu) = vEBT_Leaf((A2),(B2)) ) ).

% VEBT_internal.elim_dead.simps(1)
tff(fact_7202_less__enatE,axiom,
    ! [Nb: extended_enat,Ma: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Nb),extended_enat2(Ma))
     => ~ ! [K2: nat] :
            ( ( Nb = extended_enat2(K2) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),Ma) ) ) ).

% less_enatE
tff(fact_7203_filtermap__nhds__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [D3: A,A2: A] : ( filtermap(A,A,aTP_Lamp_adm(A,fun(A,A),D3),topolo7230453075368039082e_nhds(A,A2)) = topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D3)) ) ) ).

% filtermap_nhds_shift
tff(fact_7204_filtermap__nhds__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A] : ( filtermap(A,A,uminus_uminus(A),topolo7230453075368039082e_nhds(A,A2)) = topolo7230453075368039082e_nhds(A,aa(A,A,uminus_uminus(A),A2)) ) ) ).

% filtermap_nhds_minus
tff(fact_7205_iadd__le__enat__iff,axiom,
    ! [Xb: extended_enat,Y: extended_enat,Nb: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Xb),Y)),extended_enat2(Nb))
    <=> ? [Y7: nat,X9: nat] :
          ( ( Xb = extended_enat2(X9) )
          & ( Y = extended_enat2(Y7) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X9),Y7)),Nb) ) ) ).

% iadd_le_enat_iff
tff(fact_7206_Suc__ile__eq,axiom,
    ! [Ma: nat,Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),extended_enat2(aa(nat,nat,suc,Ma))),Nb)
    <=> aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extended_enat2(Ma)),Nb) ) ).

% Suc_ile_eq
tff(fact_7207_filtermap__nhds__times,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( filtermap(A,A,aa(A,fun(A,A),times_times(A),C2),topolo7230453075368039082e_nhds(A,A2)) = topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) ) ) ) ).

% filtermap_nhds_times
tff(fact_7208_at__bot__mirror,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add(A)
        & linorder(A) )
     => ( at_bot(A) = filtermap(A,A,uminus_uminus(A),at_top(A)) ) ) ).

% at_bot_mirror
tff(fact_7209_at__top__mirror,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add(A)
        & linorder(A) )
     => ( at_top(A) = filtermap(A,A,uminus_uminus(A),at_bot(A)) ) ) ).

% at_top_mirror
tff(fact_7210_filtermap__at__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [D3: A,A2: A] : ( filtermap(A,A,aTP_Lamp_adm(A,fun(A,A),D3),topolo174197925503356063within(A,A2,top_top(set(A)))) = topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D3),top_top(set(A))) ) ) ).

% filtermap_at_shift
tff(fact_7211_filtermap__at__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A] : ( filtermap(A,A,uminus_uminus(A),topolo174197925503356063within(A,A2,top_top(set(A)))) = topolo174197925503356063within(A,aa(A,A,uminus_uminus(A),A2),top_top(set(A))) ) ) ).

% filtermap_at_minus
tff(fact_7212_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,L: nat] : ( vEBT_VEBT_elim_dead(vEBT_Node(Info2,Dega,TreeLista,Summarya),extended_enat2(L)) = vEBT_Node(Info2,Dega,take(vEBT_VEBT,divide_divide(nat,L,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),list(vEBT_VEBT),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_adn(nat,fun(vEBT_VEBT,vEBT_VEBT),Dega)),TreeLista)),vEBT_VEBT_elim_dead(Summarya,extended_enat2(divide_divide(nat,L,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ).

% VEBT_internal.elim_dead.simps(3)
tff(fact_7213_at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A] : ( topolo174197925503356063within(A,A2,top_top(set(A))) = filtermap(A,A,aTP_Lamp_ado(A,fun(A,A),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% at_to_0
tff(fact_7214_at__left__minus,axiom,
    ! [A2: real] : ( topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)) = filtermap(real,real,uminus_uminus(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A2),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),A2)))) ) ).

% at_left_minus
tff(fact_7215_at__right__minus,axiom,
    ! [A2: real] : ( topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)) = filtermap(real,real,uminus_uminus(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A2),set_ord_lessThan(real,aa(real,real,uminus_uminus(real),A2)))) ) ).

% at_right_minus
tff(fact_7216_filtermap__times__pos__at__right,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [C2: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( filtermap(A,A,aa(A,fun(A,A),times_times(A),C2),topolo174197925503356063within(A,P3,set_ord_greaterThan(A,P3))) = topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),P3),set_ord_greaterThan(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),P3))) ) ) ) ).

% filtermap_times_pos_at_right
tff(fact_7217_at__to__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ( topolo174197925503356063within(A,zero_zero(A),top_top(set(A))) = filtermap(A,A,inverse_inverse(A),at_infinity(A)) ) ) ).

% at_to_infinity
tff(fact_7218_at__right__to__top,axiom,
    topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))) = filtermap(real,real,inverse_inverse(real),at_top(real)) ).

% at_right_to_top
tff(fact_7219_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(Xb,Xaa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( Y != vEBT_Leaf((A4),(B4)) ) )
       => ( ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info,Deg,TreeList,Summary) )
             => ( ( Xaa = extend4730790105801354508finity(extended_enat) )
               => ( Y != vEBT_Node(Info,Deg,aa(list(vEBT_VEBT),list(vEBT_VEBT),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_adn(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg)),TreeList),vEBT_VEBT_elim_dead(Summary,extend4730790105801354508finity(extended_enat))) ) ) )
         => ~ ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info,Deg,TreeList,Summary) )
               => ! [L2: nat] :
                    ( ( Xaa = extended_enat2(L2) )
                   => ( Y != vEBT_Node(Info,Deg,take(vEBT_VEBT,divide_divide(nat,L2,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),list(vEBT_VEBT),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_adn(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg)),TreeList)),vEBT_VEBT_elim_dead(Summary,extended_enat2(divide_divide(nat,L2,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
tff(fact_7220_elimcomplete,axiom,
    ! [Info2: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info2,Dega,TreeLista,Summarya),Nb)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info2,Dega,TreeLista,Summarya),extend4730790105801354508finity(extended_enat)) = vEBT_Node(Info2,Dega,TreeLista,Summarya) ) ) ).

% elimcomplete
tff(fact_7221_times__enat__simps_I3_J,axiom,
    ! [Nb: nat] :
      ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extend4730790105801354508finity(extended_enat)),extended_enat2(Nb)) = $ite(Nb = zero_zero(nat),zero_zero(extended_enat),extend4730790105801354508finity(extended_enat)) ) ).

% times_enat_simps(3)
tff(fact_7222_times__enat__simps_I4_J,axiom,
    ! [Ma: nat] :
      ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extended_enat2(Ma)),extend4730790105801354508finity(extended_enat)) = $ite(Ma = zero_zero(nat),zero_zero(extended_enat),extend4730790105801354508finity(extended_enat)) ) ).

% times_enat_simps(4)
tff(fact_7223_infinity__ne__i1,axiom,
    extend4730790105801354508finity(extended_enat) != one_one(extended_enat) ).

% infinity_ne_i1
tff(fact_7224_zero__one__enat__neq_I1_J,axiom,
    zero_zero(extended_enat) != one_one(extended_enat) ).

% zero_one_enat_neq(1)
tff(fact_7225_VEBT__internal_Oelim__dead_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,extended_enat)] :
      ( ! [A4: $o,B4: $o,Uu2: extended_enat] : ( Xb != aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Leaf((A4),(B4))),Uu2) )
     => ( ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : ( Xb != aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Node(Info,Deg,TreeList,Summary)),extend4730790105801354508finity(extended_enat)) )
       => ~ ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,L2: nat] : ( Xb != aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Node(Info,Deg,TreeList,Summary)),extended_enat2(L2)) ) ) ) ).

% VEBT_internal.elim_dead.cases
tff(fact_7226_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] : ( vEBT_VEBT_elim_dead(vEBT_Node(Info2,Dega,TreeLista,Summarya),extend4730790105801354508finity(extended_enat)) = vEBT_Node(Info2,Dega,aa(list(vEBT_VEBT),list(vEBT_VEBT),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_adn(nat,fun(vEBT_VEBT,vEBT_VEBT),Dega)),TreeLista),vEBT_VEBT_elim_dead(Summarya,extend4730790105801354508finity(extended_enat))) ) ).

% VEBT_internal.elim_dead.simps(2)
tff(fact_7227_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel,aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = vEBT_Leaf((A4),(B4)) )
               => ~ accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel,aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Leaf((A4),(B4))),Xaa)) ) )
         => ( ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info,Deg,TreeList,Summary) )
               => ( ( Xaa = extend4730790105801354508finity(extended_enat) )
                 => ( ( Y = vEBT_Node(Info,Deg,aa(list(vEBT_VEBT),list(vEBT_VEBT),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_adn(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg)),TreeList),vEBT_VEBT_elim_dead(Summary,extend4730790105801354508finity(extended_enat))) )
                   => ~ accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel,aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Node(Info,Deg,TreeList,Summary)),extend4730790105801354508finity(extended_enat))) ) ) )
           => ~ ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info,Deg,TreeList,Summary) )
                 => ! [L2: nat] :
                      ( ( Xaa = extended_enat2(L2) )
                     => ( ( Y = vEBT_Node(Info,Deg,take(vEBT_VEBT,divide_divide(nat,L2,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),list(vEBT_VEBT),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_adn(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg)),TreeList)),vEBT_VEBT_elim_dead(Summary,extended_enat2(divide_divide(nat,L2,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) )
                       => ~ accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel,aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Node(Info,Deg,TreeList,Summary)),extended_enat2(L2))) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
tff(fact_7228_diff__enat__def,axiom,
    ! [A2: extended_enat,B2: extended_enat] : ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),A2),B2) = extended_case_enat(extended_enat,aTP_Lamp_adq(extended_enat,fun(nat,extended_enat),B2),extend4730790105801354508finity(extended_enat),A2) ) ).

% diff_enat_def
tff(fact_7229_plus__enat__def,axiom,
    ! [Ma: extended_enat,Nb: extended_enat] : ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Ma),Nb) = extended_case_enat(extended_enat,aTP_Lamp_ads(extended_enat,fun(nat,extended_enat),Nb),extend4730790105801354508finity(extended_enat),Ma) ) ).

% plus_enat_def
tff(fact_7230_times__enat__def,axiom,
    ! [Ma: extended_enat,Nb: extended_enat] :
      ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),Ma),Nb) = extended_case_enat(extended_enat,aTP_Lamp_adu(extended_enat,fun(nat,extended_enat),Nb),
          $ite(Nb = zero_zero(extended_enat),zero_zero(extended_enat),extend4730790105801354508finity(extended_enat)),
          Ma) ) ).

% times_enat_def
tff(fact_7231_eSuc__def,axiom,
    ! [I: extended_enat] : ( extended_eSuc(I) = extended_case_enat(extended_enat,aTP_Lamp_adv(nat,extended_enat),extend4730790105801354508finity(extended_enat),I) ) ).

% eSuc_def
tff(fact_7232_Code__Numeral_Odup__def,axiom,
    code_dup = aa(fun(int,int),fun(code_integer,code_integer),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int),aTP_Lamp_aai(int,int)) ).

% Code_Numeral.dup_def
tff(fact_7233_eSuc__minus__1,axiom,
    ! [Nb: extended_enat] : ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),extended_eSuc(Nb)),one_one(extended_enat)) = Nb ) ).

% eSuc_minus_1
tff(fact_7234_enat__eSuc__iff,axiom,
    ! [Y: nat,Xb: extended_enat] :
      ( ( extended_enat2(Y) = extended_eSuc(Xb) )
    <=> ? [N4: nat] :
          ( ( Y = aa(nat,nat,suc,N4) )
          & ( extended_enat2(N4) = Xb ) ) ) ).

% enat_eSuc_iff
tff(fact_7235_eSuc__enat__iff,axiom,
    ! [Xb: extended_enat,Y: nat] :
      ( ( extended_eSuc(Xb) = extended_enat2(Y) )
    <=> ? [N4: nat] :
          ( ( Y = aa(nat,nat,suc,N4) )
          & ( Xb = extended_enat2(N4) ) ) ) ).

% eSuc_enat_iff
tff(fact_7236_eSuc__enat,axiom,
    ! [Nb: nat] : ( extended_eSuc(extended_enat2(Nb)) = extended_enat2(aa(nat,nat,suc,Nb)) ) ).

% eSuc_enat
tff(fact_7237_one__eSuc,axiom,
    one_one(extended_enat) = extended_eSuc(zero_zero(extended_enat)) ).

% one_eSuc
tff(fact_7238_eSuc__plus__1,axiom,
    ! [Nb: extended_enat] : ( extended_eSuc(Nb) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Nb),one_one(extended_enat)) ) ).

% eSuc_plus_1
tff(fact_7239_plus__1__eSuc_I1_J,axiom,
    ! [Q4: extended_enat] : ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),one_one(extended_enat)),Q4) = extended_eSuc(Q4) ) ).

% plus_1_eSuc(1)
tff(fact_7240_plus__1__eSuc_I2_J,axiom,
    ! [Q4: extended_enat] : ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Q4),one_one(extended_enat)) = extended_eSuc(Q4) ) ).

% plus_1_eSuc(2)
tff(fact_7241_uminus__integer__def,axiom,
    uminus_uminus(code_integer) = aa(fun(int,int),fun(code_integer,code_integer),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int),uminus_uminus(int)) ).

% uminus_integer_def
tff(fact_7242_gcd__integer__def,axiom,
    gcd_gcd(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),gcd_gcd(int)) ).

% gcd_integer_def
tff(fact_7243_minus__integer__def,axiom,
    minus_minus(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),minus_minus(int)) ).

% minus_integer_def
tff(fact_7244_plus__integer__def,axiom,
    plus_plus(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),plus_plus(int)) ).

% plus_integer_def
tff(fact_7245_List_Oset__insert,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : ( aa(list(A),set(A),set2(A),insert(A,Xb,Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),aa(list(A),set(A),set2(A),Xs)) ) ).

% List.set_insert
tff(fact_7246_has__vector__derivative__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(real,real),F7: real,Xb: real,S: set(real),G: fun(real,A),G2: A] :
          ( has_field_derivative(real,F2,F7,topolo174197925503356063within(real,Xb,S))
         => ( has_ve8173657378732805170vative(A,G,G2,topolo174197925503356063within(real,Xb,S))
           => has_ve8173657378732805170vative(A,aa(fun(real,A),fun(real,A),aTP_Lamp_adw(fun(real,real),fun(fun(real,A),fun(real,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,aa(real,real,F2,Xb),G2)),real_V8093663219630862766scaleR(A,F7,aa(real,A,G,Xb))),topolo174197925503356063within(real,Xb,S)) ) ) ) ).

% has_vector_derivative_scaleR
tff(fact_7247_in__set__insert,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( insert(A,Xb,Xs) = Xs ) ) ).

% in_set_insert
tff(fact_7248_not__in__set__insert,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( insert(A,Xb,Xs) = cons(A,Xb,Xs) ) ) ).

% not_in_set_insert
tff(fact_7249_has__vector__derivative__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(real,A),F7: A,Net: filter(real)] :
          ( has_ve8173657378732805170vative(A,F2,F7,Net)
         => has_ve8173657378732805170vative(A,aTP_Lamp_adx(fun(real,A),fun(real,A),F2),aa(A,A,uminus_uminus(A),F7),Net) ) ) ).

% has_vector_derivative_minus
tff(fact_7250_has__vector__derivative__const,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: A,Net: filter(real)] : has_ve8173657378732805170vative(A,aTP_Lamp_ady(A,fun(real,A),C2),zero_zero(A),Net) ) ).

% has_vector_derivative_const
tff(fact_7251_has__vector__derivative__id,axiom,
    ! [Net: filter(real)] : has_ve8173657378732805170vative(real,aTP_Lamp_adz(real,real),one_one(real),Net) ).

% has_vector_derivative_id
tff(fact_7252_has__vector__derivative__add,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(real,A),F7: A,Net: filter(real),G: fun(real,A),G2: A] :
          ( has_ve8173657378732805170vative(A,F2,F7,Net)
         => ( has_ve8173657378732805170vative(A,G,G2,Net)
           => has_ve8173657378732805170vative(A,aa(fun(real,A),fun(real,A),aTP_Lamp_aea(fun(real,A),fun(fun(real,A),fun(real,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),F7),G2),Net) ) ) ) ).

% has_vector_derivative_add
tff(fact_7253_has__vector__derivative__add__const,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(real,A),Z: A,F7: A,Net: filter(real)] :
          ( has_ve8173657378732805170vative(A,aa(A,fun(real,A),aTP_Lamp_aeb(fun(real,A),fun(A,fun(real,A)),G),Z),F7,Net)
        <=> has_ve8173657378732805170vative(A,G,F7,Net) ) ) ).

% has_vector_derivative_add_const
tff(fact_7254_has__vector__derivative__diff__const,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(real,A),Z: A,F7: A,Net: filter(real)] :
          ( has_ve8173657378732805170vative(A,aa(A,fun(real,A),aTP_Lamp_aec(fun(real,A),fun(A,fun(real,A)),G),Z),F7,Net)
        <=> has_ve8173657378732805170vative(A,G,F7,Net) ) ) ).

% has_vector_derivative_diff_const
tff(fact_7255_has__vector__derivative__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(real,A),F7: A,Net: filter(real),G: fun(real,A),G2: A] :
          ( has_ve8173657378732805170vative(A,F2,F7,Net)
         => ( has_ve8173657378732805170vative(A,G,G2,Net)
           => has_ve8173657378732805170vative(A,aa(fun(real,A),fun(real,A),aTP_Lamp_aed(fun(real,A),fun(fun(real,A),fun(real,A)),F2),G),aa(A,A,aa(A,fun(A,A),minus_minus(A),F7),G2),Net) ) ) ) ).

% has_vector_derivative_diff
tff(fact_7256_List_Oinsert__def,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( insert(A,Xb,Xs) = $ite(member(A,Xb,aa(list(A),set(A),set2(A),Xs)),Xs,cons(A,Xb,Xs)) ) ).

% List.insert_def
tff(fact_7257_has__vector__derivative__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(real,A),F7: A,Xb: real,S: set(real),G: fun(real,A),G2: A] :
          ( has_ve8173657378732805170vative(A,F2,F7,topolo174197925503356063within(real,Xb,S))
         => ( has_ve8173657378732805170vative(A,G,G2,topolo174197925503356063within(real,Xb,S))
           => has_ve8173657378732805170vative(A,aa(fun(real,A),fun(real,A),aTP_Lamp_aee(fun(real,A),fun(fun(real,A),fun(real,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(real,A,F2,Xb)),G2)),aa(A,A,aa(A,fun(A,A),times_times(A),F7),aa(real,A,G,Xb))),topolo174197925503356063within(real,Xb,S)) ) ) ) ).

% has_vector_derivative_mult
tff(fact_7258_bounded__bilinear_Ohas__vector__derivative,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [Prod: fun(A,fun(B,C)),F2: fun(real,A),F7: A,Xb: real,S: set(real),G: fun(real,B),G2: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( has_ve8173657378732805170vative(A,F2,F7,topolo174197925503356063within(real,Xb,S))
           => ( has_ve8173657378732805170vative(B,G,G2,topolo174197925503356063within(real,Xb,S))
             => has_ve8173657378732805170vative(C,aa(fun(real,B),fun(real,C),aa(fun(real,A),fun(fun(real,B),fun(real,C)),aTP_Lamp_aef(fun(A,fun(B,C)),fun(fun(real,A),fun(fun(real,B),fun(real,C))),Prod),F2),G),aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,aa(real,A,F2,Xb)),G2)),aa(B,C,aa(A,fun(B,C),Prod,F7),aa(real,B,G,Xb))),topolo174197925503356063within(real,Xb,S)) ) ) ) ) ).

% bounded_bilinear.has_vector_derivative
tff(fact_7259_Quotient__real,axiom,
    quotient(fun(nat,rat),real,realrel,real2,rep_real,cr_real) ).

% Quotient_real
tff(fact_7260_is__singleton__altdef,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
    <=> ( aa(set(A),nat,finite_card(A),A3) = one_one(nat) ) ) ).

% is_singleton_altdef
tff(fact_7261_inj__on__char__of__nat,axiom,
    inj_on(nat,char,unique5772411509450598832har_of(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).

% inj_on_char_of_nat
tff(fact_7262_UNIV__char__of__nat,axiom,
    top_top(set(char)) = image(nat,char,unique5772411509450598832har_of(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).

% UNIV_char_of_nat
tff(fact_7263_char__of__nat,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat] : ( aa(A,char,unique5772411509450598832har_of(A),aa(nat,A,semiring_1_of_nat(A),Nb)) = aa(nat,char,unique5772411509450598832har_of(nat),Nb) ) ) ).

% char_of_nat
tff(fact_7264_char__of__def,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: A] : ( aa(A,char,unique5772411509450598832har_of(A),Nb) = char2(~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),one_one(nat)),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2))))) ) ) ).

% char_of_def
tff(fact_7265_range__nat__of__char,axiom,
    image(char,nat,comm_s6883823935334413003f_char(nat),top_top(set(char))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ).

% range_nat_of_char
tff(fact_7266_char_Osize_I2_J,axiom,
    ! [X15: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] : ( aa(char,nat,size_size(char),char2((X15),(X2),(X32),(X42),(X52),(X62),(X72),(X82))) = zero_zero(nat) ) ).

% char.size(2)
tff(fact_7267_nat__of__char__less__256,axiom,
    ! [C2: char] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(char,nat,comm_s6883823935334413003f_char(nat),C2)),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ).

% nat_of_char_less_256
tff(fact_7268_char_Osize__gen,axiom,
    ! [X15: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] : ( size_char(char2((X15),(X2),(X32),(X42),(X52),(X62),(X72),(X82))) = zero_zero(nat) ) ).

% char.size_gen
tff(fact_7269_prod__list__def,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( groups5270119922927024881d_list(A) = groups_monoid_F(A,times_times(A),one_one(A)) ) ) ).

% prod_list_def
tff(fact_7270_sum__list__def,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ( groups8242544230860333062m_list(A) = groups_monoid_F(A,plus_plus(A),zero_zero(A)) ) ) ).

% sum_list_def
tff(fact_7271_relpow__finite__bounded1,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite(product_prod(A,A)),R)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K),R)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_aeg(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R),collect(nat,aTP_Lamp_aeh(set(product_prod(A,A)),fun(nat,$o),R))))) ) ) ).

% relpow_finite_bounded1
tff(fact_7272_cr__int__def,axiom,
    ! [X3: product_prod(nat,nat)] : ( aa(product_prod(nat,nat),fun(int,$o),cr_int,X3) = aa(int,fun(int,$o),fequal(int),aa(product_prod(nat,nat),int,abs_Integ,X3)) ) ).

% cr_int_def
tff(fact_7273_relpow__1,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),one_one(nat)),R) = R ) ).

% relpow_1
tff(fact_7274_finite__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Nb: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite(product_prod(A,A)),R)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(set(product_prod(A,A)),$o,finite_finite(product_prod(A,A)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R)) ) ) ).

% finite_relpow
tff(fact_7275_relpow__add,axiom,
    ! [A: $tType,Ma: nat,Nb: nat,R: set(product_prod(A,A))] : ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb)),R) = relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Ma),R),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R)) ) ).

% relpow_add
tff(fact_7276_relpow_Osimps_I2_J,axiom,
    ! [A: $tType,Nb: nat,R: set(product_prod(A,A))] : ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R) = relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R),R) ) ).

% relpow.simps(2)
tff(fact_7277_relpow__Suc__I2,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A)),Z: A,Nb: nat] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),R)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R))
       => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R)) ) ) ).

% relpow_Suc_I2
tff(fact_7278_relpow__Suc__E2,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R))
     => ~ ! [Y3: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y3),R)
           => ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y3),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R)) ) ) ).

% relpow_Suc_E2
tff(fact_7279_relpow__Suc__D2,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R))
     => ? [Y3: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y3),R)
          & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y3),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R)) ) ) ).

% relpow_Suc_D2
tff(fact_7280_relpow__Suc__I,axiom,
    ! [A: $tType,Xb: A,Y: A,Nb: nat,R: set(product_prod(A,A)),Z: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y),Z),R)
       => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R)) ) ) ).

% relpow_Suc_I
tff(fact_7281_relpow__Suc__E,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R))
     => ~ ! [Y3: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R))
           => ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y3),Z),R) ) ) ).

% relpow_Suc_E
tff(fact_7282_relpow__0__I,axiom,
    ! [A: $tType,Xb: A,R: set(product_prod(A,A))] : member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Xb),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R)) ).

% relpow_0_I
tff(fact_7283_relpow__0__E,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R))
     => ( Xb = Y ) ) ).

% relpow_0_E
tff(fact_7284_relpow__E,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R))
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y3: A,M: nat] :
              ( ( Nb = aa(nat,nat,suc,M) )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M),R))
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y3),Z),R) ) ) ) ) ).

% relpow_E
tff(fact_7285_relpow__E2,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R))
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y3: A,M: nat] :
              ( ( Nb = aa(nat,nat,suc,M) )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y3),R)
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y3),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M),R)) ) ) ) ) ).

% relpow_E2
tff(fact_7286_relpow__empty,axiom,
    ! [A: $tType,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ) ) ).

% relpow_empty
tff(fact_7287_int_Opcr__cr__eq,axiom,
    pcr_int = cr_int ).

% int.pcr_cr_eq
tff(fact_7288_relpow__fun__conv,axiom,
    ! [A: $tType,A2: A,B2: A,Nb: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A2),B2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R))
    <=> ? [F5: fun(nat,A)] :
          ( ( aa(nat,A,F5,zero_zero(nat)) = A2 )
          & ( aa(nat,A,F5,Nb) = B2 )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
             => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,F5,I3)),aa(nat,A,F5,aa(nat,nat,suc,I3))),R) ) ) ) ).

% relpow_fun_conv
tff(fact_7289_ntrancl__def,axiom,
    ! [A: $tType,Nb: nat,R: set(product_prod(A,A))] : ( transitive_ntrancl(A,Nb,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_aeg(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R),collect(nat,aTP_Lamp_aei(nat,fun(nat,$o),Nb)))) ) ).

% ntrancl_def
tff(fact_7290_trancl__finite__eq__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite(product_prod(A,A)),R)
     => ( transitive_trancl(A,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_aeg(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R),collect(nat,aTP_Lamp_aeh(set(product_prod(A,A)),fun(nat,$o),R)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_7291_ntrancl__Zero,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : ( transitive_ntrancl(A,zero_zero(nat),R) = R ) ).

% ntrancl_Zero
tff(fact_7292_finite__trancl__ntranl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite(product_prod(A,A)),R)
     => ( transitive_trancl(A,R) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),R)),one_one(nat)),R) ) ) ).

% finite_trancl_ntranl
tff(fact_7293_trancl__set__ntrancl,axiom,
    ! [A: $tType,Xs: list(product_prod(A,A))] : ( transitive_trancl(A,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs))),one_one(nat)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)) ) ).

% trancl_set_ntrancl
tff(fact_7294_trancl__power,axiom,
    ! [A: $tType,P3: product_prod(A,A),R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),P3,transitive_trancl(A,R))
    <=> ? [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N4)
          & member(product_prod(A,A),P3,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N4),R)) ) ) ).

% trancl_power
tff(fact_7295_less__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Ma),Nb),transitive_trancl(nat,pred_nat))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb) ) ).

% less_eq
tff(fact_7296_ntrancl__Suc,axiom,
    ! [A: $tType,Nb: nat,R: set(product_prod(A,A))] : ( transitive_ntrancl(A,aa(nat,nat,suc,Nb),R) = relcomp(A,A,A,transitive_ntrancl(A,Nb,R),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),id2(A)),R)) ) ).

% ntrancl_Suc
tff(fact_7297_Quotient__int,axiom,
    quotient(product_prod(nat,nat),int,intrel,abs_Integ,rep_Integ,cr_int) ).

% Quotient_int
tff(fact_7298_relpow_Osimps_I1_J,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R) = id2(A) ) ).

% relpow.simps(1)
tff(fact_7299_gfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),K: nat] :
          ( order_mono(A,A,F2)
         => ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,K)),F2),top_top(A)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),top_top(A)) )
           => ( complete_lattice_gfp(A,F2) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),top_top(A)) ) ) ) ) ).

% gfp_Kleene_iter
tff(fact_7300_Sup__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Xs: list(A)] : ( lattic5882676163264333800up_fin(A,aa(list(A),set(A),set2(A),cons(A,Xb,Xs))) = aa(A,A,fold(A,A,sup_sup(A),Xs),Xb) ) ) ).

% Sup_fin.set_eq_fold
tff(fact_7301_gfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),Nb: nat] :
          ( order_mono(A,A,F2)
         => ( complete_lattice_gfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Nb)),F2)) = complete_lattice_gfp(A,F2) ) ) ) ).

% gfp_funpow
tff(fact_7302_strict__sorted__equal__Uniq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : uniq(list(A),aTP_Lamp_aej(set(A),fun(list(A),$o),A3)) ) ).

% strict_sorted_equal_Uniq
tff(fact_7303_Inf__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Xs: list(A)] : ( lattic7752659483105999362nf_fin(A,aa(list(A),set(A),set2(A),cons(A,Xb,Xs))) = aa(A,A,fold(A,A,inf_inf(A),Xs),Xb) ) ) ).

% Inf_fin.set_eq_fold
tff(fact_7304_list__ex__length,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( list_ex(A,P,Xs)
    <=> ? [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(A,$o,P,aa(nat,A,nth(A,Xs),N4)) ) ) ).

% list_ex_length
tff(fact_7305_span__breakdown,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: A,S3: set(A),A2: A] :
          ( member(A,B2,S3)
         => ( member(A,A2,real_Vector_span(A,S3))
           => ? [K2: real] : member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),real_V8093663219630862766scaleR(A,K2,B2)),real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))) ) ) ) ).

% span_breakdown
tff(fact_7306_span__insert__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] : ( real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),S3)) = real_Vector_span(A,S3) ) ) ).

% span_insert_0
tff(fact_7307_span__empty,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ( real_Vector_span(A,bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A))) ) ) ).

% span_empty
tff(fact_7308_span__delete__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] : ( real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A))))) = real_Vector_span(A,S3) ) ) ).

% span_delete_0
tff(fact_7309_span__neg,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,S3: set(A)] :
          ( member(A,Xb,real_Vector_span(A,S3))
         => member(A,aa(A,A,uminus_uminus(A),Xb),real_Vector_span(A,S3)) ) ) ).

% span_neg
tff(fact_7310_list__ex__cong,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),F2: fun(A,$o),G: fun(A,$o)] :
      ( ( Xs = Ys2 )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Ys2))
           => ( aa(A,$o,F2,X)
            <=> aa(A,$o,G,X) ) )
       => ( list_ex(A,F2,Xs)
        <=> list_ex(A,G,Ys2) ) ) ) ).

% list_ex_cong
tff(fact_7311_span__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] : member(A,zero_zero(A),real_Vector_span(A,S3)) ) ).

% span_0
tff(fact_7312_span__induct__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,S3: set(A),H: fun(A,$o)] :
          ( member(A,Xb,real_Vector_span(A,S3))
         => ( aa(A,$o,H,zero_zero(A))
           => ( ! [C3: real,X: A,Y3: A] :
                  ( member(A,X,S3)
                 => ( aa(A,$o,H,Y3)
                   => aa(A,$o,H,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,C3,X)),Y3)) ) )
             => aa(A,$o,H,Xb) ) ) ) ) ).

% span_induct_alt
tff(fact_7313_eq__span__insert__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,S3: set(A)] :
          ( member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y),real_Vector_span(A,S3))
         => ( real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),S3)) = real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),S3)) ) ) ) ).

% eq_span_insert_eq
tff(fact_7314_span__add,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,S3: set(A),Y: A] :
          ( member(A,Xb,real_Vector_span(A,S3))
         => ( member(A,Y,real_Vector_span(A,S3))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y),real_Vector_span(A,S3)) ) ) ) ).

% span_add
tff(fact_7315_span__add__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,S3: set(A),Y: A] :
          ( member(A,Xb,real_Vector_span(A,S3))
         => ( member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y),real_Vector_span(A,S3))
          <=> member(A,Y,real_Vector_span(A,S3)) ) ) ) ).

% span_add_eq
tff(fact_7316_span__add__eq2,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Y: A,S3: set(A),Xb: A] :
          ( member(A,Y,real_Vector_span(A,S3))
         => ( member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y),real_Vector_span(A,S3))
          <=> member(A,Xb,real_Vector_span(A,S3)) ) ) ) ).

% span_add_eq2
tff(fact_7317_span__diff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,S3: set(A),Y: A] :
          ( member(A,Xb,real_Vector_span(A,S3))
         => ( member(A,Y,real_Vector_span(A,S3))
           => member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y),real_Vector_span(A,S3)) ) ) ) ).

% span_diff
tff(fact_7318_span__breakdown__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,A2: A,S3: set(A)] :
          ( member(A,Xb,real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),S3)))
        <=> ? [K3: real] : member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),real_V8093663219630862766scaleR(A,K3,A2)),real_Vector_span(A,S3)) ) ) ).

% span_breakdown_eq
tff(fact_7319_span__Un,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A),T4: set(A)] : ( real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S3),T4)) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_aek(set(A),fun(set(A),fun(A,$o)),S3),T4)) ) ) ).

% span_Un
tff(fact_7320_span__insert,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: A,S3: set(A)] : ( real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),S3)) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_ael(A,fun(set(A),fun(A,$o)),A2),S3)) ) ) ).

% span_insert
tff(fact_7321_dim__def,axiom,
    ! [V: set(a)] :
      ( real_Vector_dim(a,V) = $ite(
          ? [B5: set(a)] :
            ( ~ real_V358717886546972837endent(a,B5)
            & ( real_Vector_span(a,B5) = real_Vector_span(a,V) ) ),
          aa(set(a),nat,finite_card(a),fChoice(set(a),aTP_Lamp_aem(set(a),fun(set(a),$o),V))),
          zero_zero(nat) ) ) ).

% dim_def
tff(fact_7322_linear__indep__image__lemma,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),B3: set(A),Xb: A] :
          ( real_Vector_linear(A,B,F2)
         => ( aa(set(A),$o,finite_finite(A),B3)
           => ( ~ real_V358717886546972837endent(B,image(A,B,F2,B3))
             => ( inj_on(A,B,F2,B3)
               => ( member(A,Xb,real_Vector_span(A,B3))
                 => ( ( aa(A,B,F2,Xb) = zero_zero(B) )
                   => ( Xb = zero_zero(A) ) ) ) ) ) ) ) ) ).

% linear_indep_image_lemma
tff(fact_7323_linear__eq__0__on__span,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [F2: fun(A,B),B2: set(A),Xb: A] :
          ( real_Vector_linear(A,B,F2)
         => ( ! [X: A] :
                ( member(A,X,B2)
               => ( aa(A,B,F2,X) = zero_zero(B) ) )
           => ( member(A,Xb,real_Vector_span(A,B2))
             => ( aa(A,B,F2,Xb) = zero_zero(B) ) ) ) ) ) ).

% linear_eq_0_on_span
tff(fact_7324_linear__injective__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_Vector_linear(A,B,F2)
         => ( inj_on(A,B,F2,top_top(set(A)))
          <=> ! [X4: A] :
                ( ( aa(A,B,F2,X4) = zero_zero(B) )
               => ( X4 = zero_zero(A) ) ) ) ) ) ).

% linear_injective_0
tff(fact_7325_linear__compose__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( real_Vector_linear(A,B,F2)
         => ( real_Vector_linear(A,B,G)
           => real_Vector_linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aen(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% linear_compose_add
tff(fact_7326_linear__compose__sub,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( real_Vector_linear(A,B,F2)
         => ( real_Vector_linear(A,B,G)
           => real_Vector_linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aeo(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% linear_compose_sub
tff(fact_7327_linear__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( real_Vector_linear(A,B,F2)
         => ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)) ) ) ) ).

% linear_diff
tff(fact_7328_linear__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),B1: A,B22: A] :
          ( real_Vector_linear(A,B,F2)
         => ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),B22)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,B1)),aa(A,B,F2,B22)) ) ) ) ).

% linear_add
tff(fact_7329_linearI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B)] :
          ( ! [B12: A,B23: A] : ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),B12),B23)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,B12)),aa(A,B,F2,B23)) )
         => ( ! [R4: real,B4: A] : ( aa(A,B,F2,real_V8093663219630862766scaleR(A,R4,B4)) = real_V8093663219630862766scaleR(B,R4,aa(A,B,F2,B4)) )
           => real_Vector_linear(A,B,F2) ) ) ) ).

% linearI
tff(fact_7330_Real__Vector__Spaces_Olinear__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_Vector_linear(A,B,F2)
        <=> ( ! [X4: A,Y5: A] : ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Y5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X4)),aa(A,B,F2,Y5)) )
            & ! [C4: real,X4: A] : ( aa(A,B,F2,real_V8093663219630862766scaleR(A,C4,X4)) = real_V8093663219630862766scaleR(B,C4,aa(A,B,F2,X4)) ) ) ) ) ).

% Real_Vector_Spaces.linear_iff
tff(fact_7331_module__hom__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => real_Vector_linear(A,B,aTP_Lamp_aep(A,B)) ) ).

% module_hom_zero
tff(fact_7332_linear__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [F2: fun(A,B)] :
          ( real_Vector_linear(A,B,F2)
         => ( aa(A,B,F2,zero_zero(A)) = zero_zero(B) ) ) ) ).

% linear_0
tff(fact_7333_linear__compose__neg,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_Vector_linear(A,B,F2)
         => real_Vector_linear(A,B,aTP_Lamp_aeq(fun(A,B),fun(A,B),F2)) ) ) ).

% linear_compose_neg
tff(fact_7334_linear__neg,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),Xb: A] :
          ( real_Vector_linear(A,B,F2)
         => ( aa(A,B,F2,aa(A,A,uminus_uminus(A),Xb)) = aa(B,B,uminus_uminus(B),aa(A,B,F2,Xb)) ) ) ) ).

% linear_neg
tff(fact_7335_module__hom__uminus,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => real_Vector_linear(A,A,uminus_uminus(A)) ) ).

% module_hom_uminus
tff(fact_7336_representation__diff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V2: A,U: A] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( member(A,V2,real_Vector_span(A,Basis))
           => ( member(A,U,real_Vector_span(A,Basis))
             => ! [X3: A] : ( real_V7696804695334737415tation(A,Basis,aa(A,A,aa(A,fun(A,A),minus_minus(A),U),V2),X3) = aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7696804695334737415tation(A,Basis,U,X3)),real_V7696804695334737415tation(A,Basis,V2,X3)) ) ) ) ) ) ).

% representation_diff
tff(fact_7337_representation__neg,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V2: A] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( member(A,V2,real_Vector_span(A,Basis))
           => ! [X3: A] : ( real_V7696804695334737415tation(A,Basis,aa(A,A,uminus_uminus(A),V2),X3) = aa(real,real,uminus_uminus(real),real_V7696804695334737415tation(A,Basis,V2,X3)) ) ) ) ) ).

% representation_neg
tff(fact_7338_representation__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),X3: A] : ( real_V7696804695334737415tation(A,Basis,zero_zero(A),X3) = zero_zero(real) ) ) ).

% representation_zero
tff(fact_7339_representation__basis,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),B2: A] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( member(A,B2,Basis)
           => ! [X3: A] :
                ( real_V7696804695334737415tation(A,Basis,B2,X3) = $ite(X3 = B2,one_one(real),zero_zero(real)) ) ) ) ) ).

% representation_basis
tff(fact_7340_representation__add,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V2: A,U: A] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( member(A,V2,real_Vector_span(A,Basis))
           => ( member(A,U,real_Vector_span(A,Basis))
             => ! [X3: A] : ( real_V7696804695334737415tation(A,Basis,aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2),X3) = aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7696804695334737415tation(A,Basis,U,X3)),real_V7696804695334737415tation(A,Basis,V2,X3)) ) ) ) ) ) ).

% representation_add
tff(fact_7341_and_Osemilattice__neutr__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => semilattice_neutr(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% and.semilattice_neutr_axioms
tff(fact_7342_finite__sequence__to__countable__set,axiom,
    ! [A: $tType,X6: set(A)] :
      ( countable_countable(A,X6)
     => ~ ! [F8: fun(nat,set(A))] :
            ( ! [I4: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F8,I4)),X6)
           => ( ! [I4: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F8,I4)),aa(nat,set(A),F8,aa(nat,nat,suc,I4)))
             => ( ! [I4: nat] : aa(set(A),$o,finite_finite(A),aa(nat,set(A),F8,I4))
               => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),F8,top_top(set(nat)))) != X6 ) ) ) ) ) ).

% finite_sequence_to_countable_set
tff(fact_7343_semilattice__neutr__order_Oaxioms_I1_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semila1105856199041335345_order(A,F2,Z,Less_eq,Less)
     => semilattice_neutr(A,F2,Z) ) ).

% semilattice_neutr_order.axioms(1)
tff(fact_7344_ccInf__less__iff,axiom,
    ! [A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [S3: set(A),A2: A] :
          ( countable_countable(A,S3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S3)),A2)
          <=> ? [X4: A] :
                ( member(A,X4,S3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),A2) ) ) ) ) ).

% ccInf_less_iff
tff(fact_7345_less__ccSup__iff,axiom,
    ! [A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [S3: set(A),A2: A] :
          ( countable_countable(A,S3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),S3))
          <=> ? [X4: A] :
                ( member(A,X4,S3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X4) ) ) ) ) ).

% less_ccSup_iff
tff(fact_7346_inf__top_Osemilattice__neutr__axioms,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => semilattice_neutr(A,inf_inf(A),top_top(A)) ) ).

% inf_top.semilattice_neutr_axioms
tff(fact_7347_sup__bot_Osemilattice__neutr__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semilattice_neutr(A,sup_sup(A),bot_bot(A)) ) ).

% sup_bot.semilattice_neutr_axioms
tff(fact_7348_gcd__nat_Osemilattice__neutr__axioms,axiom,
    semilattice_neutr(nat,gcd_gcd(nat),zero_zero(nat)) ).

% gcd_nat.semilattice_neutr_axioms
tff(fact_7349_less__ccSUP__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta3822494911875563373attice(B)
        & linorder(B) )
     => ! [A3: set(A),A2: B,F2: fun(A,B)] :
          ( countable_countable(A,A3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A3)))
          <=> ? [X4: A] :
                ( member(A,X4,A3)
                & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(A,B,F2,X4)) ) ) ) ) ).

% less_ccSUP_iff
tff(fact_7350_ccINF__less__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( counta3822494911875563373attice(B)
        & linorder(B) )
     => ! [A3: set(A),F2: fun(A,B),A2: B] :
          ( countable_countable(A,A3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,A3))),A2)
          <=> ? [X4: A] :
                ( member(A,X4,A3)
                & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),A2) ) ) ) ) ).

% ccINF_less_iff
tff(fact_7351_or_Osemilattice__neutr__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => semilattice_neutr(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).

% or.semilattice_neutr_axioms
tff(fact_7352_max__nat_Osemilattice__neutr__axioms,axiom,
    semilattice_neutr(nat,ord_max(nat),zero_zero(nat)) ).

% max_nat.semilattice_neutr_axioms
tff(fact_7353_Collect__finite__subset__eq__lists,axiom,
    ! [A: $tType,T4: set(A)] : ( collect(set(A),aTP_Lamp_aer(set(A),fun(set(A),$o),T4)) = image(list(A),set(A),set2(A),lists(A,T4)) ) ).

% Collect_finite_subset_eq_lists
tff(fact_7354_Collect__finite__eq__lists,axiom,
    ! [A: $tType] : ( collect(set(A),finite_finite(A)) = image(list(A),set(A),set2(A),lists(A,top_top(set(A)))) ) ).

% Collect_finite_eq_lists
tff(fact_7355_in__listsI,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => member(A,X,A3) )
     => member(list(A),Xs,lists(A,A3)) ) ).

% in_listsI
tff(fact_7356_in__lists__conv__set,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] :
      ( member(list(A),Xs,lists(A,A3))
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => member(A,X4,A3) ) ) ).

% in_lists_conv_set
tff(fact_7357_in__listsD,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] :
      ( member(list(A),Xs,lists(A,A3))
     => ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => member(A,X3,A3) ) ) ).

% in_listsD
tff(fact_7358_lists__eq__set,axiom,
    ! [A: $tType,A3: set(A)] : ( lists(A,A3) = collect(list(A),aTP_Lamp_aes(set(A),fun(list(A),$o),A3)) ) ).

% lists_eq_set
tff(fact_7359_at__right__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( topolo174197925503356063within(A,Xb,set_ord_greaterThan(A,Xb)) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),image(A,filter(A),aTP_Lamp_aet(A,fun(A,filter(A)),Xb),set_ord_greaterThan(A,Xb))) ) ) ) ).

% at_right_eq
tff(fact_7360_at__left__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( topolo174197925503356063within(A,Xb,set_ord_lessThan(A,Xb)) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),image(A,filter(A),aTP_Lamp_aeu(A,fun(A,filter(A)),Xb),set_ord_lessThan(A,Xb))) ) ) ) ).

% at_left_eq
tff(fact_7361_card__Un__disjnt,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( aa(set(A),$o,finite_finite(A),B3)
       => ( disjnt(A,A3,B3)
         => ( aa(set(A),nat,finite_card(A),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),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)) ) ) ) ) ).

% card_Un_disjnt
tff(fact_7362_construct__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [B3: set(B),G: fun(B,A),V2: B] : ( real_V4425403222259421789struct(B,A,B3,G,V2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(B,fun(B,A),aa(fun(B,A),fun(B,fun(B,A)),aTP_Lamp_aev(set(B),fun(fun(B,A),fun(B,fun(B,A))),B3),G),V2)),collect(B,aa(B,fun(B,$o),aTP_Lamp_aew(set(B),fun(B,fun(B,$o)),B3),V2))) ) ) ).

% construct_def
tff(fact_7363_construct__outside,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [B3: set(A),V2: A,F2: fun(A,B)] :
          ( ~ real_V358717886546972837endent(A,B3)
         => ( member(A,V2,real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),real_V4986007116245087402_basis(A,B3)),B3)))
           => ( real_V4425403222259421789struct(A,B,B3,F2,V2) = zero_zero(B) ) ) ) ) ).

% construct_outside
tff(fact_7364_disjnt__ge__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y6: set(A),X6: set(A)] :
          ( aa(set(A),$o,finite_finite(A),Y6)
         => ( ! [X: A] :
                ( member(A,X,X6)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),lattic643756798349783984er_Max(A,Y6)),X) )
           => disjnt(A,X6,Y6) ) ) ) ).

% disjnt_ge_max
tff(fact_7365_construct__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [B3: set(A),F2: fun(A,B),G: fun(A,B),V2: A] :
          ( ~ real_V358717886546972837endent(A,B3)
         => ( real_V4425403222259421789struct(A,B,B3,aa(fun(A,B),fun(A,B),aTP_Lamp_aen(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),V2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),real_V4425403222259421789struct(A,B,B3,F2,V2)),real_V4425403222259421789struct(A,B,B3,G,V2)) ) ) ) ).

% construct_add
tff(fact_7366_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,ord_less(nat))) ).

% natLess_def
tff(fact_7367_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_aex(A,B)) = G ) ) ) ).

% fun_cong_unused_0
tff(fact_7368_Restr__natLeq,axiom,
    ! [Nb: nat] : ( aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aa(nat,fun(nat,$o)),Nb)),aTP_Lamp_aey(nat,fun(nat,set(nat)),Nb))) = collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_aez(nat,fun(nat,fun(nat,$o)),Nb))) ) ).

% Restr_natLeq
tff(fact_7369_add_Ogroup__axioms,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => group(A,plus_plus(A),zero_zero(A),uminus_uminus(A)) ) ).

% add.group_axioms
tff(fact_7370_group_Oinverse__distrib__swap,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A2: A,B2: A] :
      ( group(A,F2,Z,Inverse)
     => ( aa(A,A,Inverse,aa(A,A,aa(A,fun(A,A),F2,A2),B2)) = aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,B2)),aa(A,A,Inverse,A2)) ) ) ).

% group.inverse_distrib_swap
tff(fact_7371_group_Ogroup__left__neutral,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A2: A] :
      ( group(A,F2,Z,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F2,Z),A2) = A2 ) ) ).

% group.group_left_neutral
tff(fact_7372_group_Oinverse__neutral,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( group(A,F2,Z,Inverse)
     => ( aa(A,A,Inverse,Z) = Z ) ) ).

% group.inverse_neutral
tff(fact_7373_group_Oinverse__inverse,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A2: A] :
      ( group(A,F2,Z,Inverse)
     => ( aa(A,A,Inverse,aa(A,A,Inverse,A2)) = A2 ) ) ).

% group.inverse_inverse
tff(fact_7374_group_Oinverse__unique,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A2: A,B2: A] :
      ( group(A,F2,Z,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F2,A2),B2) = Z )
       => ( aa(A,A,Inverse,A2) = B2 ) ) ) ).

% group.inverse_unique
tff(fact_7375_group_Oright__inverse,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A2: A] :
      ( group(A,F2,Z,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F2,A2),aa(A,A,Inverse,A2)) = Z ) ) ).

% group.right_inverse
tff(fact_7376_group_Oright__cancel,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),B2: A,A2: A,C2: A] :
      ( group(A,F2,Z,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F2,B2),A2) = aa(A,A,aa(A,fun(A,A),F2,C2),A2) )
      <=> ( B2 = C2 ) ) ) ).

% group.right_cancel
tff(fact_7377_group_Oleft__inverse,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A2: A] :
      ( group(A,F2,Z,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,A2)),A2) = Z ) ) ).

% group.left_inverse
tff(fact_7378_group_Oleft__cancel,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A2: A,B2: A,C2: A] :
      ( group(A,F2,Z,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F2,A2),B2) = aa(A,A,aa(A,fun(A,A),F2,A2),C2) )
      <=> ( B2 = C2 ) ) ) ).

% group.left_cancel
tff(fact_7379_Restr__natLeq2,axiom,
    ! [Nb: nat] : ( aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,order_underS(nat,bNF_Ca8665028551170535155natLeq,Nb),aTP_Lamp_afa(nat,fun(nat,set(nat)),Nb))) = collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_aez(nat,fun(nat,fun(nat,$o)),Nb))) ) ).

% Restr_natLeq2
tff(fact_7380_rcis__cnj,axiom,
    ! [A2: complex] : ( cnj(A2) = rcis(real_V7770717601297561774m_norm(complex,A2),aa(real,real,uminus_uminus(real),arg(A2))) ) ).

% rcis_cnj
tff(fact_7381_complex__cnj__one__iff,axiom,
    ! [Z: complex] :
      ( ( cnj(Z) = one_one(complex) )
    <=> ( Z = one_one(complex) ) ) ).

% complex_cnj_one_iff
tff(fact_7382_complex__cnj__one,axiom,
    cnj(one_one(complex)) = one_one(complex) ).

% complex_cnj_one
tff(fact_7383_complex__cnj__minus,axiom,
    ! [Xb: complex] : ( cnj(aa(complex,complex,uminus_uminus(complex),Xb)) = aa(complex,complex,uminus_uminus(complex),cnj(Xb)) ) ).

% complex_cnj_minus
tff(fact_7384_complex__cnj__inverse,axiom,
    ! [Xb: complex] : ( cnj(aa(complex,complex,inverse_inverse(complex),Xb)) = aa(complex,complex,inverse_inverse(complex),cnj(Xb)) ) ).

% complex_cnj_inverse
tff(fact_7385_complex__cnj__of__int,axiom,
    ! [Z: int] : ( cnj(aa(int,complex,ring_1_of_int(complex),Z)) = aa(int,complex,ring_1_of_int(complex),Z) ) ).

% complex_cnj_of_int
tff(fact_7386_complex__cnj__i,axiom,
    cnj(imaginary_unit) = aa(complex,complex,uminus_uminus(complex),imaginary_unit) ).

% complex_cnj_i
tff(fact_7387_complex__cnj__neg__numeral,axiom,
    ! [W: num] : ( cnj(aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W))) = aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W)) ) ).

% complex_cnj_neg_numeral
tff(fact_7388_natLeq__underS__less,axiom,
    ! [Nb: nat] : ( order_underS(nat,bNF_Ca8665028551170535155natLeq,Nb) = collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aa(nat,fun(nat,$o)),Nb)) ) ).

% natLeq_underS_less
tff(fact_7389_cnj_Osimps_I2_J,axiom,
    ! [Z: complex] : ( im(cnj(Z)) = aa(real,real,uminus_uminus(real),im(Z)) ) ).

% cnj.simps(2)
tff(fact_7390_complex__cnj,axiom,
    ! [A2: real,B2: real] : ( cnj(complex2(A2,B2)) = complex2(A2,aa(real,real,uminus_uminus(real),B2)) ) ).

% complex_cnj
tff(fact_7391_cis__cnj,axiom,
    ! [Ta: real] : ( cnj(cis(Ta)) = cis(aa(real,real,uminus_uminus(real),Ta)) ) ).

% cis_cnj
tff(fact_7392_cnj_Ocode,axiom,
    ! [Z: complex] : ( cnj(Z) = complex2(re(Z),aa(real,real,uminus_uminus(real),im(Z))) ) ).

% cnj.code
tff(fact_7393_splice__replicate,axiom,
    ! [A: $tType,Ma: nat,Xb: A,Nb: nat] : ( splice(A,replicate(A,Ma,Xb),replicate(A,Nb,Xb)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Nb),Xb) ) ).

% splice_replicate
tff(fact_7394_and_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => monoid(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% and.monoid_axioms
tff(fact_7395_length__splice,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : ( aa(list(A),nat,size_size(list(A)),splice(A,Xs,Ys2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys2)) ) ).

% length_splice
tff(fact_7396_inf__top_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => monoid(A,inf_inf(A),top_top(A)) ) ).

% inf_top.monoid_axioms
tff(fact_7397_mult_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => monoid(A,times_times(A),one_one(A)) ) ).

% mult.monoid_axioms
tff(fact_7398_add_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => monoid(A,plus_plus(A),zero_zero(A)) ) ).

% add.monoid_axioms
tff(fact_7399_monoid_Oright__neutral,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,A2: A] :
      ( monoid(A,F2,Z)
     => ( aa(A,A,aa(A,fun(A,A),F2,A2),Z) = A2 ) ) ).

% monoid.right_neutral
tff(fact_7400_monoid_Oleft__neutral,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,A2: A] :
      ( monoid(A,F2,Z)
     => ( aa(A,A,aa(A,fun(A,A),F2,Z),A2) = A2 ) ) ).

% monoid.left_neutral
tff(fact_7401_sup__bot_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => monoid(A,sup_sup(A),bot_bot(A)) ) ).

% sup_bot.monoid_axioms
tff(fact_7402_gcd__nat_Omonoid__axioms,axiom,
    monoid(nat,gcd_gcd(nat),zero_zero(nat)) ).

% gcd_nat.monoid_axioms
tff(fact_7403_or_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => monoid(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).

% or.monoid_axioms
tff(fact_7404_xor_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => monoid(A,bit_se5824344971392196577ns_xor(A),zero_zero(A)) ) ).

% xor.monoid_axioms
tff(fact_7405_max__nat_Omonoid__axioms,axiom,
    monoid(nat,ord_max(nat),zero_zero(nat)) ).

% max_nat.monoid_axioms
tff(fact_7406_inv__o__cancel,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(fun(A,B),fun(A,A),comp(B,A,A,hilbert_inv_into(A,B,top_top(set(A)),F2)),F2) = id(A) ) ) ).

% inv_o_cancel
tff(fact_7407_less__cSUP__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [A3: set(A),F2: fun(A,B),A2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,image(A,B,F2,A3))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F2,A3)))
            <=> ? [X4: A] :
                  ( member(A,X4,A3)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(A,B,F2,X4)) ) ) ) ) ) ).

% less_cSUP_iff
tff(fact_7408_inv__into__f__f,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),Xb: A] :
      ( inj_on(A,B,F2,A3)
     => ( member(A,Xb,A3)
       => ( aa(B,A,hilbert_inv_into(A,B,A3,F2),aa(A,B,F2,Xb)) = Xb ) ) ) ).

% inv_into_f_f
tff(fact_7409_inv__identity,axiom,
    ! [A: $tType,X3: A] : ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),aTP_Lamp_afb(A,A)),X3) = X3 ) ).

% inv_identity
tff(fact_7410_inv__id,axiom,
    ! [A: $tType] : ( hilbert_inv_into(A,A,top_top(set(A)),id(A)) = id(A) ) ).

% inv_id
tff(fact_7411_inv__into__image__cancel,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),S3: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),A3)
       => ( image(B,A,hilbert_inv_into(A,B,A3,F2),image(A,B,F2,S3)) = S3 ) ) ) ).

% inv_into_image_cancel
tff(fact_7412_o__inv__o__cancel,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(A,B),G: fun(A,C)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(fun(A,B),fun(A,C),comp(B,C,A,aa(fun(B,A),fun(B,C),comp(A,C,B,G),hilbert_inv_into(A,B,top_top(set(A)),F2))),F2) = G ) ) ).

% o_inv_o_cancel
tff(fact_7413_less__cSup__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Y: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X6)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6))
            <=> ? [X4: A] :
                  ( member(A,X4,X6)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X4) ) ) ) ) ) ).

% less_cSup_iff
tff(fact_7414_inv__equality,axiom,
    ! [A: $tType,B: $tType,G: fun(B,A),F2: fun(A,B)] :
      ( ! [X: A] : ( aa(B,A,G,aa(A,B,F2,X)) = X )
     => ( ! [Y3: B] : ( aa(A,B,F2,aa(B,A,G,Y3)) = Y3 )
       => ( hilbert_inv_into(A,B,top_top(set(A)),F2) = G ) ) ) ).

% inv_equality
tff(fact_7415_inv__unique__comp,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),G: fun(A,B)] :
      ( ( aa(fun(A,B),fun(A,A),comp(B,A,A,F2),G) = id(A) )
     => ( ( aa(fun(B,A),fun(B,B),comp(A,B,B,G),F2) = id(B) )
       => ( hilbert_inv_into(B,A,top_top(set(B)),F2) = G ) ) ) ).

% inv_unique_comp
tff(fact_7416_inv__f__f,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xb: A] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),aa(A,B,F2,Xb)) = Xb ) ) ).

% inv_f_f
tff(fact_7417_inv__f__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xb: A,Y: B] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( ( aa(A,B,F2,Xb) = Y )
       => ( aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),Y) = Xb ) ) ) ).

% inv_f_eq
tff(fact_7418_inj__imp__inv__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),G: fun(B,A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( ! [X: B] : ( aa(A,B,F2,aa(B,A,G,X)) = X )
       => ( hilbert_inv_into(A,B,top_top(set(A)),F2) = G ) ) ) ).

% inj_imp_inv_eq
tff(fact_7419_inv__into__f__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),Xb: A,Y: B] :
      ( inj_on(A,B,F2,A3)
     => ( member(A,Xb,A3)
       => ( ( aa(A,B,F2,Xb) = Y )
         => ( aa(B,A,hilbert_inv_into(A,B,A3,F2),Y) = Xb ) ) ) ) ).

% inv_into_f_eq
tff(fact_7420_bij__imp__bij__inv,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => bij_betw(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),top_top(set(B)),top_top(set(A))) ) ).

% bij_imp_bij_inv
tff(fact_7421_bij__inv__eq__iff,axiom,
    ! [A: $tType,B: $tType,P3: fun(A,B),Xb: A,Y: B] :
      ( bij_betw(A,B,P3,top_top(set(A)),top_top(set(B)))
     => ( ( Xb = aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),P3),Y) )
      <=> ( aa(A,B,P3,Xb) = Y ) ) ) ).

% bij_inv_eq_iff
tff(fact_7422_inv__inv__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => ( hilbert_inv_into(B,A,top_top(set(B)),hilbert_inv_into(A,B,top_top(set(A)),F2)) = F2 ) ) ).

% inv_inv_eq
tff(fact_7423_o__inv__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,B),G: fun(C,A)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => ( bij_betw(C,A,G,top_top(set(C)),top_top(set(A)))
       => ( hilbert_inv_into(C,B,top_top(set(C)),aa(fun(C,A),fun(C,B),comp(A,B,C,F2),G)) = aa(fun(B,A),fun(B,C),comp(A,C,B,hilbert_inv_into(C,A,top_top(set(C)),G)),hilbert_inv_into(A,B,top_top(set(A)),F2)) ) ) ) ).

% o_inv_distrib
tff(fact_7424_mono__inv,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B)] :
          ( order_mono(A,B,F2)
         => ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
           => order_mono(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2)) ) ) ) ).

% mono_inv
tff(fact_7425_bij__betw__inv__into__right,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),A9: set(B),A6: B] :
      ( bij_betw(A,B,F2,A3,A9)
     => ( member(B,A6,A9)
       => ( aa(A,B,F2,aa(B,A,hilbert_inv_into(A,B,A3,F2),A6)) = A6 ) ) ) ).

% bij_betw_inv_into_right
tff(fact_7426_bij__betw__inv__into__left,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),A9: set(B),A2: A] :
      ( bij_betw(A,B,F2,A3,A9)
     => ( member(A,A2,A3)
       => ( aa(B,A,hilbert_inv_into(A,B,A3,F2),aa(A,B,F2,A2)) = A2 ) ) ) ).

% bij_betw_inv_into_left
tff(fact_7427_inv__into__inv__into__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),A9: set(B),A2: A] :
      ( bij_betw(A,B,F2,A3,A9)
     => ( member(A,A2,A3)
       => ( aa(A,B,hilbert_inv_into(B,A,A9,hilbert_inv_into(A,B,A3,F2)),A2) = aa(A,B,F2,A2) ) ) ) ).

% inv_into_inv_into_eq
tff(fact_7428_bij__betw__inv__into,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B3: set(B)] :
      ( bij_betw(A,B,F2,A3,B3)
     => bij_betw(B,A,hilbert_inv_into(A,B,A3,F2),B3,A3) ) ).

% bij_betw_inv_into
tff(fact_7429_inv__fn,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat] :
      ( bij_betw(A,A,F2,top_top(set(A)),top_top(set(A)))
     => ( hilbert_inv_into(A,A,top_top(set(A)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),hilbert_inv_into(A,A,top_top(set(A)),F2)) ) ) ).

% inv_fn
tff(fact_7430_inv__into__def,axiom,
    ! [B: $tType,A: $tType,A3: set(B),F2: fun(B,A),X3: A] : ( aa(A,B,hilbert_inv_into(B,A,A3,F2),X3) = fChoice(B,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_afc(set(B),fun(fun(B,A),fun(A,fun(B,$o))),A3),F2),X3)) ) ).

% inv_into_def
tff(fact_7431_inv__into__def2,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),Xb: B] : ( aa(B,A,hilbert_inv_into(A,B,A3,F2),Xb) = fChoice(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_mj(set(A),fun(fun(A,B),fun(B,fun(A,$o))),A3),F2),Xb)) ) ).

% inv_into_def2
tff(fact_7432_inv__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),X3: A] : ( aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F2),X3) = fChoice(B,aa(A,fun(B,$o),aTP_Lamp_afd(fun(B,A),fun(A,fun(B,$o)),F2),X3)) ) ).

% inv_def
tff(fact_7433_bij__betw__inv__into__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),A9: set(B),B3: set(A),B13: set(B)] :
      ( bij_betw(A,B,F2,A3,A9)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
       => ( ( image(A,B,F2,B3) = B13 )
         => bij_betw(B,A,hilbert_inv_into(A,B,A3,F2),B13,B3) ) ) ) ).

% bij_betw_inv_into_subset
tff(fact_7434_inj__on__inv__into,axiom,
    ! [B: $tType,A: $tType,B3: set(A),F2: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),image(B,A,F2,A3))
     => inj_on(A,B,hilbert_inv_into(B,A,A3,F2),B3) ) ).

% inj_on_inv_into
tff(fact_7435_image__inv__into__cancel,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),A9: set(A),B13: set(A)] :
      ( ( image(B,A,F2,A3) = A9 )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B13),A9)
       => ( image(B,A,F2,image(A,B,hilbert_inv_into(B,A,A3,F2),B13)) = B13 ) ) ) ).

% image_inv_into_cancel
tff(fact_7436_inv__into__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(A,B),G: fun(C,A),A3: set(C),Xb: B] :
      ( inj_on(A,B,F2,image(C,A,G,A3))
     => ( inj_on(C,A,G,A3)
       => ( member(B,Xb,image(A,B,F2,image(C,A,G,A3)))
         => ( aa(B,C,hilbert_inv_into(C,B,A3,aa(fun(C,A),fun(C,B),comp(A,B,C,F2),G)),Xb) = aa(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,hilbert_inv_into(C,A,A3,G)),hilbert_inv_into(A,B,image(C,A,G,A3),F2)),Xb) ) ) ) ) ).

% inv_into_comp
tff(fact_7437_surj__imp__inj__inv,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] :
      ( ( image(B,A,F2,top_top(set(B))) = top_top(set(A)) )
     => inj_on(A,B,hilbert_inv_into(B,A,top_top(set(B)),F2),top_top(set(A))) ) ).

% surj_imp_inj_inv
tff(fact_7438_inj__imp__surj__inv,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( image(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),top_top(set(B))) = top_top(set(A)) ) ) ).

% inj_imp_surj_inv
tff(fact_7439_image__inv__f__f,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( image(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),image(A,B,F2,A3)) = A3 ) ) ).

% image_inv_f_f
tff(fact_7440_inj__transfer,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),P: fun(A,$o),Xb: A] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( ! [Y3: B] :
            ( member(B,Y3,image(A,B,F2,top_top(set(A))))
           => aa(A,$o,P,aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),Y3)) )
       => aa(A,$o,P,Xb) ) ) ).

% inj_transfer
tff(fact_7441_f__inv__into__f,axiom,
    ! [B: $tType,A: $tType,Y: A,F2: fun(B,A),A3: set(B)] :
      ( member(A,Y,image(B,A,F2,A3))
     => ( aa(B,A,F2,aa(A,B,hilbert_inv_into(B,A,A3,F2),Y)) = Y ) ) ).

% f_inv_into_f
tff(fact_7442_inv__into__into,axiom,
    ! [A: $tType,B: $tType,Xb: A,F2: fun(B,A),A3: set(B)] :
      ( member(A,Xb,image(B,A,F2,A3))
     => member(B,aa(A,B,hilbert_inv_into(B,A,A3,F2),Xb),A3) ) ).

% inv_into_into
tff(fact_7443_inv__into__injective,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),Xb: B,Y: B] :
      ( ( aa(B,A,hilbert_inv_into(A,B,A3,F2),Xb) = aa(B,A,hilbert_inv_into(A,B,A3,F2),Y) )
     => ( member(B,Xb,image(A,B,F2,A3))
       => ( member(B,Y,image(A,B,F2,A3))
         => ( Xb = Y ) ) ) ) ).

% inv_into_injective
tff(fact_7444_surj__f__inv__f,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Y: A] :
      ( ( image(B,A,F2,top_top(set(B))) = top_top(set(A)) )
     => ( aa(B,A,F2,aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F2),Y)) = Y ) ) ).

% surj_f_inv_f
tff(fact_7445_surj__iff__all,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] :
      ( ( image(B,A,F2,top_top(set(B))) = top_top(set(A)) )
    <=> ! [X4: A] : ( aa(B,A,F2,aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F2),X4)) = X4 ) ) ).

% surj_iff_all
tff(fact_7446_image__f__inv__f,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(A)] :
      ( ( image(B,A,F2,top_top(set(B))) = top_top(set(A)) )
     => ( image(B,A,F2,image(A,B,hilbert_inv_into(B,A,top_top(set(B)),F2),A3)) = A3 ) ) ).

% image_f_inv_f
tff(fact_7447_surj__imp__inv__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),G: fun(A,B)] :
      ( ( image(B,A,F2,top_top(set(B))) = top_top(set(A)) )
     => ( ! [X: B] : ( aa(A,B,G,aa(B,A,F2,X)) = X )
       => ( hilbert_inv_into(B,A,top_top(set(B)),F2) = G ) ) ) ).

% surj_imp_inv_eq
tff(fact_7448_bij__image__Collect__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),P: fun(A,$o)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => ( image(A,B,F2,collect(A,P)) = collect(B,aa(fun(A,$o),fun(B,$o),aTP_Lamp_afe(fun(A,B),fun(fun(A,$o),fun(B,$o)),F2),P)) ) ) ).

% bij_image_Collect_eq
tff(fact_7449_cSUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),Y: A,I: B] :
          ( condit941137186595557371_above(A,image(B,A,F2,A3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F2,A3))),Y)
           => ( member(B,I,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,I)),Y) ) ) ) ) ).

% cSUP_lessD
tff(fact_7450_surj__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] :
      ( ( image(B,A,F2,top_top(set(B))) = top_top(set(A)) )
    <=> ( aa(fun(A,B),fun(A,A),comp(B,A,A,F2),hilbert_inv_into(B,A,top_top(set(B)),F2)) = id(A) ) ) ).

% surj_iff
tff(fact_7451_inj__imp__bij__betw__inv,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),M9: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => bij_betw(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),image(A,B,F2,M9),M9) ) ).

% inj_imp_bij_betw_inv
tff(fact_7452_inj__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
    <=> ( aa(fun(A,B),fun(A,A),comp(B,A,A,hilbert_inv_into(A,B,top_top(set(A)),F2)),F2) = id(A) ) ) ).

% inj_iff
tff(fact_7453_bij__vimage__eq__inv__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => ( vimage(A,B,F2,A3) = image(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),A3) ) ) ).

% bij_vimage_eq_inv_image
tff(fact_7454_fn__o__inv__fn__is__id,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat] :
      ( bij_betw(A,A,F2,top_top(set(A)),top_top(set(A)))
     => ! [X3: A] : ( aa(A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),hilbert_inv_into(A,A,top_top(set(A)),F2))),X3) = X3 ) ) ).

% fn_o_inv_fn_is_id
tff(fact_7455_inv__fn__o__fn__is__id,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat] :
      ( bij_betw(A,A,F2,top_top(set(A)),top_top(set(A)))
     => ! [X3: A] : ( aa(A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),hilbert_inv_into(A,A,top_top(set(A)),F2))),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)),X3) = X3 ) ) ).

% inv_fn_o_fn_is_id
tff(fact_7456_strict__mono__inv__on__range,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_strict_mono(A,B,F2)
         => strict_mono_on(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2),image(A,B,F2,top_top(set(A)))) ) ) ).

% strict_mono_inv_on_range
tff(fact_7457_bijection_Oinv__comp__right,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
     => ( aa(fun(A,A),fun(A,A),comp(A,A,A,F2),hilbert_inv_into(A,A,top_top(set(A)),F2)) = id(A) ) ) ).

% bijection.inv_comp_right
tff(fact_7458_bijection_Oinv__comp__left,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
     => ( aa(fun(A,A),fun(A,A),comp(A,A,A,hilbert_inv_into(A,A,top_top(set(A)),F2)),F2) = id(A) ) ) ).

% bijection.inv_comp_left
tff(fact_7459_bijection_Osurj,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
     => ( image(A,A,F2,top_top(set(A))) = top_top(set(A)) ) ) ).

% bijection.surj
tff(fact_7460_bijection_Oeq__iff,axiom,
    ! [A: $tType,F2: fun(A,A),A2: A,B2: A] :
      ( hilbert_bijection(A,F2)
     => ( ( aa(A,A,F2,A2) = aa(A,A,F2,B2) )
      <=> ( A2 = B2 ) ) ) ).

% bijection.eq_iff
tff(fact_7461_bijection_OeqI,axiom,
    ! [A: $tType,F2: fun(A,A),A2: A,B2: A] :
      ( hilbert_bijection(A,F2)
     => ( ( aa(A,A,F2,A2) = aa(A,A,F2,B2) )
       => ( A2 = B2 ) ) ) ).

% bijection.eqI
tff(fact_7462_bijection_Oinj,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
     => inj_on(A,A,F2,top_top(set(A))) ) ).

% bijection.inj
tff(fact_7463_bijection_Obij,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
     => bij_betw(A,A,F2,top_top(set(A)),top_top(set(A))) ) ).

% bijection.bij
tff(fact_7464_bijection_Ointro,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( bij_betw(A,A,F2,top_top(set(A)),top_top(set(A)))
     => hilbert_bijection(A,F2) ) ).

% bijection.intro
tff(fact_7465_bijection__def,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
    <=> bij_betw(A,A,F2,top_top(set(A)),top_top(set(A))) ) ).

% bijection_def
tff(fact_7466_bijection_Oeq__invI,axiom,
    ! [A: $tType,F2: fun(A,A),A2: A,B2: A] :
      ( hilbert_bijection(A,F2)
     => ( ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),A2) = aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),B2) )
       => ( A2 = B2 ) ) ) ).

% bijection.eq_invI
tff(fact_7467_bijection_Oinv__left,axiom,
    ! [A: $tType,F2: fun(A,A),A2: A] :
      ( hilbert_bijection(A,F2)
     => ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),aa(A,A,F2,A2)) = A2 ) ) ).

% bijection.inv_left
tff(fact_7468_bijection_Oinv__right,axiom,
    ! [A: $tType,F2: fun(A,A),A2: A] :
      ( hilbert_bijection(A,F2)
     => ( aa(A,A,F2,aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),A2)) = A2 ) ) ).

% bijection.inv_right
tff(fact_7469_bijection_Oeq__inv__iff,axiom,
    ! [A: $tType,F2: fun(A,A),A2: A,B2: A] :
      ( hilbert_bijection(A,F2)
     => ( ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),A2) = aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),B2) )
      <=> ( A2 = B2 ) ) ) ).

% bijection.eq_inv_iff
tff(fact_7470_bijection_Oinv__left__eq__iff,axiom,
    ! [A: $tType,F2: fun(A,A),A2: A,B2: A] :
      ( hilbert_bijection(A,F2)
     => ( ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),A2) = B2 )
      <=> ( aa(A,A,F2,B2) = A2 ) ) ) ).

% bijection.inv_left_eq_iff
tff(fact_7471_bijection_Oinv__right__eq__iff,axiom,
    ! [A: $tType,F2: fun(A,A),B2: A,A2: A] :
      ( hilbert_bijection(A,F2)
     => ( ( B2 = aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),A2) )
      <=> ( aa(A,A,F2,B2) = A2 ) ) ) ).

% bijection.inv_right_eq_iff
tff(fact_7472_bijection_Osurj__inv,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
     => ( image(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),top_top(set(A))) = top_top(set(A)) ) ) ).

% bijection.surj_inv
tff(fact_7473_bijection_Oinj__inv,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
     => inj_on(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),top_top(set(A))) ) ).

% bijection.inj_inv
tff(fact_7474_bijection_Obij__inv,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( hilbert_bijection(A,F2)
     => bij_betw(A,A,hilbert_inv_into(A,A,top_top(set(A)),F2),top_top(set(A)),top_top(set(A))) ) ).

% bijection.bij_inv
tff(fact_7475_cINF__less__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [A3: set(A),F2: fun(A,B),A2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,image(A,B,F2,A3))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F2,A3))),A2)
            <=> ? [X4: A] :
                  ( member(A,X4,A3)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),A2) ) ) ) ) ) ).

% cINF_less_iff
tff(fact_7476_filter__set,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : ( filter3(A,P,aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),filter2(A,P,Xs)) ) ).

% filter_set
tff(fact_7477_bdd__below__uminus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X6: set(A)] :
          ( condit1013018076250108175_below(A,image(A,A,uminus_uminus(A),X6))
        <=> condit941137186595557371_above(A,X6) ) ) ).

% bdd_below_uminus
tff(fact_7478_bdd__above__uminus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X6: set(A)] :
          ( condit941137186595557371_above(A,image(A,A,uminus_uminus(A),X6))
        <=> condit1013018076250108175_below(A,X6) ) ) ).

% bdd_above_uminus
tff(fact_7479_cInf__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Y: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,X6)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Y)
            <=> ? [X4: A] :
                  ( member(A,X4,X6)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y) ) ) ) ) ) ).

% cInf_less_iff
tff(fact_7480_less__cINF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),Y: A,I: B] :
          ( condit1013018076250108175_below(A,image(B,A,F2,A3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F2,A3)))
           => ( member(B,I,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(B,A,F2,I)) ) ) ) ) ).

% less_cINF_D
tff(fact_7481_Lcm__Gcd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] : ( gcd_Lcm(A,A3) = gcd_Gcd(A,collect(A,aTP_Lamp_aff(set(A),fun(A,$o),A3))) ) ) ).

% Lcm_Gcd
tff(fact_7482_Gcd__Lcm,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] : ( gcd_Gcd(A,A3) = gcd_Lcm(A,collect(A,aTP_Lamp_afg(set(A),fun(A,$o),A3))) ) ) ).

% Gcd_Lcm
tff(fact_7483_Lcm__eq__0__I__nat,axiom,
    ! [A3: set(nat)] :
      ( member(nat,zero_zero(nat),A3)
     => ( gcd_Lcm(nat,A3) = zero_zero(nat) ) ) ).

% Lcm_eq_0_I_nat
tff(fact_7484_abs__Lcm__eq,axiom,
    ! [K6: set(int)] : ( aa(int,int,abs_abs(int),gcd_Lcm(int,K6)) = gcd_Lcm(int,K6) ) ).

% abs_Lcm_eq
tff(fact_7485_Lcm__0__iff__nat,axiom,
    ! [A3: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),A3)
     => ( ( gcd_Lcm(nat,A3) = zero_zero(nat) )
      <=> member(nat,zero_zero(nat),A3) ) ) ).

% Lcm_0_iff_nat
tff(fact_7486_Lcm__abs__eq,axiom,
    ! [K6: set(int)] : ( gcd_Lcm(int,image(int,int,abs_abs(int),K6)) = gcd_Lcm(int,K6) ) ).

% Lcm_abs_eq
tff(fact_7487_Lcm__int__eq,axiom,
    ! [N5: set(nat)] : ( gcd_Lcm(int,image(nat,int,semiring_1_of_nat(int),N5)) = aa(nat,int,semiring_1_of_nat(int),gcd_Lcm(nat,N5)) ) ).

% Lcm_int_eq
tff(fact_7488_Lcm__UNIV,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Lcm(A,top_top(set(A))) = zero_zero(A) ) ) ).

% Lcm_UNIV
tff(fact_7489_Lcm__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Lcm(A,bot_bot(set(A))) = one_one(A) ) ) ).

% Lcm_empty
tff(fact_7490_Lcm__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( ( gcd_Lcm(A,A3) = one_one(A) )
        <=> ! [X4: A] :
              ( member(A,X4,A3)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X4),one_one(A)) ) ) ) ).

% Lcm_1_iff
tff(fact_7491_Lcm__nat__abs__eq,axiom,
    ! [K6: set(int)] : ( gcd_Lcm(nat,image(int,nat,aTP_Lamp_ml(int,nat),K6)) = aa(int,nat,nat2,gcd_Lcm(int,K6)) ) ).

% Lcm_nat_abs_eq
tff(fact_7492_Lcm__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_Gcd(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A3)
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(A,B,F2,X)),aa(A,B,G,X)) )
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),gcd_Lcm(B,image(A,B,F2,A3))),gcd_Lcm(B,image(A,B,G,A3))) ) ) ).

% Lcm_mono
tff(fact_7493_Lcm__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ( gcd_Lcm(A,A3) = zero_zero(A) )
          <=> member(A,zero_zero(A),A3) ) ) ) ).

% Lcm_0_iff
tff(fact_7494_Lcm__nat__infinite,axiom,
    ! [M9: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite(nat),M9)
     => ( gcd_Lcm(nat,M9) = zero_zero(nat) ) ) ).

% Lcm_nat_infinite
tff(fact_7495_Lcm__least__int,axiom,
    ! [A3: set(int),A2: int] :
      ( ! [B4: int] :
          ( member(int,B4,A3)
         => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),B4),A2) )
     => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),gcd_Lcm(int,A3)),A2) ) ).

% Lcm_least_int
tff(fact_7496_dvd__Lcm__int,axiom,
    ! [Ma: int,M9: set(int)] :
      ( member(int,Ma,M9)
     => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Ma),gcd_Lcm(int,M9)) ) ).

% dvd_Lcm_int
tff(fact_7497_dvd__Lcm,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( member(A,A2,A3)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),gcd_Lcm(A,A3)) ) ) ).

% dvd_Lcm
tff(fact_7498_Lcm__dvdD,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A),Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),gcd_Lcm(A,A3)),Xb)
         => ( member(A,Y,A3)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Y),Xb) ) ) ) ).

% Lcm_dvdD
tff(fact_7499_Lcm__least,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A),A2: A] :
          ( ! [B4: A] :
              ( member(A,B4,A3)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B4),A2) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),gcd_Lcm(A,A3)),A2) ) ) ).

% Lcm_least
tff(fact_7500_Lcm__dvd__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),gcd_Lcm(A,A3)),Xb)
        <=> ! [X4: A] :
              ( member(A,X4,A3)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X4),Xb) ) ) ) ).

% Lcm_dvd_iff
tff(fact_7501_dvd__Lcm__nat,axiom,
    ! [Ma: nat,M9: set(nat)] :
      ( member(nat,Ma,M9)
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),gcd_Lcm(nat,M9)) ) ).

% dvd_Lcm_nat
tff(fact_7502_Lcm__dvd__nat,axiom,
    ! [M9: set(nat),Nb: nat] :
      ( ! [X: nat] :
          ( member(nat,X,M9)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),X),Nb) )
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),gcd_Lcm(nat,M9)),Nb) ) ).

% Lcm_dvd_nat
tff(fact_7503_Lcm__int__greater__eq__0,axiom,
    ! [K6: set(int)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),gcd_Lcm(int,K6)) ).

% Lcm_int_greater_eq_0
tff(fact_7504_Lcm__eq__0__I,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( member(A,zero_zero(A),A3)
         => ( gcd_Lcm(A,A3) = zero_zero(A) ) ) ) ).

% Lcm_eq_0_I
tff(fact_7505_Lcm__no__multiple,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( ! [M: A] :
              ( ( M != zero_zero(A) )
             => ? [X3: A] :
                  ( member(A,X3,A3)
                  & ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X3),M) ) )
         => ( gcd_Lcm(A,A3) = zero_zero(A) ) ) ) ).

% Lcm_no_multiple
tff(fact_7506_Lcm__0__iff_H,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( ( gcd_Lcm(A,A3) = zero_zero(A) )
        <=> ~ ? [L3: A] :
                ( ( L3 != zero_zero(A) )
                & ! [X4: A] :
                    ( member(A,X4,A3)
                   => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X4),L3) ) ) ) ) ).

% Lcm_0_iff'
tff(fact_7507_Lcm__subset,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),gcd_Lcm(A,A3)),gcd_Lcm(A,B3)) ) ) ).

% Lcm_subset
tff(fact_7508_Lcm__nat__empty,axiom,
    gcd_Lcm(nat,bot_bot(set(nat))) = one_one(nat) ).

% Lcm_nat_empty
tff(fact_7509_Gcd__nat__def,axiom,
    ! [M9: set(nat)] : ( gcd_Gcd(nat,M9) = gcd_Lcm(nat,collect(nat,aTP_Lamp_afh(set(nat),fun(nat,$o),M9))) ) ).

% Gcd_nat_def
tff(fact_7510_Lcm__int__def,axiom,
    ! [K6: set(int)] : ( gcd_Lcm(int,K6) = aa(nat,int,semiring_1_of_nat(int),gcd_Lcm(nat,image(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int)),K6))) ) ).

% Lcm_int_def
tff(fact_7511_Lcm__no__units,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] : ( gcd_Lcm(A,A3) = gcd_Lcm(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),collect(A,aTP_Lamp_afi(A,$o)))) ) ) ).

% Lcm_no_units
tff(fact_7512_Lcm__coprime_H,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( ( aa(set(A),nat,finite_card(A),A3) != zero_zero(nat) )
         => ( ! [A4: A,B4: A] :
                ( member(A,A4,A3)
               => ( member(A,B4,A3)
                 => ( ( A4 != B4 )
                   => algebr8660921524188924756oprime(A,A4,B4) ) ) )
           => ( gcd_Lcm(A,A3) = aa(A,A,normal6383669964737779283malize(A),aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_afj(A,A)),A3)) ) ) ) ) ).

% Lcm_coprime'
tff(fact_7513_Lcm__coprime,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A,B4: A] :
                  ( member(A,A4,A3)
                 => ( member(A,B4,A3)
                   => ( ( A4 != B4 )
                     => algebr8660921524188924756oprime(A,A4,B4) ) ) )
             => ( gcd_Lcm(A,A3) = aa(A,A,normal6383669964737779283malize(A),aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_afj(A,A)),A3)) ) ) ) ) ) ).

% Lcm_coprime
tff(fact_7514_normalize__idem,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] : ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,normal6383669964737779283malize(A),A2)) = aa(A,A,normal6383669964737779283malize(A),A2) ) ) ).

% normalize_idem
tff(fact_7515_normalize__eq__0__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% normalize_eq_0_iff
tff(fact_7516_normalize__0,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( aa(A,A,normal6383669964737779283malize(A),zero_zero(A)) = zero_zero(A) ) ) ).

% normalize_0
tff(fact_7517_lcm_Onormalize__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(A,A,normal6383669964737779283malize(A),zero_zero(A)) = zero_zero(A) ) ) ).

% lcm.normalize_bottom
tff(fact_7518_normalize__mult__normalize__left,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A,B2: A] : ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A2)),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ).

% normalize_mult_normalize_left
tff(fact_7519_normalize__mult__normalize__right,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A,B2: A] : ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,normal6383669964737779283malize(A),B2))) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ).

% normalize_mult_normalize_right
tff(fact_7520_gcd_Onormalize__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).

% gcd.normalize_bottom
tff(fact_7521_normalize__1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).

% normalize_1
tff(fact_7522_normalize__dvd__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,normal6383669964737779283malize(A),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% normalize_dvd_iff
tff(fact_7523_dvd__normalize__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,normal6383669964737779283malize(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% dvd_normalize_iff
tff(fact_7524_gcd_Onormalize__right__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(A,A,normal6383669964737779283malize(A),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ).

% gcd.normalize_right_idem
tff(fact_7525_gcd_Onormalize__left__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,normal6383669964737779283malize(A),A2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ).

% gcd.normalize_left_idem
tff(fact_7526_gcd_Onormalize__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ).

% gcd.normalize_idem
tff(fact_7527_gcd_Oidem__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),A2) = aa(A,A,normal6383669964737779283malize(A),A2) ) ) ).

% gcd.idem_normalize
tff(fact_7528_coprime__normalize__right__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,aa(A,A,normal6383669964737779283malize(A),B2))
        <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% coprime_normalize_right_iff
tff(fact_7529_coprime__normalize__left__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,normal6383669964737779283malize(A),A2),B2)
        <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% coprime_normalize_left_iff
tff(fact_7530_normalize__Lcm,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] : ( aa(A,A,normal6383669964737779283malize(A),gcd_Lcm(A,A3)) = gcd_Lcm(A,A3) ) ) ).

% normalize_Lcm
tff(fact_7531_normalize__Gcd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] : ( aa(A,A,normal6383669964737779283malize(A),gcd_Gcd(A,A3)) = gcd_Gcd(A,A3) ) ) ).

% normalize_Gcd
tff(fact_7532_Gcd__fin_Onormalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] : ( aa(A,A,normal6383669964737779283malize(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A3)) = aa(set(A),A,semiring_gcd_Gcd_fin(A),A3) ) ) ).

% Gcd_fin.normalize
tff(fact_7533_gcd_Otop__right__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),zero_zero(A)) = aa(A,A,normal6383669964737779283malize(A),A2) ) ) ).

% gcd.top_right_normalize
tff(fact_7534_gcd_Otop__left__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),zero_zero(A)),A2) = aa(A,A,normal6383669964737779283malize(A),A2) ) ) ).

% gcd.top_left_normalize
tff(fact_7535_Gcd__image__normalize,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] : ( gcd_Gcd(A,image(A,A,normal6383669964737779283malize(A),A3)) = gcd_Gcd(A,A3) ) ) ).

% Gcd_image_normalize
tff(fact_7536_normalize__mult__unit__left,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,normal6383669964737779283malize(A),B2) ) ) ) ).

% normalize_mult_unit_left
tff(fact_7537_normalize__mult__unit__right,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,normal6383669964737779283malize(A),A2) ) ) ) ).

% normalize_mult_unit_right
tff(fact_7538_Lcm__singleton,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A] : ( gcd_Lcm(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) = aa(A,A,normal6383669964737779283malize(A),A2) ) ) ).

% Lcm_singleton
tff(fact_7539_Gcd__singleton,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A] : ( gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) = aa(A,A,normal6383669964737779283malize(A),A2) ) ) ).

% Gcd_singleton
tff(fact_7540_Lcm__eqI,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A2) = A2 )
         => ( ! [B4: A] :
                ( member(A,B4,A3)
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B4),A2) )
           => ( ! [C3: A] :
                  ( ! [B14: A] :
                      ( member(A,B14,A3)
                     => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B14),C3) )
                 => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C3) )
             => ( gcd_Lcm(A,A3) = A2 ) ) ) ) ) ).

% Lcm_eqI
tff(fact_7541_LcmI,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A),B2: A] :
          ( ! [A4: A] :
              ( member(A,A4,A3)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A4),B2) )
         => ( ! [C3: A] :
                ( ! [A8: A] :
                    ( member(A,A8,A3)
                   => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A8),C3) )
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C3) )
           => ( ( aa(A,A,normal6383669964737779283malize(A),B2) = B2 )
             => ( B2 = gcd_Lcm(A,A3) ) ) ) ) ) ).

% LcmI
tff(fact_7542_associated__unit,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A2) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A)) ) ) ) ).

% associated_unit
tff(fact_7543_normalize__1__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A2) = one_one(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A)) ) ) ).

% normalize_1_iff
tff(fact_7544_is__unit__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,normal6383669964737779283malize(A),A2) = one_one(A) ) ) ) ).

% is_unit_normalize
tff(fact_7545_normalize__idem__imp__is__unit__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A2) = A2 )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
          <=> ( A2 = one_one(A) ) ) ) ) ).

% normalize_idem_imp_is_unit_iff
tff(fact_7546_gcd__exp__weak,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,Nb: nat,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) = aa(A,A,normal6383669964737779283malize(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),Nb)) ) ) ).

% gcd_exp_weak
tff(fact_7547_gcd__proj2__if__dvd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) = aa(A,A,normal6383669964737779283malize(A),B2) ) ) ) ).

% gcd_proj2_if_dvd
tff(fact_7548_gcd__proj1__if__dvd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) = aa(A,A,normal6383669964737779283malize(A),A2) ) ) ) ).

% gcd_proj1_if_dvd
tff(fact_7549_gcd__proj2__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Ma: A,Nb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Ma),Nb) = aa(A,A,normal6383669964737779283malize(A),Nb) )
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Nb),Ma) ) ) ).

% gcd_proj2_iff
tff(fact_7550_gcd__proj1__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Ma: A,Nb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Ma),Nb) = aa(A,A,normal6383669964737779283malize(A),Ma) )
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Ma),Nb) ) ) ).

% gcd_proj1_iff
tff(fact_7551_gcd__unique,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [D3: A,A2: A,B2: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),A2)
            & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),B2)
            & ( aa(A,A,normal6383669964737779283malize(A),D3) = D3 )
            & ! [E3: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E3),A2)
                  & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E3),B2) )
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),E3),D3) ) )
        <=> ( D3 = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ) ).

% gcd_unique
tff(fact_7552_gcdI,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
           => ( ! [D5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D5),A2)
                 => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D5),B2)
                   => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D5),C2) ) )
             => ( ( aa(A,A,normal6383669964737779283malize(A),C2) = C2 )
               => ( C2 = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ) ) ) ) ) ).

% gcdI
tff(fact_7553_Gcd__eqI,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A2) = A2 )
         => ( ! [B4: A] :
                ( member(A,B4,A3)
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B4) )
           => ( ! [C3: A] :
                  ( ! [B14: A] :
                      ( member(A,B14,A3)
                     => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B14) )
                 => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),A2) )
             => ( gcd_Gcd(A,A3) = A2 ) ) ) ) ) ).

% Gcd_eqI
tff(fact_7554_GcdI,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A),B2: A] :
          ( ! [A4: A] :
              ( member(A,A4,A3)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A4) )
         => ( ! [C3: A] :
                ( ! [A8: A] :
                    ( member(A,A8,A3)
                   => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),A8) )
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B2) )
           => ( ( aa(A,A,normal6383669964737779283malize(A),B2) = B2 )
             => ( B2 = gcd_Gcd(A,A3) ) ) ) ) ) ).

% GcdI
tff(fact_7555_associated__iff__dvd,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A2) = aa(A,A,normal6383669964737779283malize(A),B2) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ) ).

% associated_iff_dvd
tff(fact_7556_associated__eqI,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
           => ( ( aa(A,A,normal6383669964737779283malize(A),A2) = A2 )
             => ( ( aa(A,A,normal6383669964737779283malize(A),B2) = B2 )
               => ( A2 = B2 ) ) ) ) ) ) ).

% associated_eqI
tff(fact_7557_associatedD2,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A2) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ).

% associatedD2
tff(fact_7558_associatedD1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A2) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% associatedD1
tff(fact_7559_associatedI,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
           => ( aa(A,A,normal6383669964737779283malize(A),A2) = aa(A,A,normal6383669964737779283malize(A),B2) ) ) ) ) ).

% associatedI
tff(fact_7560_dvd__normalize__div,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
         => ( aa(A,A,normal6383669964737779283malize(A),divide_divide(A,A2,B2)) = divide_divide(A,aa(A,A,normal6383669964737779283malize(A),A2),aa(A,A,normal6383669964737779283malize(A),B2)) ) ) ) ).

% dvd_normalize_div
tff(fact_7561_coprime__crossproduct,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,D3: A,B2: A,C2: A] :
          ( algebr8660921524188924756oprime(A,A2,D3)
         => ( algebr8660921524188924756oprime(A,B2,C2)
           => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A2)),aa(A,A,normal6383669964737779283malize(A),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),B2)),aa(A,A,normal6383669964737779283malize(A),D3)) )
            <=> ( ( aa(A,A,normal6383669964737779283malize(A),A2) = aa(A,A,normal6383669964737779283malize(A),B2) )
                & ( aa(A,A,normal6383669964737779283malize(A),C2) = aa(A,A,normal6383669964737779283malize(A),D3) ) ) ) ) ) ) ).

% coprime_crossproduct
tff(fact_7562_normalize__mult,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [A2: A,B2: A] : ( aa(A,A,normal6383669964737779283malize(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,normal6383669964737779283malize(A),A2)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ) ).

% normalize_mult
tff(fact_7563_gcd__mult__distrib_H,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C2: A,A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),C2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(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)) ) ) ).

% gcd_mult_distrib'
tff(fact_7564_gcd__mult__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),A2)),C2)) ) ) ).

% gcd_mult_right
tff(fact_7565_gcd__mult__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2))) ) ) ).

% gcd_mult_left
tff(fact_7566_Gcd__mult,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [C2: A,A3: set(A)] : ( gcd_Gcd(A,image(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),gcd_Gcd(A,A3))) ) ) ).

% Gcd_mult
tff(fact_7567_Lcm__mult,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A),C2: A] :
          ( ( A3 != bot_bot(set(A)) )
         => ( gcd_Lcm(A,image(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),gcd_Lcm(A,A3))) ) ) ) ).

% Lcm_mult
tff(fact_7568_Gcd__fin__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A),B2: A] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),image(A,A,aa(A,fun(A,A),times_times(A),B2),A3)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(set(A),A,semiring_gcd_Gcd_fin(A),A3))) ) ) ) ).

% Gcd_fin_mult
tff(fact_7569_Gcd__fin_Obounded__quasi__semilattice__set__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => bounde6485984586167503788ce_set(A,gcd_gcd(A),zero_zero(A),one_one(A),normal6383669964737779283malize(A)) ) ).

% Gcd_fin.bounded_quasi_semilattice_set_axioms
tff(fact_7570_Lcm__eq__Max__nat,axiom,
    ! [M9: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),M9)
     => ( ( M9 != bot_bot(set(nat)) )
       => ( ~ member(nat,zero_zero(nat),M9)
         => ( ! [M: nat,N: nat] :
                ( member(nat,M,M9)
               => ( member(nat,N,M9)
                 => member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N),M9) ) )
           => ( gcd_Lcm(nat,M9) = lattic643756798349783984er_Max(nat,M9) ) ) ) ) ) ).

% Lcm_eq_Max_nat
tff(fact_7571_Lcm__int__set__eq__fold,axiom,
    ! [Xs: list(int)] : ( gcd_Lcm(int,aa(list(int),set(int),set2(int),Xs)) = aa(int,int,fold(int,int,gcd_lcm(int),Xs),one_one(int)) ) ).

% Lcm_int_set_eq_fold
tff(fact_7572_lcm__right__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) ) ) ).

% lcm_right_idem
tff(fact_7573_lcm__left__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) ) ) ).

% lcm_left_idem
tff(fact_7574_lcm_Obottom__right__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),zero_zero(A)) = zero_zero(A) ) ) ).

% lcm.bottom_right_bottom
tff(fact_7575_lcm_Obottom__left__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),zero_zero(A)),A2) = zero_zero(A) ) ) ).

% lcm.bottom_left_bottom
tff(fact_7576_lcm__neg2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) ) ) ).

% lcm_neg2
tff(fact_7577_lcm__neg1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) ) ) ).

% lcm_neg1
tff(fact_7578_dvd__lcm1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)) ) ).

% dvd_lcm1
tff(fact_7579_dvd__lcm2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,A2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)) ) ).

% dvd_lcm2
tff(fact_7580_lcm__least__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)),C2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
            & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2) ) ) ) ).

% lcm_least_iff
tff(fact_7581_lcm_Oidem__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),A2) = aa(A,A,normal6383669964737779283malize(A),A2) ) ) ).

% lcm.idem_normalize
tff(fact_7582_lcm_Onormalize__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) ) ) ).

% lcm.normalize_idem
tff(fact_7583_lcm_Onormalize__left__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,normal6383669964737779283malize(A),A2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) ) ) ).

% lcm.normalize_left_idem
tff(fact_7584_lcm_Onormalize__right__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),aa(A,A,normal6383669964737779283malize(A),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) ) ) ).

% lcm.normalize_right_idem
tff(fact_7585_lcm__0__iff__int,axiom,
    ! [Ma: int,Nb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Ma),Nb) = zero_zero(int) )
    <=> ( ( Ma = zero_zero(int) )
        | ( Nb = zero_zero(int) ) ) ) ).

% lcm_0_iff_int
tff(fact_7586_lcm__0__iff__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Ma),Nb) = zero_zero(nat) )
    <=> ( ( Ma = zero_zero(nat) )
        | ( Nb = zero_zero(nat) ) ) ) ).

% lcm_0_iff_nat
tff(fact_7587_abs__lcm__int,axiom,
    ! [I: int,J: int] : ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),I),J)) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),I),J) ) ).

% abs_lcm_int
tff(fact_7588_lcm__abs1__int,axiom,
    ! [Xb: int,Y: int] : ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,abs_abs(int),Xb)),Y) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Xb),Y) ) ).

% lcm_abs1_int
tff(fact_7589_lcm__abs2__int,axiom,
    ! [Xb: int,Y: int] : ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Xb),aa(int,int,abs_abs(int),Y)) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Xb),Y) ) ).

% lcm_abs2_int
tff(fact_7590_lcm__proj2__if__dvd__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Xb),Y)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Xb),Y) = Y ) ) ).

% lcm_proj2_if_dvd_nat
tff(fact_7591_lcm__proj1__if__dvd__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Xb),Y)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Y),Xb) = Y ) ) ).

% lcm_proj1_if_dvd_nat
tff(fact_7592_lcm__proj2__iff__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Ma),Nb) = Nb )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),Nb) ) ).

% lcm_proj2_iff_nat
tff(fact_7593_lcm__proj1__iff__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Ma),Nb) = Ma )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),Ma) ) ).

% lcm_proj1_iff_nat
tff(fact_7594_lcm__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A,Nb: num] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),aa(num,A,numeral_numeral(A),Nb)) ) ) ).

% lcm_neg_numeral_2
tff(fact_7595_lcm__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Nb: num,A2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))),A2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(num,A,numeral_numeral(A),Nb)),A2) ) ) ).

% lcm_neg_numeral_1
tff(fact_7596_lcm__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) = one_one(A) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
            & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A)) ) ) ) ).

% lcm_eq_1_iff
tff(fact_7597_lcm_Otop__left__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),one_one(A)),A2) = aa(A,A,normal6383669964737779283malize(A),A2) ) ) ).

% lcm.top_left_normalize
tff(fact_7598_lcm_Otop__right__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),one_one(A)) = aa(A,A,normal6383669964737779283malize(A),A2) ) ) ).

% lcm.top_right_normalize
tff(fact_7599_Lcm__insert,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A3: set(A)] : ( gcd_Lcm(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),gcd_Lcm(A,A3)) ) ) ).

% Lcm_insert
tff(fact_7600_lcm__1__iff__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Ma),Nb) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( Ma = aa(nat,nat,suc,zero_zero(nat)) )
        & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% lcm_1_iff_nat
tff(fact_7601_lcm__1__iff__int,axiom,
    ! [Ma: int,Nb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Ma),Nb) = one_one(int) )
    <=> ( ( ( Ma = one_one(int) )
          | ( Ma = aa(int,int,uminus_uminus(int),one_one(int)) ) )
        & ( ( Nb = one_one(int) )
          | ( Nb = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% lcm_1_iff_int
tff(fact_7602_lcm__proj1__iff__int,axiom,
    ! [Ma: int,Nb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Ma),Nb) = aa(int,int,abs_abs(int),Ma) )
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Nb),Ma) ) ).

% lcm_proj1_iff_int
tff(fact_7603_lcm__proj2__iff__int,axiom,
    ! [Ma: int,Nb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Ma),Nb) = aa(int,int,abs_abs(int),Nb) )
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Ma),Nb) ) ).

% lcm_proj2_iff_int
tff(fact_7604_lcm__proj1__if__dvd__int,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Xb),Y)
     => ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Y),Xb) = aa(int,int,abs_abs(int),Y) ) ) ).

% lcm_proj1_if_dvd_int
tff(fact_7605_lcm__proj2__if__dvd__int,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Xb),Y)
     => ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Xb),Y) = aa(int,int,abs_abs(int),Y) ) ) ).

% lcm_proj2_if_dvd_int
tff(fact_7606_lcm__int__int__eq,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(nat,int,semiring_1_of_nat(int),Ma)),aa(nat,int,semiring_1_of_nat(int),Nb)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Ma),Nb)) ) ).

% lcm_int_int_eq
tff(fact_7607_gcd__mult__lcm,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A2)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ) ).

% gcd_mult_lcm
tff(fact_7608_lcm__mult__gcd,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A2)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ) ).

% lcm_mult_gcd
tff(fact_7609_Lcm__2,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,B2: A] : ( gcd_Lcm(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) ) ) ).

% Lcm_2
tff(fact_7610_lcm__nat__abs__right__eq,axiom,
    ! [Nb: nat,K: int] : ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Nb),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(nat,int,semiring_1_of_nat(int),Nb)),K)) ) ).

% lcm_nat_abs_right_eq
tff(fact_7611_lcm__nat__abs__left__eq,axiom,
    ! [K: int,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),Nb) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),K),aa(nat,int,semiring_1_of_nat(int),Nb))) ) ).

% lcm_nat_abs_left_eq
tff(fact_7612_Lcm__Un,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A),B3: set(A)] : ( gcd_Lcm(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),gcd_lcm(A),gcd_Lcm(A,A3)),gcd_Lcm(A,B3)) ) ) ).

% Lcm_Un
tff(fact_7613_Lcm__nat__insert,axiom,
    ! [Nb: nat,M9: set(nat)] : ( gcd_Lcm(nat,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),Nb),M9)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Nb),gcd_Lcm(nat,M9)) ) ).

% Lcm_nat_insert
tff(fact_7614_lcmI,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2)
           => ( ! [D5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),D5)
                 => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),D5)
                   => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),D5) ) )
             => ( ( aa(A,A,normal6383669964737779283malize(A),C2) = C2 )
               => ( C2 = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) ) ) ) ) ) ) ).

% lcmI
tff(fact_7615_lcm__unique,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,D3: A,B2: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),D3)
            & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),D3)
            & ( aa(A,A,normal6383669964737779283malize(A),D3) = D3 )
            & ! [E3: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),E3)
                  & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),E3) )
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),E3) ) )
        <=> ( D3 = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) ) ) ) ).

% lcm_unique
tff(fact_7616_lcm__proj1__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Ma: A,Nb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),Ma),Nb) = aa(A,A,normal6383669964737779283malize(A),Ma) )
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Nb),Ma) ) ) ).

% lcm_proj1_iff
tff(fact_7617_lcm__proj2__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Ma: A,Nb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),Ma),Nb) = aa(A,A,normal6383669964737779283malize(A),Nb) )
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Ma),Nb) ) ) ).

% lcm_proj2_iff
tff(fact_7618_lcm__proj1__if__dvd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) = aa(A,A,normal6383669964737779283malize(A),A2) ) ) ) ).

% lcm_proj1_if_dvd
tff(fact_7619_lcm__proj2__if__dvd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) = aa(A,A,normal6383669964737779283malize(A),B2) ) ) ) ).

% lcm_proj2_if_dvd
tff(fact_7620_lcm__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,B2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% lcm_coprime
tff(fact_7621_lcm__mult__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2))) ) ) ).

% lcm_mult_left
tff(fact_7622_lcm__mult__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),A2)),C2)) ) ) ).

% lcm_mult_right
tff(fact_7623_lcm__mult__distrib_H,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C2: A,A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),C2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(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)) ) ) ).

% lcm_mult_distrib'
tff(fact_7624_lcm__gcd__prod,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ).

% lcm_gcd_prod
tff(fact_7625_lcm__nat__def,axiom,
    ! [Xb: nat,Y: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Xb),Y) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Xb),Y),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Y)) ) ).

% lcm_nat_def
tff(fact_7626_lcm__unique__int,axiom,
    ! [D3: int,A2: int,B2: int] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),D3)
        & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),D3)
        & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),B2),D3)
        & ! [E3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),E3)
              & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),B2),E3) )
           => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),E3) ) )
    <=> ( D3 = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),A2),B2) ) ) ).

% lcm_unique_int
tff(fact_7627_lcm__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).

% lcm_mult_unit1
tff(fact_7628_lcm__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).

% lcm_mult_unit2
tff(fact_7629_prod__gcd__lcm__nat,axiom,
    ! [Ma: nat,Nb: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ma),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Ma),Nb)) ) ).

% prod_gcd_lcm_nat
tff(fact_7630_lcm__neg1__int,axiom,
    ! [Xb: int,Y: int] : ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,uminus_uminus(int),Xb)),Y) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Xb),Y) ) ).

% lcm_neg1_int
tff(fact_7631_lcm__neg2__int,axiom,
    ! [Xb: int,Y: int] : ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Xb),aa(int,int,uminus_uminus(int),Y)) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Xb),Y) ) ).

% lcm_neg2_int
tff(fact_7632_lcm__int__def,axiom,
    ! [Xb: int,Y: int] : ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Xb),Y) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),Xb))),aa(int,nat,nat2,aa(int,int,abs_abs(int),Y)))) ) ).

% lcm_int_def
tff(fact_7633_lcm__unique__nat,axiom,
    ! [A2: nat,D3: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),D3)
        & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),D3)
        & ! [E3: nat] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),E3)
              & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),E3) )
           => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),E3) ) )
    <=> ( D3 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),A2),B2) ) ) ).

% lcm_unique_nat
tff(fact_7634_dvd__lcm__I2__nat,axiom,
    ! [K: nat,Nb: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Ma),Nb)) ) ).

% dvd_lcm_I2_nat
tff(fact_7635_dvd__lcm__I1__nat,axiom,
    ! [K: nat,Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Ma)
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Ma),Nb)) ) ).

% dvd_lcm_I1_nat
tff(fact_7636_lcm__mono,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),D3)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),C2),D3)) ) ) ) ).

% lcm_mono
tff(fact_7637_dvd__lcmI1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2)) ) ) ).

% dvd_lcmI1
tff(fact_7638_dvd__lcmI2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2)) ) ) ).

% dvd_lcmI2
tff(fact_7639_lcm__dvdD1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)),C2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ).

% lcm_dvdD1
tff(fact_7640_lcm__dvdD2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)),C2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2) ) ) ).

% lcm_dvdD2
tff(fact_7641_lcm__least,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)),C2) ) ) ) ).

% lcm_least
tff(fact_7642_lcm_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,A2: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2)) ) ) ).

% lcm.left_commute
tff(fact_7643_lcm_Ocommute,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),A2) ) ) ).

% lcm.commute
tff(fact_7644_lcm_Oassoc,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2)) ) ) ).

% lcm.assoc
tff(fact_7645_dvd__lcm__I2__int,axiom,
    ! [I: int,Nb: int,Ma: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),I),Nb)
     => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),I),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Ma),Nb)) ) ).

% dvd_lcm_I2_int
tff(fact_7646_dvd__lcm__I1__int,axiom,
    ! [I: int,Ma: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),I),Ma)
     => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),I),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Ma),Nb)) ) ).

% dvd_lcm_I1_int
tff(fact_7647_gcd__dvd__lcm,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)) ) ).

% gcd_dvd_lcm
tff(fact_7648_lcm__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))),gcd_lcm(int)),gcd_lcm(int)) ).

% lcm_integer.rsp
tff(fact_7649_lcm__cases__int,axiom,
    ! [Xb: int,Y: int,P: fun(int,$o)] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Xb),Y)) ) )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),zero_zero(int))
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Xb),aa(int,int,uminus_uminus(int),Y))) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),zero_zero(int))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,uminus_uminus(int),Xb)),Y)) ) )
         => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),zero_zero(int))
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),zero_zero(int))
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,uminus_uminus(int),Xb)),aa(int,int,uminus_uminus(int),Y))) ) )
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Xb),Y)) ) ) ) ) ).

% lcm_cases_int
tff(fact_7650_lcm__ge__0__int,axiom,
    ! [Xb: int,Y: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Xb),Y)) ).

% lcm_ge_0_int
tff(fact_7651_lcm__pos__int,axiom,
    ! [Ma: int,Nb: int] :
      ( ( Ma != zero_zero(int) )
     => ( ( Nb != zero_zero(int) )
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Ma),Nb)) ) ) ).

% lcm_pos_int
tff(fact_7652_lcm__pos__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Ma),Nb)) ) ) ).

% lcm_pos_nat
tff(fact_7653_zero__eq__lcm__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_eq_lcm_iff
tff(fact_7654_lcm__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% lcm_eq_0_iff
tff(fact_7655_lcm__div__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),divide_divide(A,C2,A2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).

% lcm_div_unit2
tff(fact_7656_lcm__div__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),divide_divide(A,B2,A2)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).

% lcm_div_unit1
tff(fact_7657_prod__gcd__lcm__int,axiom,
    ! [Ma: int,Nb: int] : ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),Ma)),aa(int,int,abs_abs(int),Nb)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Ma),Nb)),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Ma),Nb)) ) ).

% prod_gcd_lcm_int
tff(fact_7658_Lcm__in__lcm__closed__set__nat,axiom,
    ! [M9: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),M9)
     => ( ( M9 != bot_bot(set(nat)) )
       => ( ! [M: nat,N: nat] :
              ( member(nat,M,M9)
             => ( member(nat,N,M9)
               => member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N),M9) ) )
         => member(nat,gcd_Lcm(nat,M9),M9) ) ) ) ).

% Lcm_in_lcm_closed_set_nat
tff(fact_7659_lcm__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2) = aa(A,A,normal6383669964737779283malize(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2))) ) ) ).

% lcm_gcd
tff(fact_7660_Lcm__set__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Xs: list(A)] : ( gcd_Lcm(A,aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,gcd_lcm(A),Xs),one_one(A)) ) ) ).

% Lcm_set_eq_fold
tff(fact_7661_Lcm__fin_Obounded__quasi__semilattice__set__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => bounde6485984586167503788ce_set(A,gcd_lcm(A),one_one(A),zero_zero(A),normal6383669964737779283malize(A)) ) ).

% Lcm_fin.bounded_quasi_semilattice_set_axioms
tff(fact_7662_lcm__altdef__int,axiom,
    ! [A2: int,B2: int] : ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),A2),B2) = divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),A2)),aa(int,int,abs_abs(int),B2)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2)) ) ).

% lcm_altdef_int
tff(fact_7663_gcd__lcm,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) = aa(A,A,normal6383669964737779283malize(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2))) ) ) ) ) ).

% gcd_lcm
tff(fact_7664_Lcm__nat__set__eq__fold,axiom,
    ! [Xs: list(nat)] : ( gcd_Lcm(nat,aa(list(nat),set(nat),set2(nat),Xs)) = aa(nat,nat,fold(nat,nat,gcd_lcm(nat),Xs),one_one(nat)) ) ).

% Lcm_nat_set_eq_fold
tff(fact_7665_Lcm__nat__def,axiom,
    ! [M9: set(nat)] :
      ( gcd_Lcm(nat,M9) = $ite(aa(set(nat),$o,finite_finite(nat),M9),lattic5214292709420241887eutr_F(nat,gcd_lcm(nat),one_one(nat),M9),zero_zero(nat)) ) ).

% Lcm_nat_def
tff(fact_7666_Lcm__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A3) = $ite(aa(set(A),$o,finite_finite(A),A3),finite_fold(A,A,gcd_lcm(A),one_one(A),A3),zero_zero(A)) ) ) ).

% Lcm_fin.eq_fold
tff(fact_7667_Lcm__fin_Onormalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] : ( aa(A,A,normal6383669964737779283malize(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A3)) = aa(set(A),A,semiring_gcd_Lcm_fin(A),A3) ) ) ).

% Lcm_fin.normalize
tff(fact_7668_Lcm__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( ~ aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A3) = zero_zero(A) ) ) ) ).

% Lcm_fin.infinite
tff(fact_7669_Lcm__fin_Oempty,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),bot_bot(set(A))) = one_one(A) ) ) ).

% Lcm_fin.empty
tff(fact_7670_is__unit__Lcm__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A3)),one_one(A))
        <=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A3) = one_one(A) ) ) ) ).

% is_unit_Lcm_fin_iff
tff(fact_7671_Lcm__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A3: set(A)] : ( aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),aa(set(A),A,semiring_gcd_Lcm_fin(A),A3)) ) ) ).

% Lcm_fin.insert
tff(fact_7672_Lcm__fin__eq__Lcm,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A3) = gcd_Lcm(A,A3) ) ) ) ).

% Lcm_fin_eq_Lcm
tff(fact_7673_Lcm__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A),B3: set(A)] : ( aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A3)),aa(set(A),A,semiring_gcd_Lcm_fin(A),B3)) ) ) ).

% Lcm_fin.union
tff(fact_7674_lcm__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : ( aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_lcm(code_integer),Xb),Xaa)) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(code_integer,int,code_int_of_integer,Xb)),aa(code_integer,int,code_int_of_integer,Xaa)) ) ).

% lcm_integer.rep_eq
tff(fact_7675_Lcm__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( member(A,A2,A3)
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),aa(set(A),A,semiring_gcd_Lcm_fin(A),A3)) = aa(set(A),A,semiring_gcd_Lcm_fin(A),A3) ) ) ) ).

% Lcm_fin.in_idem
tff(fact_7676_lcm__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_lcm(code_integer),aa(int,code_integer,code_integer_of_int,Xaa)),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Xaa),Xb)) ) ).

% lcm_integer.abs_eq
tff(fact_7677_Lcm__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B3: set(A),A3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),B3)),aa(set(A),A,semiring_gcd_Lcm_fin(A),A3)) = aa(set(A),A,semiring_gcd_Lcm_fin(A),A3) ) ) ) ).

% Lcm_fin.subset
tff(fact_7678_lcm__list__dvd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A),B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(list(A),set(A),set2(A),Xs))),B2)
        <=> ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X4),B2) ) ) ) ).

% lcm_list_dvd_iff
tff(fact_7679_lcm__list__least,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Bs: list(A),A2: A] :
          ( ! [B4: A] :
              ( member(A,B4,aa(list(A),set(A),set2(A),Bs))
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B4),A2) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(list(A),set(A),set2(A),Bs))),A2) ) ) ).

% lcm_list_least
tff(fact_7680_dvd__Lcm__fin,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( member(A,A2,A3)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(set(A),A,semiring_gcd_Lcm_fin(A),A3)) ) ) ).

% dvd_Lcm_fin
tff(fact_7681_Lcm__fin__least,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ! [B4: A] :
                ( member(A,B4,A3)
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B4),A2) )
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A3)),A2) ) ) ) ).

% Lcm_fin_least
tff(fact_7682_Lcm__fin__dvd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A),B2: A] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A3)),B2)
          <=> ! [X4: A] :
                ( member(A,X4,A3)
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X4),B2) ) ) ) ) ).

% Lcm_fin_dvd_iff
tff(fact_7683_Lcm__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A3) = zero_zero(A) )
          <=> member(A,zero_zero(A),A3) ) ) ) ).

% Lcm_fin_0_iff
tff(fact_7684_Lcm__fin__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A3) = one_one(A) )
        <=> ( ! [X4: A] :
                ( member(A,X4,A3)
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X4),one_one(A)) )
            & aa(set(A),$o,finite_finite(A),A3) ) ) ) ).

% Lcm_fin_1_iff
tff(fact_7685_lcm__code__integer,axiom,
    ! [A2: code_integer,B2: code_integer] : ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_lcm(code_integer),A2),B2) = divide_divide(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),A2)),aa(code_integer,code_integer,abs_abs(code_integer),B2)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),A2),B2)) ) ).

% lcm_code_integer
tff(fact_7686_lcm__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(code_integer,code_integer)),$o,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),$o),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),gcd_lcm(int)),gcd_lcm(code_integer)) ).

% lcm_integer.transfer
tff(fact_7687_Lcm__fin__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A),B2: A] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),image(A,A,aa(A,fun(A,A),times_times(A),B2),A3)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(set(A),A,semiring_gcd_Lcm_fin(A),A3))) ) ) ) ).

% Lcm_fin_mult
tff(fact_7688_Lcm__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( member(A,A2,A3)
         => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A3) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))) ) ) ) ).

% Lcm_fin.remove
tff(fact_7689_Lcm__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A3: set(A)] : ( aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))) ) ) ).

% Lcm_fin.insert_remove
tff(fact_7690_Lcm__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A)] : ( aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,gcd_lcm(A),Xs),one_one(A)) ) ) ).

% Lcm_fin.set_eq_fold
tff(fact_7691_Lcm__fin__def,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( semiring_gcd_Lcm_fin(A) = bounde2362111253966948842tice_F(A,gcd_lcm(A),one_one(A),zero_zero(A)) ) ) ).

% Lcm_fin_def
tff(fact_7692_lcm__integer__def,axiom,
    gcd_lcm(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),gcd_lcm(int)) ).

% lcm_integer_def
tff(fact_7693_lcm_Obounded__quasi__semilattice__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => bounde8507323023520639062attice(A,gcd_lcm(A),one_one(A),zero_zero(A),normal6383669964737779283malize(A)) ) ).

% lcm.bounded_quasi_semilattice_axioms
tff(fact_7694_gcd_Obounded__quasi__semilattice__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => bounde8507323023520639062attice(A,gcd_gcd(A),zero_zero(A),one_one(A),normal6383669964737779283malize(A)) ) ).

% gcd.bounded_quasi_semilattice_axioms
tff(fact_7695_bounded__quasi__semilattice__set_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde8507323023520639062attice(A,F2,Top,Bot,Normalize)
     => bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize) ) ).

% bounded_quasi_semilattice_set.intro
tff(fact_7696_bounded__quasi__semilattice__set_Oaxioms,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => bounde8507323023520639062attice(A,F2,Top,Bot,Normalize) ) ).

% bounded_quasi_semilattice_set.axioms
tff(fact_7697_bounded__quasi__semilattice__set__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
    <=> bounde8507323023520639062attice(A,F2,Top,Bot,Normalize) ) ).

% bounded_quasi_semilattice_set_def
tff(fact_7698_bounded__quasi__semilattice_Onormalize__right__idem,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A2: A,B2: A] :
      ( bounde8507323023520639062attice(A,F2,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F2,A2),aa(A,A,Normalize,B2)) = aa(A,A,aa(A,fun(A,A),F2,A2),B2) ) ) ).

% bounded_quasi_semilattice.normalize_right_idem
tff(fact_7699_bounded__quasi__semilattice_Otop__right__normalize,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A2: A] :
      ( bounde8507323023520639062attice(A,F2,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F2,A2),Top) = aa(A,A,Normalize,A2) ) ) ).

% bounded_quasi_semilattice.top_right_normalize
tff(fact_7700_bounded__quasi__semilattice_Onormalize__left__idem,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A2: A,B2: A] :
      ( bounde8507323023520639062attice(A,F2,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Normalize,A2)),B2) = aa(A,A,aa(A,fun(A,A),F2,A2),B2) ) ) ).

% bounded_quasi_semilattice.normalize_left_idem
tff(fact_7701_bounded__quasi__semilattice_Obottom__right__bottom,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A2: A] :
      ( bounde8507323023520639062attice(A,F2,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F2,A2),Bot) = Bot ) ) ).

% bounded_quasi_semilattice.bottom_right_bottom
tff(fact_7702_bounded__quasi__semilattice_Otop__left__normalize,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A2: A] :
      ( bounde8507323023520639062attice(A,F2,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F2,Top),A2) = aa(A,A,Normalize,A2) ) ) ).

% bounded_quasi_semilattice.top_left_normalize
tff(fact_7703_bounded__quasi__semilattice_Obottom__left__bottom,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A2: A] :
      ( bounde8507323023520639062attice(A,F2,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F2,Bot),A2) = Bot ) ) ).

% bounded_quasi_semilattice.bottom_left_bottom
tff(fact_7704_bounded__quasi__semilattice_Onormalize__bottom,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde8507323023520639062attice(A,F2,Top,Bot,Normalize)
     => ( aa(A,A,Normalize,Bot) = Bot ) ) ).

% bounded_quasi_semilattice.normalize_bottom
tff(fact_7705_bounded__quasi__semilattice_Onormalize__idem,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A2: A,B2: A] :
      ( bounde8507323023520639062attice(A,F2,Top,Bot,Normalize)
     => ( aa(A,A,Normalize,aa(A,A,aa(A,fun(A,A),F2,A2),B2)) = aa(A,A,aa(A,fun(A,A),F2,A2),B2) ) ) ).

% bounded_quasi_semilattice.normalize_idem
tff(fact_7706_bounded__quasi__semilattice_Oidem__normalize,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A2: A] :
      ( bounde8507323023520639062attice(A,F2,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F2,A2),A2) = aa(A,A,Normalize,A2) ) ) ).

% bounded_quasi_semilattice.idem_normalize
tff(fact_7707_bounded__quasi__semilattice_Onormalize__top,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde8507323023520639062attice(A,F2,Top,Bot,Normalize)
     => ( aa(A,A,Normalize,Top) = Top ) ) ).

% bounded_quasi_semilattice.normalize_top
tff(fact_7708_bounded__quasi__semilattice_Oright__idem,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A2: A,B2: A] :
      ( bounde8507323023520639062attice(A,F2,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),F2,A2),B2) ) ) ).

% bounded_quasi_semilattice.right_idem
tff(fact_7709_bounded__quasi__semilattice_Oleft__idem,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A2: A,B2: A] :
      ( bounde8507323023520639062attice(A,F2,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F2,A2),aa(A,A,aa(A,fun(A,A),F2,A2),B2)) = aa(A,A,aa(A,fun(A,A),F2,A2),B2) ) ) ).

% bounded_quasi_semilattice.left_idem
tff(fact_7710_normalize__div,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] : ( divide_divide(A,aa(A,A,normal6383669964737779283malize(A),A2),A2) = divide_divide(A,one_one(A),unit_f5069060285200089521factor(A,A2)) ) ) ).

% normalize_div
tff(fact_7711_normalize__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,normal6383669964737779283malize(A),unit_f5069060285200089521factor(A,A2)) = one_one(A) ) ) ) ).

% normalize_unit_factor
tff(fact_7712_unit__factor__simps_I1_J,axiom,
    unit_f5069060285200089521factor(nat,zero_zero(nat)) = zero_zero(nat) ).

% unit_factor_simps(1)
tff(fact_7713_unit__factor__idem,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] : ( unit_f5069060285200089521factor(A,unit_f5069060285200089521factor(A,A2)) = unit_f5069060285200089521factor(A,A2) ) ) ).

% unit_factor_idem
tff(fact_7714_unit__factor__simps_I2_J,axiom,
    ! [Nb: nat] : ( unit_f5069060285200089521factor(nat,aa(nat,nat,suc,Nb)) = one_one(nat) ) ).

% unit_factor_simps(2)
tff(fact_7715_unit__factor__eq__0__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] :
          ( ( unit_f5069060285200089521factor(A,A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% unit_factor_eq_0_iff
tff(fact_7716_unit__factor__0,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ( unit_f5069060285200089521factor(A,zero_zero(A)) = zero_zero(A) ) ) ).

% unit_factor_0
tff(fact_7717_unit__factor__1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( unit_f5069060285200089521factor(A,one_one(A)) = one_one(A) ) ) ).

% unit_factor_1
tff(fact_7718_unit__factor__mult__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,A2)),aa(A,A,normal6383669964737779283malize(A),A2)) = A2 ) ) ).

% unit_factor_mult_normalize
tff(fact_7719_normalize__mult__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A2)),unit_f5069060285200089521factor(A,A2)) = A2 ) ) ).

% normalize_mult_unit_factor
tff(fact_7720_div__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] : ( divide_divide(A,A2,unit_f5069060285200089521factor(A,A2)) = aa(A,A,normal6383669964737779283malize(A),A2) ) ) ).

% div_unit_factor
tff(fact_7721_div__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] : ( divide_divide(A,A2,aa(A,A,normal6383669964737779283malize(A),A2)) = unit_f5069060285200089521factor(A,A2) ) ) ).

% div_normalize
tff(fact_7722_inv__unit__factor__eq__0__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] :
          ( ( divide_divide(A,one_one(A),unit_f5069060285200089521factor(A,A2)) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% inv_unit_factor_eq_0_iff
tff(fact_7723_mult__one__div__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,one_one(A),unit_f5069060285200089521factor(A,B2))) = divide_divide(A,A2,unit_f5069060285200089521factor(A,B2)) ) ) ).

% mult_one_div_unit_factor
tff(fact_7724_unit__factor__mult__unit__right,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( unit_f5069060285200089521factor(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),unit_f5069060285200089521factor(A,B2)),A2) ) ) ) ).

% unit_factor_mult_unit_right
tff(fact_7725_unit__factor__mult__unit__left,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( unit_f5069060285200089521factor(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),unit_f5069060285200089521factor(A,B2)) ) ) ) ).

% unit_factor_mult_unit_left
tff(fact_7726_unit__factor__lcm,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)) = $ite(
              ( ( A2 = zero_zero(A) )
              | ( B2 = zero_zero(A) ) ),
              zero_zero(A),
              one_one(A) ) ) ) ).

% unit_factor_lcm
tff(fact_7727_unit__factor__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( unit_f5069060285200089521factor(A,aa(A,A,normal6383669964737779283malize(A),A2)) = one_one(A) ) ) ) ).

% unit_factor_normalize
tff(fact_7728_lcm__mult__distrib,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [K: A,A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2))),unit_f5069060285200089521factor(A,K)) ) ) ).

% lcm_mult_distrib
tff(fact_7729_mult__lcm__right,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(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),gcd_lcm(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),unit_f5069060285200089521factor(A,C2)) ) ) ).

% mult_lcm_right
tff(fact_7730_mult__lcm__left,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(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),gcd_lcm(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,C2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ).

% mult_lcm_left
tff(fact_7731_dvd__unit__factor__div,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
         => ( unit_f5069060285200089521factor(A,divide_divide(A,A2,B2)) = divide_divide(A,unit_f5069060285200089521factor(A,A2),unit_f5069060285200089521factor(A,B2)) ) ) ) ).

% dvd_unit_factor_div
tff(fact_7732_unit__factor__is__unit,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),unit_f5069060285200089521factor(A,A2)),one_one(A)) ) ) ).

% unit_factor_is_unit
tff(fact_7733_is__unit__unit__factor,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( unit_f5069060285200089521factor(A,A2) = A2 ) ) ) ).

% is_unit_unit_factor
tff(fact_7734_unit__factor__nat__def,axiom,
    ! [Nb: nat] :
      ( unit_f5069060285200089521factor(nat,Nb) = $ite(Nb = zero_zero(nat),zero_zero(nat),one_one(nat)) ) ).

% unit_factor_nat_def
tff(fact_7735_unit__factor__dvd,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),unit_f5069060285200089521factor(A,A2)),B2) ) ) ).

% unit_factor_dvd
tff(fact_7736_unit__factor__self,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),unit_f5069060285200089521factor(A,A2)),A2) ) ).

% unit_factor_self
tff(fact_7737_unit__factor__mult,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [A2: A,B2: A] : ( unit_f5069060285200089521factor(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),unit_f5069060285200089521factor(A,A2)),unit_f5069060285200089521factor(A,B2)) ) ) ).

% unit_factor_mult
tff(fact_7738_gcd__mult__distrib,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [K: A,A2: A,B2: A] : ( aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2))),unit_f5069060285200089521factor(A,K)) ) ) ).

% gcd_mult_distrib
tff(fact_7739_mult__gcd__right,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(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),gcd_gcd(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),unit_f5069060285200089521factor(A,C2)) ) ) ).

% mult_gcd_right
tff(fact_7740_mult__gcd__left,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(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),gcd_gcd(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,C2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ).

% mult_gcd_left
tff(fact_7741_normalize__unit__factor__eqI,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A2) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => ( ( unit_f5069060285200089521factor(A,A2) = unit_f5069060285200089521factor(A,B2) )
           => ( A2 = B2 ) ) ) ) ).

% normalize_unit_factor_eqI
tff(fact_7742_unit__factor__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) = $ite(
              ( ( A2 = zero_zero(A) )
              & ( B2 = zero_zero(A) ) ),
              zero_zero(A),
              one_one(A) ) ) ) ).

% unit_factor_gcd
tff(fact_7743_coprime__crossproduct_H,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [B2: A,D3: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( unit_f5069060285200089521factor(A,B2) = unit_f5069060285200089521factor(A,D3) )
           => ( algebr8660921524188924756oprime(A,A2,B2)
             => ( algebr8660921524188924756oprime(A,C2,D3)
               => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),D3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
                <=> ( ( A2 = C2 )
                    & ( B2 = D3 ) ) ) ) ) ) ) ) ).

% coprime_crossproduct'
tff(fact_7744_unit__factor__Lcm,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( unit_f5069060285200089521factor(A,gcd_Lcm(A,A3)) = $ite(gcd_Lcm(A,A3) = zero_zero(A),zero_zero(A),one_one(A)) ) ) ).

% unit_factor_Lcm
tff(fact_7745_unit__factor__Gcd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( unit_f5069060285200089521factor(A,gcd_Gcd(A,A3)) = $ite(gcd_Gcd(A,A3) = zero_zero(A),zero_zero(A),one_one(A)) ) ) ).

% unit_factor_Gcd
tff(fact_7746_normalize__idem__imp__unit__factor__eq,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A2) = A2 )
         => ( unit_f5069060285200089521factor(A,A2) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ) ) ).

% normalize_idem_imp_unit_factor_eq
tff(fact_7747_unit__factor__Lcm__fin,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] : ( unit_f5069060285200089521factor(A,aa(set(A),A,semiring_gcd_Lcm_fin(A),A3)) = aa($o,A,zero_neq_one_of_bool(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A3) != zero_zero(A)) ) ) ).

% unit_factor_Lcm_fin
tff(fact_7748_unit__factor__Gcd__fin,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] : ( unit_f5069060285200089521factor(A,aa(set(A),A,semiring_gcd_Gcd_fin(A),A3)) = aa($o,A,zero_neq_one_of_bool(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A3) != zero_zero(A)) ) ) ).

% unit_factor_Gcd_fin
tff(fact_7749_and_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => comm_monoid(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% and.comm_monoid_axioms
tff(fact_7750_card__Plus__conv__if,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B)] :
      ( aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A3,B3)) = $ite(
          ( aa(set(A),$o,finite_finite(A),A3)
          & aa(set(B),$o,finite_finite(B),B3) ),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B3)),
          zero_zero(nat) ) ) ).

% card_Plus_conv_if
tff(fact_7751_card__Plus,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( aa(set(A),$o,finite_finite(A),A3)
     => ( aa(set(B),$o,finite_finite(B),B3)
       => ( aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A3,B3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B3)) ) ) ) ).

% card_Plus
tff(fact_7752_sup__bot_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => comm_monoid(A,sup_sup(A),bot_bot(A)) ) ).

% sup_bot.comm_monoid_axioms
tff(fact_7753_mult_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => comm_monoid(A,times_times(A),one_one(A)) ) ).

% mult.comm_monoid_axioms
tff(fact_7754_comm__monoid_Ocomm__neutral,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,A2: A] :
      ( comm_monoid(A,F2,Z)
     => ( aa(A,A,aa(A,fun(A,A),F2,A2),Z) = A2 ) ) ).

% comm_monoid.comm_neutral
tff(fact_7755_gcd__nat_Ocomm__monoid__axioms,axiom,
    comm_monoid(nat,gcd_gcd(nat),zero_zero(nat)) ).

% gcd_nat.comm_monoid_axioms
tff(fact_7756_add_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => comm_monoid(A,plus_plus(A),zero_zero(A)) ) ).

% add.comm_monoid_axioms
tff(fact_7757_or_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => comm_monoid(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).

% or.comm_monoid_axioms
tff(fact_7758_xor_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => comm_monoid(A,bit_se5824344971392196577ns_xor(A),zero_zero(A)) ) ).

% xor.comm_monoid_axioms
tff(fact_7759_max__nat_Ocomm__monoid__axioms,axiom,
    comm_monoid(nat,ord_max(nat),zero_zero(nat)) ).

% max_nat.comm_monoid_axioms
tff(fact_7760_semilattice__neutr_Oaxioms_I2_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A] :
      ( semilattice_neutr(A,F2,Z)
     => comm_monoid(A,F2,Z) ) ).

% semilattice_neutr.axioms(2)
tff(fact_7761_inf__top_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => comm_monoid(A,inf_inf(A),top_top(A)) ) ).

% inf_top.comm_monoid_axioms
tff(fact_7762_is__num_Osimps,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A2: A] :
          ( neg_numeral_is_num(A,A2)
        <=> ( ( A2 = one_one(A) )
            | ? [X4: A] :
                ( ( A2 = aa(A,A,uminus_uminus(A),X4) )
                & neg_numeral_is_num(A,X4) )
            | ? [X4: A,Y5: A] :
                ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Y5) )
                & neg_numeral_is_num(A,X4)
                & neg_numeral_is_num(A,Y5) ) ) ) ) ).

% is_num.simps
tff(fact_7763_is__num__normalize_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => neg_numeral_is_num(A,one_one(A)) ) ).

% is_num_normalize(4)
tff(fact_7764_is__num__add__left__commute,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( neg_numeral_is_num(A,Xb)
         => ( neg_numeral_is_num(A,Y)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z)) ) ) ) ) ).

% is_num_add_left_commute
tff(fact_7765_is__num__add__commute,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xb: A,Y: A] :
          ( neg_numeral_is_num(A,Xb)
         => ( neg_numeral_is_num(A,Y)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Xb) ) ) ) ) ).

% is_num_add_commute
tff(fact_7766_is__num__normalize_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xb: A,Y: A] :
          ( neg_numeral_is_num(A,Xb)
         => ( neg_numeral_is_num(A,Y)
           => neg_numeral_is_num(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) ) ) ) ).

% is_num_normalize(6)
tff(fact_7767_is__num__numeral,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_is_num(A,aa(num,A,numeral_numeral(A),K)) ) ).

% is_num_numeral
tff(fact_7768_is__num__normalize_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xb: A] :
          ( neg_numeral_is_num(A,Xb)
         => neg_numeral_is_num(A,aa(A,A,uminus_uminus(A),Xb)) ) ) ).

% is_num_normalize(5)
tff(fact_7769_is__num_Ocases,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A2: A] :
          ( neg_numeral_is_num(A,A2)
         => ( ( A2 != one_one(A) )
           => ( ! [X: A] :
                  ( ( A2 = aa(A,A,uminus_uminus(A),X) )
                 => ~ neg_numeral_is_num(A,X) )
             => ~ ! [X: A,Y3: A] :
                    ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3) )
                   => ( neg_numeral_is_num(A,X)
                     => ~ neg_numeral_is_num(A,Y3) ) ) ) ) ) ) ).

% is_num.cases
tff(fact_7770_times__num__def,axiom,
    ! [Ma: num,Nb: num] : ( aa(num,num,aa(num,fun(num,num),times_times(num),Ma),Nb) = aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,nat_of_num,Ma)),aa(num,nat,nat_of_num,Nb))) ) ).

% times_num_def
tff(fact_7771_arg__max__nat__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,F2: fun(A,nat),B2: nat] :
      ( aa(A,$o,P,Xb)
     => ( ! [Y3: A] :
            ( aa(A,$o,P,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y3)),B2) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,lattices_ord_arg_max(A,nat,F2,P))) ) ) ).

% arg_max_nat_le
tff(fact_7772_arg__maxI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [P: fun(A,$o),Xb: A,F2: fun(A,B),Q: fun(A,$o)] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y3: A] :
                ( aa(A,$o,P,Y3)
               => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y3)) )
           => ( ! [X: A] :
                  ( aa(A,$o,P,X)
                 => ( ! [Y4: A] :
                        ( aa(A,$o,P,Y4)
                       => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y4)) )
                   => aa(A,$o,Q,X) ) )
             => aa(A,$o,Q,lattices_ord_arg_max(A,B,F2,P)) ) ) ) ) ).

% arg_maxI
tff(fact_7773_nat__of__num__mult,axiom,
    ! [Xb: num,Y: num] : ( aa(num,nat,nat_of_num,aa(num,num,aa(num,fun(num,num),times_times(num),Xb),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,nat_of_num,Xb)),aa(num,nat,nat_of_num,Y)) ) ).

% nat_of_num_mult
tff(fact_7774_nat__of__num__sqr,axiom,
    ! [Xb: num] : ( aa(num,nat,nat_of_num,sqr(Xb)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,nat_of_num,Xb)),aa(num,nat,nat_of_num,Xb)) ) ).

% nat_of_num_sqr
tff(fact_7775_nat__of__num__inc,axiom,
    ! [Xb: num] : ( aa(num,nat,nat_of_num,inc(Xb)) = aa(nat,nat,suc,aa(num,nat,nat_of_num,Xb)) ) ).

% nat_of_num_inc
tff(fact_7776_arg__max__natI,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) )
       => aa(A,$o,P,lattices_ord_arg_max(A,nat,F2,P)) ) ) ).

% arg_max_natI
tff(fact_7777_nat__of__num__inverse,axiom,
    ! [Xb: num] : ( aa(nat,num,num_of_nat,aa(num,nat,nat_of_num,Xb)) = Xb ) ).

% nat_of_num_inverse
tff(fact_7778_num__eq__iff,axiom,
    ! [Xb: num,Y: num] :
      ( ( Xb = Y )
    <=> ( aa(num,nat,nat_of_num,Xb) = aa(num,nat,nat_of_num,Y) ) ) ).

% num_eq_iff
tff(fact_7779_nat__of__num__numeral,axiom,
    nat_of_num = numeral_numeral(nat) ).

% nat_of_num_numeral
tff(fact_7780_less__num__def,axiom,
    ! [Ma: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),Ma),Nb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,nat_of_num,Ma)),aa(num,nat,nat_of_num,Nb)) ) ).

% less_num_def
tff(fact_7781_nat__of__num__pos,axiom,
    ! [Xb: num] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(num,nat,nat_of_num,Xb)) ).

% nat_of_num_pos
tff(fact_7782_nat__of__num__neq__0,axiom,
    ! [Xb: num] : ( aa(num,nat,nat_of_num,Xb) != zero_zero(nat) ) ).

% nat_of_num_neq_0
tff(fact_7783_nat__of__num__code_I2_J,axiom,
    ! [Nb: num] :
      ( aa(num,nat,nat_of_num,bit0(Nb)) = $let(
          m2: nat,
          m2:= aa(num,nat,nat_of_num,Nb),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),m2),m2) ) ) ).

% nat_of_num_code(2)
tff(fact_7784_nat__of__num__code_I1_J,axiom,
    aa(num,nat,nat_of_num,one2) = one_one(nat) ).

% nat_of_num_code(1)
tff(fact_7785_nat__of__num_Osimps_I2_J,axiom,
    ! [Xb: num] : ( aa(num,nat,nat_of_num,bit0(Xb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,Xb)),aa(num,nat,nat_of_num,Xb)) ) ).

% nat_of_num.simps(2)
tff(fact_7786_nat__of__num__add,axiom,
    ! [Xb: num,Y: num] : ( aa(num,nat,nat_of_num,aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,Xb)),aa(num,nat,nat_of_num,Y)) ) ).

% nat_of_num_add
tff(fact_7787_less__eq__num__def,axiom,
    ! [Ma: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Ma),Nb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,nat_of_num,Ma)),aa(num,nat,nat_of_num,Nb)) ) ).

% less_eq_num_def
tff(fact_7788_nat__of__num_Osimps_I1_J,axiom,
    aa(num,nat,nat_of_num,one2) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_of_num.simps(1)
tff(fact_7789_nat__of__num_Osimps_I3_J,axiom,
    ! [Xb: num] : ( aa(num,nat,nat_of_num,aa(num,num,bit1,Xb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,Xb)),aa(num,nat,nat_of_num,Xb))) ) ).

% nat_of_num.simps(3)
tff(fact_7790_num__of__nat__inverse,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(num,nat,nat_of_num,aa(nat,num,num_of_nat,Nb)) = Nb ) ) ).

% num_of_nat_inverse
tff(fact_7791_nat__of__num__code_I3_J,axiom,
    ! [Nb: num] :
      ( aa(num,nat,nat_of_num,aa(num,num,bit1,Nb)) = $let(
          m2: nat,
          m2:= aa(num,nat,nat_of_num,Nb),
          aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),m2),m2)) ) ) ).

% nat_of_num_code(3)
tff(fact_7792_plus__num__def,axiom,
    ! [Ma: num,Nb: num] : ( aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),Nb) = aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,Ma)),aa(num,nat,nat_of_num,Nb))) ) ).

% plus_num_def
tff(fact_7793_arg__max__nat__lemma,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) )
       => ( aa(A,$o,P,lattices_ord_arg_max(A,nat,F2,P))
          & ! [Y4: A] :
              ( aa(A,$o,P,Y4)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y4)),aa(A,nat,F2,lattices_ord_arg_max(A,nat,F2,P))) ) ) ) ) ).

% arg_max_nat_lemma
tff(fact_7794_distinct__adj__map__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xs: list(A)] :
      ( inj_on(A,B,F2,aa(list(A),set(A),set2(A),Xs))
     => ( distinct_adj(B,aa(list(A),list(B),map(A,B,F2),Xs))
      <=> distinct_adj(A,Xs) ) ) ).

% distinct_adj_map_iff
tff(fact_7795_distinct__adj__mapI,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,B)] :
      ( distinct_adj(A,Xs)
     => ( inj_on(A,B,F2,aa(list(A),set(A),set2(A),Xs))
       => distinct_adj(B,aa(list(A),list(B),map(A,B,F2),Xs)) ) ) ).

% distinct_adj_mapI
tff(fact_7796_real__floor__code,axiom,
    ! [Xb: rat] : ( aa(real,int,archim6421214686448440834_floor(real),aa(rat,real,ratreal,Xb)) = aa(rat,int,archim6421214686448440834_floor(rat),Xb) ) ).

% real_floor_code
tff(fact_7797_prod_H__def,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ( groups1962203154675924110t_prod(A,B) = groups_comm_monoid_G(B,A,times_times(B),one_one(B)) ) ) ).

% prod'_def
tff(fact_7798_real__minus__code,axiom,
    ! [Xb: rat,Y: rat] : ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(rat,real,ratreal,Xb)),aa(rat,real,ratreal,Y)) = aa(rat,real,ratreal,aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Xb),Y)) ) ).

% real_minus_code
tff(fact_7799_real__less__eq__code,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(rat,real,ratreal,Xb)),aa(rat,real,ratreal,Y))
    <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),Xb),Y) ) ).

% real_less_eq_code
tff(fact_7800_real__less__code,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(rat,real,ratreal,Xb)),aa(rat,real,ratreal,Y))
    <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Xb),Y) ) ).

% real_less_code
tff(fact_7801_zero__real__code,axiom,
    zero_zero(real) = aa(rat,real,ratreal,zero_zero(rat)) ).

% zero_real_code
tff(fact_7802_Ratreal__def,axiom,
    ratreal = field_char_0_of_rat(real) ).

% Ratreal_def
tff(fact_7803_real__uminus__code,axiom,
    ! [Xb: rat] : ( aa(real,real,uminus_uminus(real),aa(rat,real,ratreal,Xb)) = aa(rat,real,ratreal,aa(rat,rat,uminus_uminus(rat),Xb)) ) ).

% real_uminus_code
tff(fact_7804_one__real__code,axiom,
    one_one(real) = aa(rat,real,ratreal,one_one(rat)) ).

% one_real_code
tff(fact_7805_real__times__code,axiom,
    ! [Xb: rat,Y: rat] : ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(rat,real,ratreal,Xb)),aa(rat,real,ratreal,Y)) = aa(rat,real,ratreal,aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),Xb),Y)) ) ).

% real_times_code
tff(fact_7806_real__plus__code,axiom,
    ! [Xb: rat,Y: rat] : ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(rat,real,ratreal,Xb)),aa(rat,real,ratreal,Y)) = aa(rat,real,ratreal,aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),Xb),Y)) ) ).

% real_plus_code
tff(fact_7807_real__inverse__code,axiom,
    ! [Xb: rat] : ( aa(real,real,inverse_inverse(real),aa(rat,real,ratreal,Xb)) = aa(rat,real,ratreal,aa(rat,rat,inverse_inverse(rat),Xb)) ) ).

% real_inverse_code
tff(fact_7808_real__divide__code,axiom,
    ! [Xb: rat,Y: rat] : ( divide_divide(real,aa(rat,real,ratreal,Xb),aa(rat,real,ratreal,Y)) = aa(rat,real,ratreal,divide_divide(rat,Xb,Y)) ) ).

% real_divide_code
tff(fact_7809_sum_H__def,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ( groups1027152243600224163dd_sum(A,B) = groups_comm_monoid_G(B,A,plus_plus(B),zero_zero(B)) ) ) ).

% sum'_def
tff(fact_7810_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc: A] :
      ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),F2),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),A2),aa(A,product_prod(nat,A),product_Pair(nat,A,B2),Acc))))
     => ( 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.psimps
tff(fact_7811_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,Xb: fun(nat,fun(A,A)),Xaa: nat,Xba: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xb,Xaa,Xba,Xc) = Y )
     => ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),Xb),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),Xaa),aa(A,product_prod(nat,A),product_Pair(nat,A,Xba),Xc))))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xba),Xaa),Xc,set_fo6178422350223883121st_nat(A,Xb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xaa),one_one(nat)),Xba,aa(A,A,aa(nat,fun(A,A),Xb,Xaa),Xc))) )
           => ~ accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),Xb),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),Xaa),aa(A,product_prod(nat,A),product_Pair(nat,A,Xba),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_7812_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A: $tType,A0: fun(nat,fun(A,A)),A1: nat,A22: nat,A32: A,P: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))))] :
      ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),A0),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),A1),aa(A,product_prod(nat,A),product_Pair(nat,A,A22),A32))))
     => ( ! [F3: fun(nat,fun(A,A)),A4: nat,B4: nat,Acc2: A] :
            ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),A4),aa(A,product_prod(nat,A),product_Pair(nat,A,B4),Acc2))))
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B4),A4)
               => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),one_one(nat))),B4),aa(A,A,aa(nat,fun(A,A),F3,A4),Acc2)) )
             => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F3),A4),B4),Acc2) ) )
       => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,A0),A1),A22),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_7813_prod_Ocomm__monoid__list__set__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => groups4802862169904069756st_set(A,times_times(A),one_one(A)) ) ).

% prod.comm_monoid_list_set_axioms
tff(fact_7814_sum_Ocomm__monoid__list__set__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => groups4802862169904069756st_set(A,plus_plus(A),zero_zero(A)) ) ).

% sum.comm_monoid_list_set_axioms
tff(fact_7815_comm__monoid__list__set_Oset__conv__list,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(B,A),Xs: list(B)] :
      ( groups4802862169904069756st_set(A,F2,Z)
     => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z),G),aa(list(B),set(B),set2(B),Xs)) = aa(list(A),A,groups_monoid_F(A,F2,Z),aa(list(B),list(A),map(B,A,G),remdups(B,Xs))) ) ) ).

% comm_monoid_list_set.set_conv_list
tff(fact_7816_comm__monoid__list__set_Odistinct__set__conv__list,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z: A,Xs: list(B),G: fun(B,A)] :
      ( groups4802862169904069756st_set(A,F2,Z)
     => ( distinct(B,Xs)
       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z),G),aa(list(B),set(B),set2(B),Xs)) = aa(list(A),A,groups_monoid_F(A,F2,Z),aa(list(B),list(A),map(B,A,G),Xs)) ) ) ) ).

% comm_monoid_list_set.distinct_set_conv_list
tff(fact_7817_comm__monoid__add__class_Osum__def,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ( groups7311177749621191930dd_sum(A,B) = groups_comm_monoid_F(B,A,plus_plus(B),zero_zero(B)) ) ) ).

% comm_monoid_add_class.sum_def
tff(fact_7818_comm__monoid__mult__class_Oprod__def,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ( groups7121269368397514597t_prod(A,B) = groups_comm_monoid_F(B,A,times_times(B),one_one(B)) ) ) ).

% comm_monoid_mult_class.prod_def
tff(fact_7819_comm__monoid__set_Otriangle__reindex,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,fun(nat,A)),Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups_comm_monoid_F(A,product_prod(nat,nat),F2,Z),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_hu(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_afl(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F2),Z),G)),set_ord_lessThan(nat,Nb)) ) ) ).

% comm_monoid_set.triangle_reindex
tff(fact_7820_comm__monoid__set_Ozero__middle,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,P3: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P3)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aa(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),aTP_Lamp_afm(A,fun(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A)))),Z),K),G),H)),set_ord_atMost(nat,P3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_afn(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),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% comm_monoid_set.zero_middle
tff(fact_7821_comm__monoid__set_OSuc__reindex__ivl,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Ma: nat,Nb: nat,G: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
       => ( aa(A,A,aa(A,fun(A,A),F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,Ma,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),F2,aa(nat,A,G,Ma)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aTP_Lamp_on(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Ma,Nb))) ) ) ) ).

% comm_monoid_set.Suc_reindex_ivl
tff(fact_7822_comm__monoid__set_OatLeast__Suc__atMost,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Ma: nat,Nb: nat,G: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,Ma,Nb)) = aa(A,A,aa(A,fun(A,A),F2,aa(nat,A,G,Ma)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),Nb))) ) ) ) ).

% comm_monoid_set.atLeast_Suc_atMost
tff(fact_7823_comm__monoid__set_Onat__ivl__Suc_H,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Ma: nat,Nb: nat,G: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,Nb))
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),F2,aa(nat,A,G,aa(nat,nat,suc,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,Ma,Nb))) ) ) ) ).

% comm_monoid_set.nat_ivl_Suc'
tff(fact_7824_comm__monoid__set_OatLeastLessThan__Suc,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,A2: nat,B2: nat,G: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% comm_monoid_set.atLeastLessThan_Suc
tff(fact_7825_comm__monoid__set_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Ma: nat,K: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% comm_monoid_set.atLeastLessThan_shift_bounds
tff(fact_7826_comm__monoid__set_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Ma: nat,K: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% comm_monoid_set.atLeastAtMost_shift_bounds
tff(fact_7827_comm__monoid__set_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Ma: nat,K: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(nat,fun(nat,A),aTP_Lamp_afo(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% comm_monoid_set.shift_bounds_cl_nat_ivl
tff(fact_7828_comm__monoid__set_OatLeastAtMost__rev,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Nb: nat,Ma: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,Nb,Ma)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_afp(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Ma)),set_or1337092689740270186AtMost(nat,Nb,Ma)) ) ) ).

% comm_monoid_set.atLeastAtMost_rev
tff(fact_7829_comm__monoid__set_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Ma: nat,K: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(nat,fun(nat,A),aTP_Lamp_afo(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% comm_monoid_set.shift_bounds_nat_ivl
tff(fact_7830_comm__monoid__set_Onat__diff__reindex,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(nat,fun(nat,A),aTP_Lamp_afq(fun(nat,A),fun(nat,fun(nat,A)),G),Nb)),set_ord_lessThan(nat,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_ord_lessThan(nat,Nb)) ) ) ).

% comm_monoid_set.nat_diff_reindex
tff(fact_7831_comm__monoid__set_OlessThan__Suc__shift,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),F2,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aTP_Lamp_on(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb))) ) ) ).

% comm_monoid_set.lessThan_Suc_shift
tff(fact_7832_comm__monoid__set_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(nat,A,G,Nb)) ) ) ).

% comm_monoid_set.atLeast0_lessThan_Suc
tff(fact_7833_comm__monoid__set_OatMost__Suc__shift,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_ord_atMost(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),F2,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aTP_Lamp_on(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Nb))) ) ) ).

% comm_monoid_set.atMost_Suc_shift
tff(fact_7834_comm__monoid__set_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ) ).

% comm_monoid_set.atLeast0_atMost_Suc
tff(fact_7835_comm__monoid__set_OlessThan__Suc,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_ord_lessThan(nat,Nb))),aa(nat,A,G,Nb)) ) ) ).

% comm_monoid_set.lessThan_Suc
tff(fact_7836_comm__monoid__set_Oop__ivl__Suc,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Ma: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,Ma,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma),Z,aa(A,A,aa(A,fun(A,A),F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,Ma,Nb))),aa(nat,A,G,Nb))) ) ) ).

% comm_monoid_set.op_ivl_Suc
tff(fact_7837_comm__monoid__set_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Ma: nat,Nb: nat,G: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Nb)
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,Ma,Nb)) = aa(A,A,aa(A,fun(A,A),F2,aa(nat,A,G,Ma)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),Nb))) ) ) ) ).

% comm_monoid_set.atLeast_Suc_lessThan
tff(fact_7838_comm__monoid__set_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Ma: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aTP_Lamp_on(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% comm_monoid_set.shift_bounds_Suc_ivl
tff(fact_7839_comm__monoid__set_OatMost__Suc,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_ord_atMost(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_ord_atMost(nat,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ) ).

% comm_monoid_set.atMost_Suc
tff(fact_7840_comm__monoid__set_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Ma: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aTP_Lamp_on(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% comm_monoid_set.shift_bounds_cl_Suc_ivl
tff(fact_7841_comm__monoid__set_Ocl__ivl__Suc,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Ma: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),Ma),Z,aa(A,A,aa(A,fun(A,A),F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,Ma,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb)))) ) ) ).

% comm_monoid_set.cl_ivl_Suc
tff(fact_7842_comm__monoid__set_Ohead__if,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Ma: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,Ma,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Ma),Z,aa(A,A,aa(A,fun(A,A),F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,Ma,Nb))),aa(nat,A,G,Nb))) ) ) ).

% comm_monoid_set.head_if
tff(fact_7843_comm__monoid__set_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Ma: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% comm_monoid_set.atLeast_Suc_atMost_Suc_shift
tff(fact_7844_comm__monoid__set_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Ma: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% comm_monoid_set.atLeast_Suc_lessThan_Suc_shift
tff(fact_7845_prod_Ocomm__monoid__set__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => groups778175481326437816id_set(A,times_times(A),one_one(A)) ) ).

% prod.comm_monoid_set_axioms
tff(fact_7846_sum_Ocomm__monoid__set__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => groups778175481326437816id_set(A,plus_plus(A),zero_zero(A)) ) ).

% sum.comm_monoid_set_axioms
tff(fact_7847_comm__monoid__set_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(B)
     => ! [F2: fun(A,fun(A,A)),Z: A,A2: B,C2: B,B2: B,D3: B,G: fun(B,A),H: fun(B,A)] :
          ( groups778175481326437816id_set(A,F2,Z)
         => ( ( A2 = C2 )
           => ( ( B2 = D3 )
             => ( ! [X: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),X)
                   => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),D3)
                     => ( aa(B,A,G,X) = aa(B,A,H,X) ) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z),G),set_or7035219750837199246ssThan(B,A2,B2)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z),H),set_or7035219750837199246ssThan(B,C2,D3)) ) ) ) ) ) ) ).

% comm_monoid_set.ivl_cong
tff(fact_7848_comm__monoid__set_Oub__add__nat,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Ma: nat,Nb: nat,G: fun(nat,A),P3: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P3))) = aa(A,A,aa(A,fun(A,A),F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,Ma,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P3)))) ) ) ) ).

% comm_monoid_set.ub_add_nat
tff(fact_7849_comm__monoid__set_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),F2,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ) ) ).

% comm_monoid_set.atLeast0_atMost_Suc_shift
tff(fact_7850_comm__monoid__set_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),F2,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ) ).

% comm_monoid_set.atLeast0_lessThan_Suc_shift
tff(fact_7851_comm__monoid__set_OatLeastLessThan__rev,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Nb: nat,Ma: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,Nb,Ma)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_afr(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Ma)),set_or7035219750837199246ssThan(nat,Nb,Ma)) ) ) ).

% comm_monoid_set.atLeastLessThan_rev
tff(fact_7852_comm__monoid__set_Onested__swap,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,A2: fun(nat,fun(nat,A)),Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_afs(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F2),Z),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(nat,fun(nat,A),aa(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A))),aTP_Lamp_afu(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)))),F2),Z),A2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ) ).

% comm_monoid_set.nested_swap
tff(fact_7853_comm__monoid__set_OatLeast1__atMost__eq,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aTP_Lamp_on(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb)) ) ) ).

% comm_monoid_set.atLeast1_atMost_eq
tff(fact_7854_comm__monoid__set_OatMost__shift,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_ord_atMost(nat,Nb)) = aa(A,A,aa(A,fun(A,A),F2,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aTP_Lamp_on(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb))) ) ) ).

% comm_monoid_set.atMost_shift
tff(fact_7855_comm__monoid__set_OIf__cases,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z: A,A3: set(B),P: fun(B,$o),H: fun(B,A),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(B),$o,finite_finite(B),A3)
       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_afv(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H),G)),A3) = aa(A,A,aa(A,fun(A,A),F2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z),H),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),collect(B,P)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),aa(set(B),set(B),uminus_uminus(set(B)),collect(B,P))))) ) ) ) ).

% comm_monoid_set.If_cases
tff(fact_7856_comm__monoid__set_Onat__group,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),K: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,A),fun(nat,fun(nat,A))),aTP_Lamp_afw(fun(A,fun(A,A)),fun(A,fun(fun(nat,A),fun(nat,fun(nat,A)))),F2),Z),G),K)),set_ord_lessThan(nat,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))) ) ) ).

% comm_monoid_set.nat_group
tff(fact_7857_comm__monoid__set_Onested__swap_H,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,A2: fun(nat,fun(nat,A)),Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_afx(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F2),Z),A2)),set_ord_atMost(nat,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(nat,fun(nat,A),aa(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A))),aTP_Lamp_afu(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)))),F2),Z),A2),Nb)),set_ord_lessThan(nat,Nb)) ) ) ).

% comm_monoid_set.nested_swap'
tff(fact_7858_comm__monoid__set_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(int,A),Ma: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups_comm_monoid_F(A,int,F2,Z),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),Ma),aa(nat,int,semiring_1_of_nat(int),Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% comm_monoid_set.atLeast_int_atMost_int_shift
tff(fact_7859_comm__monoid__set_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(int,A),Ma: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups_comm_monoid_F(A,int,F2,Z),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),Ma),aa(nat,int,semiring_1_of_nat(int),Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% comm_monoid_set.atLeast_int_lessThan_int_shift
tff(fact_7860_comm__monoid__set_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Ma: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,Ma,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),Ma))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma))) ) ) ).

% comm_monoid_set.atLeastLessThan_shift_0
tff(fact_7861_comm__monoid__set_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Nb: nat,Ma: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,Nb,Ma)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_afp(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Ma)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Ma)) ) ) ).

% comm_monoid_set.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_7862_comm__monoid__set_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Ma: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_kv(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% comm_monoid_set.atLeast_atMost_pred_shift
tff(fact_7863_comm__monoid__set_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Ma: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_kv(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or7035219750837199246ssThan(nat,Ma,Nb)) ) ) ).

% comm_monoid_set.atLeast_lessThan_pred_shift
tff(fact_7864_comm__monoid__set_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Ma: nat,Nb: nat,G: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Nb)
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,Ma,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),Ma))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ma))) ) ) ) ).

% comm_monoid_set.atLeastAtMost_shift_0
tff(fact_7865_comm__monoid__set_Oin__pairs,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Ma: nat,Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ma),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,A),fun(nat,A),aTP_Lamp_afy(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,A)),F2),G)),set_or1337092689740270186AtMost(nat,Ma,Nb)) ) ) ).

% comm_monoid_set.in_pairs
tff(fact_7866_comm__monoid__set_Oin__pairs__0,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,A),Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),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))),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,A),fun(nat,A),aTP_Lamp_afy(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,A)),F2),G)),set_ord_atMost(nat,Nb)) ) ) ).

% comm_monoid_set.in_pairs_0
tff(fact_7867_comm__monoid__set_Otriangle__reindex__eq,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,G: fun(nat,fun(nat,A)),Nb: nat] :
      ( groups778175481326437816id_set(A,F2,Z)
     => ( aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups_comm_monoid_F(A,product_prod(nat,nat),F2,Z),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_hp(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_afl(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F2),Z),G)),set_ord_atMost(nat,Nb)) ) ) ).

% comm_monoid_set.triangle_reindex_eq
tff(fact_7868_natLeq__on__wo__rel,axiom,
    ! [Nb: nat] : bNF_Wellorder_wo_rel(nat,collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_aez(nat,fun(nat,fun(nat,$o)),Nb)))) ).

% natLeq_on_wo_rel
tff(fact_7869_max_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semilattice_order(A,ord_max(A),aTP_Lamp_ade(A,fun(A,$o)),aTP_Lamp_adf(A,fun(A,$o))) ) ).

% max.semilattice_order_axioms
tff(fact_7870_semilattice__neutr__order__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semila1105856199041335345_order(A,F2,Z,Less_eq,Less)
    <=> ( semilattice_neutr(A,F2,Z)
        & semilattice_order(A,F2,Less_eq,Less) ) ) ).

% semilattice_neutr_order_def
tff(fact_7871_semilattice__neutr__order_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semilattice_neutr(A,F2,Z)
     => ( semilattice_order(A,F2,Less_eq,Less)
       => semila1105856199041335345_order(A,F2,Z,Less_eq,Less) ) ) ).

% semilattice_neutr_order.intro
tff(fact_7872_gcd__nat_Osemilattice__order__axioms,axiom,
    semilattice_order(nat,gcd_gcd(nat),dvd_dvd(nat),aTP_Lamp_abc(nat,fun(nat,$o))) ).

% gcd_nat.semilattice_order_axioms
tff(fact_7873_semilattice__order_Omono,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,C2: A,B2: A,D3: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),C2)
       => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),D3)
         => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A2),B2)),aa(A,A,aa(A,fun(A,A),F2,C2),D3)) ) ) ) ).

% semilattice_order.mono
tff(fact_7874_semilattice__order_OorderE,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),B2)
       => ( A2 = aa(A,A,aa(A,fun(A,A),F2,A2),B2) ) ) ) ).

% semilattice_order.orderE
tff(fact_7875_semilattice__order_OorderI,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( ( A2 = aa(A,A,aa(A,fun(A,A),F2,A2),B2) )
       => aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),B2) ) ) ).

% semilattice_order.orderI
tff(fact_7876_semilattice__order_Oabsorb1,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),B2)
       => ( aa(A,A,aa(A,fun(A,A),F2,A2),B2) = A2 ) ) ) ).

% semilattice_order.absorb1
tff(fact_7877_semilattice__order_Oabsorb2,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B2: A,A2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),A2)
       => ( aa(A,A,aa(A,fun(A,A),F2,A2),B2) = B2 ) ) ) ).

% semilattice_order.absorb2
tff(fact_7878_semilattice__order_Oabsorb3,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A2),B2)
       => ( aa(A,A,aa(A,fun(A,A),F2,A2),B2) = A2 ) ) ) ).

% semilattice_order.absorb3
tff(fact_7879_semilattice__order_Oabsorb4,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B2: A,A2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,B2),A2)
       => ( aa(A,A,aa(A,fun(A,A),F2,A2),B2) = B2 ) ) ) ).

% semilattice_order.absorb4
tff(fact_7880_semilattice__order_OboundedE,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A,C2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),aa(A,A,aa(A,fun(A,A),F2,B2),C2))
       => ~ ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),B2)
           => ~ aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),C2) ) ) ) ).

% semilattice_order.boundedE
tff(fact_7881_semilattice__order_OboundedI,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A,C2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),B2)
       => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),aa(A,A,aa(A,fun(A,A),F2,B2),C2)) ) ) ) ).

% semilattice_order.boundedI
tff(fact_7882_semilattice__order_Oorder__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),B2)
      <=> ( A2 = aa(A,A,aa(A,fun(A,A),F2,A2),B2) ) ) ) ).

% semilattice_order.order_iff
tff(fact_7883_semilattice__order_Ocobounded1,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A2),B2)),A2) ) ).

% semilattice_order.cobounded1
tff(fact_7884_semilattice__order_Ocobounded2,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A2),B2)),B2) ) ).

% semilattice_order.cobounded2
tff(fact_7885_semilattice__order_Oabsorb__iff1,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),B2)
      <=> ( aa(A,A,aa(A,fun(A,A),F2,A2),B2) = A2 ) ) ) ).

% semilattice_order.absorb_iff1
tff(fact_7886_semilattice__order_Oabsorb__iff2,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B2: A,A2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),A2)
      <=> ( aa(A,A,aa(A,fun(A,A),F2,A2),B2) = B2 ) ) ) ).

% semilattice_order.absorb_iff2
tff(fact_7887_semilattice__order_Obounded__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A,C2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),aa(A,A,aa(A,fun(A,A),F2,B2),C2))
      <=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),B2)
          & aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),C2) ) ) ) ).

% semilattice_order.bounded_iff
tff(fact_7888_semilattice__order_OcoboundedI1,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,C2: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),C2)
       => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A2),B2)),C2) ) ) ).

% semilattice_order.coboundedI1
tff(fact_7889_semilattice__order_OcoboundedI2,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B2: A,C2: A,A2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),C2)
       => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A2),B2)),C2) ) ) ).

% semilattice_order.coboundedI2
tff(fact_7890_semilattice__order_Ostrict__boundedE,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A,C2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A2),aa(A,A,aa(A,fun(A,A),F2,B2),C2))
       => ~ ( aa(A,$o,aa(A,fun(A,$o),Less,A2),B2)
           => ~ aa(A,$o,aa(A,fun(A,$o),Less,A2),C2) ) ) ) ).

% semilattice_order.strict_boundedE
tff(fact_7891_semilattice__order_Ostrict__order__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A2),B2)
      <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),F2,A2),B2) )
          & ( A2 != B2 ) ) ) ) ).

% semilattice_order.strict_order_iff
tff(fact_7892_semilattice__order_Ostrict__coboundedI1,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,C2: A,B2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,A2),C2)
       => aa(A,$o,aa(A,fun(A,$o),Less,aa(A,A,aa(A,fun(A,A),F2,A2),B2)),C2) ) ) ).

% semilattice_order.strict_coboundedI1
tff(fact_7893_semilattice__order_Ostrict__coboundedI2,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B2: A,C2: A,A2: A] :
      ( semilattice_order(A,F2,Less_eq,Less)
     => ( aa(A,$o,aa(A,fun(A,$o),Less,B2),C2)
       => aa(A,$o,aa(A,fun(A,$o),Less,aa(A,A,aa(A,fun(A,A),F2,A2),B2)),C2) ) ) ).

% semilattice_order.strict_coboundedI2
tff(fact_7894_semilattice__neutr__order_Oaxioms_I2_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( semila1105856199041335345_order(A,F2,Z,Less_eq,Less)
     => semilattice_order(A,F2,Less_eq,Less) ) ).

% semilattice_neutr_order.axioms(2)
tff(fact_7895_inf_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => semilattice_order(A,inf_inf(A),ord_less_eq(A),ord_less(A)) ) ).

% inf.semilattice_order_axioms
tff(fact_7896_min_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semilattice_order(A,ord_min(A),ord_less_eq(A),ord_less(A)) ) ).

% min.semilattice_order_axioms
tff(fact_7897_sup_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => semilattice_order(A,sup_sup(A),aTP_Lamp_adg(A,fun(A,$o)),aTP_Lamp_adh(A,fun(A,$o))) ) ).

% sup.semilattice_order_axioms
tff(fact_7898_lexordp__eq_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( ord_lexordp_eq(A,A1,A22)
        <=> ( ? [Ys4: list(A)] :
                ( ( A1 = nil(A) )
                & ( A22 = Ys4 ) )
            | ? [X4: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A1 = cons(A,X4,Xs3) )
                & ( A22 = cons(A,Y5,Ys4) )
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y5) )
            | ? [X4: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A1 = cons(A,X4,Xs3) )
                & ( A22 = cons(A,Y5,Ys4) )
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y5)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X4)
                & ord_lexordp_eq(A,Xs3,Ys4) ) ) ) ) ).

% lexordp_eq.simps
tff(fact_7899_lexordp__eq__simps_I4_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Xs: list(A),Y: A,Ys2: list(A)] :
          ( ord_lexordp_eq(A,cons(A,Xb,Xs),cons(A,Y,Ys2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
            | ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
              & ord_lexordp_eq(A,Xs,Ys2) ) ) ) ) ).

% lexordp_eq_simps(4)
tff(fact_7900_lexordp__eq_OCons__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A,Xs: list(A),Ys2: list(A)] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
           => ( ord_lexordp_eq(A,Xs,Ys2)
             => ord_lexordp_eq(A,cons(A,Xb,Xs),cons(A,Y,Ys2)) ) ) ) ) ).

% lexordp_eq.Cons_eq
tff(fact_7901_lexordp__eq_OCons,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A,Xs: list(A),Ys2: list(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ord_lexordp_eq(A,cons(A,Xb,Xs),cons(A,Y,Ys2)) ) ) ).

% lexordp_eq.Cons
tff(fact_7902_lexordp__eq_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( ord_lexordp_eq(A,A1,A22)
         => ( ( A1 != nil(A) )
           => ( ! [X: A] :
                  ( ? [Xs2: list(A)] : ( A1 = cons(A,X,Xs2) )
                 => ! [Y3: A] :
                      ( ? [Ys3: list(A)] : ( A22 = cons(A,Y3,Ys3) )
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3) ) )
             => ~ ! [X: A,Y3: A,Xs2: list(A)] :
                    ( ( A1 = cons(A,X,Xs2) )
                   => ! [Ys3: list(A)] :
                        ( ( A22 = cons(A,Y3,Ys3) )
                       => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
                         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X)
                           => ~ ord_lexordp_eq(A,Xs2,Ys3) ) ) ) ) ) ) ) ) ).

% lexordp_eq.cases
tff(fact_7903_bdd__below_Opreordering__bdd__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit622319405099724424ng_bdd(A,aTP_Lamp_afz(A,fun(A,$o)),aTP_Lamp_act(A,fun(A,$o))) ) ).

% bdd_below.preordering_bdd_axioms
tff(fact_7904_card_Ofolding__on__axioms,axiom,
    ! [A: $tType] : finite_folding_on(A,nat,top_top(set(A)),aTP_Lamp_adk(A,fun(nat,nat))) ).

% card.folding_on_axioms
tff(fact_7905_bdd__above_Opreordering__bdd__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit622319405099724424ng_bdd(A,ord_less_eq(A),ord_less(A)) ) ).

% bdd_above.preordering_bdd_axioms
tff(fact_7906_tendsto__Zfun__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
        <=> zfun(A,B,aa(B,fun(A,B),aTP_Lamp_tm(fun(A,B),fun(B,fun(A,B)),F2),A2),F4) ) ) ).

% tendsto_Zfun_iff
tff(fact_7907_natural__decr,axiom,
    ! [Nb: code_natural] :
      ( ( Nb != zero_zero(code_natural) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(code_natural,nat,code_nat_of_natural,Nb)),aa(nat,nat,suc,zero_zero(nat)))),aa(code_natural,nat,code_nat_of_natural,Nb)) ) ).

% natural_decr
tff(fact_7908_plus__natural_Orep__eq,axiom,
    ! [Xb: code_natural,Xaa: code_natural] : ( aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),Xb),Xaa)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(code_natural,nat,code_nat_of_natural,Xb)),aa(code_natural,nat,code_nat_of_natural,Xaa)) ) ).

% plus_natural.rep_eq
tff(fact_7909_minus__natural_Orep__eq,axiom,
    ! [Xb: code_natural,Xaa: code_natural] : ( aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),Xb),Xaa)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(code_natural,nat,code_nat_of_natural,Xb)),aa(code_natural,nat,code_nat_of_natural,Xaa)) ) ).

% minus_natural.rep_eq
tff(fact_7910_one__natural_Orep__eq,axiom,
    aa(code_natural,nat,code_nat_of_natural,one_one(code_natural)) = one_one(nat) ).

% one_natural.rep_eq
tff(fact_7911_zero__natural_Orep__eq,axiom,
    aa(code_natural,nat,code_nat_of_natural,zero_zero(code_natural)) = zero_zero(nat) ).

% zero_natural.rep_eq
tff(fact_7912_Zfun__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( zfun(A,B,F2,F4)
         => zfun(A,B,aTP_Lamp_aga(fun(A,B),fun(A,B),F2),F4) ) ) ).

% Zfun_minus
tff(fact_7913_less__natural_Orep__eq,axiom,
    ! [Xb: code_natural,Xaa: code_natural] :
      ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),Xb),Xaa)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(code_natural,nat,code_nat_of_natural,Xb)),aa(code_natural,nat,code_nat_of_natural,Xaa)) ) ).

% less_natural.rep_eq
tff(fact_7914_Zfun__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F4: filter(A)] : zfun(A,B,aTP_Lamp_agb(A,B),F4) ) ).

% Zfun_zero
tff(fact_7915_natural__zero__minus__one,axiom,
    aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),zero_zero(code_natural)),one_one(code_natural)) = zero_zero(code_natural) ).

% natural_zero_minus_one
tff(fact_7916_Zfun__add,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( zfun(A,B,F2,F4)
         => ( zfun(A,B,G,F4)
           => zfun(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xp(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F4) ) ) ) ).

% Zfun_add
tff(fact_7917_Zfun__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( zfun(A,B,F2,F4)
         => ( zfun(A,B,G,F4)
           => zfun(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_to(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F4) ) ) ) ).

% Zfun_diff
tff(fact_7918_integer__of__natural_Orep__eq,axiom,
    ! [Xb: code_natural] : ( aa(code_integer,int,code_int_of_integer,aa(code_natural,code_integer,code_i5400310926305786745atural,Xb)) = aa(nat,int,semiring_1_of_nat(int),aa(code_natural,nat,code_nat_of_natural,Xb)) ) ).

% integer_of_natural.rep_eq
tff(fact_7919_int__of__integer__of__natural,axiom,
    ! [Nb: code_natural] : ( aa(code_integer,int,code_int_of_integer,aa(code_natural,code_integer,code_i5400310926305786745atural,Nb)) = aa(nat,int,semiring_1_of_nat(int),aa(code_natural,nat,code_nat_of_natural,Nb)) ) ).

% int_of_integer_of_natural
tff(fact_7920_log_Oelims,axiom,
    ! [Xb: code_natural,Xaa: code_natural,Y: code_natural] :
      ( ( log(Xb,Xaa) = Y )
     => ( Y = $ite(
            ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less_eq(code_natural),Xb),one_one(code_natural))
            | aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),Xaa),Xb) ),
            one_one(code_natural),
            aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(Xb,divide_divide(code_natural,Xaa,Xb))) ) ) ) ).

% log.elims
tff(fact_7921_log_Osimps,axiom,
    ! [B2: code_natural,I: code_natural] :
      ( log(B2,I) = $ite(
          ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less_eq(code_natural),B2),one_one(code_natural))
          | aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),I),B2) ),
          one_one(code_natural),
          aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(B2,divide_divide(code_natural,I,B2))) ) ) ).

% log.simps
tff(fact_7922_next_Osimps,axiom,
    ! [V2: code_natural,W: code_natural] :
      ( aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,aa(code_natural,product_prod(code_natural,code_natural),product_Pair(code_natural,code_natural,V2),W)) = $let(
          v: code_natural,
          v:= minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,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,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,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,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,V2,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),divide_divide(code_natural,V2,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,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,aa(num,num,bit1,bit0(one2)))))))))))))))),
          $let(
            w: code_natural,
            w:= minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,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,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,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(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,W,aa(num,code_natural,numeral_numeral(code_natural),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(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,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(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,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),divide_divide(code_natural,W,aa(num,code_natural,numeral_numeral(code_natural),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(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,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),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(aa(num,num,bit1,aa(num,num,bit1,one2)))))))))))))),
            aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,product_prod(code_natural,code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,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,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,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,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),v,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),w),one_one(code_natural)))),one_one(code_natural))),aa(code_natural,product_prod(code_natural,code_natural),product_Pair(code_natural,code_natural,v),w)) ) ) ) ).

% next.simps
tff(fact_7923_Code__Numeral_ONat_Orep__eq,axiom,
    ! [Xb: code_integer] : ( aa(code_natural,nat,code_nat_of_natural,aa(code_integer,code_natural,code_Nat,Xb)) = aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,Xb)) ) ).

% Code_Numeral.Nat.rep_eq
tff(fact_7924_Random_Orange__def,axiom,
    ! [K: code_natural] : ( range(K) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),iterate(code_natural,product_prod(code_natural,code_natural),log(aa(num,code_natural,numeral_numeral(code_natural),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,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,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,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),K),aTP_Lamp_agd(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))))),one_one(code_natural)),aTP_Lamp_age(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),K)) ) ).

% Random.range_def
tff(fact_7925_inc__shift__def,axiom,
    ! [V2: code_natural,K: code_natural] :
      ( inc_shift(V2,K) = $ite(V2 = K,one_one(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),K),one_one(code_natural))) ) ).

% inc_shift_def
tff(fact_7926_iterate_Oelims,axiom,
    ! [A: $tType,B: $tType,Xb: code_natural,Xaa: fun(B,fun(A,product_prod(B,A))),Xba: B,Y: fun(A,product_prod(B,A))] :
      ( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,Xb,Xaa),Xba) = Y )
     => ( Y = $ite(Xb = zero_zero(code_natural),product_Pair(B,A,Xba),product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xaa,Xba),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),Xb),one_one(code_natural)),Xaa))) ) ) ).

% iterate.elims
tff(fact_7927_iterate_Osimps,axiom,
    ! [A: $tType,B: $tType,K: code_natural,F2: fun(B,fun(A,product_prod(B,A))),Xb: B] :
      ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,K,F2),Xb) = $ite(K = zero_zero(code_natural),product_Pair(B,A,Xb),product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),F2,Xb),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K),one_one(code_natural)),F2))) ) ).

% iterate.simps
tff(fact_7928_log_Opelims,axiom,
    ! [Xb: code_natural,Xaa: code_natural,Y: code_natural] :
      ( ( log(Xb,Xaa) = Y )
     => ( accp(product_prod(code_natural,code_natural),log_rel,aa(code_natural,product_prod(code_natural,code_natural),product_Pair(code_natural,code_natural,Xb),Xaa))
       => ~ ( ( Y = $ite(
                  ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less_eq(code_natural),Xb),one_one(code_natural))
                  | aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),Xaa),Xb) ),
                  one_one(code_natural),
                  aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(Xb,divide_divide(code_natural,Xaa,Xb))) ) )
           => ~ accp(product_prod(code_natural,code_natural),log_rel,aa(code_natural,product_prod(code_natural,code_natural),product_Pair(code_natural,code_natural,Xb),Xaa)) ) ) ) ).

% log.pelims
tff(fact_7929_natural__of__integer_Orep__eq,axiom,
    ! [Xb: code_integer] : ( aa(code_natural,nat,code_nat_of_natural,aa(code_integer,code_natural,code_n4118661773612635043nteger,Xb)) = aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,Xb)) ) ).

% natural_of_integer.rep_eq
tff(fact_7930_select__weight__member,axiom,
    ! [A: $tType,Xs: list(product_prod(code_natural,A)),S: product_prod(code_natural,code_natural)] :
      ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),zero_zero(code_natural)),aa(list(code_natural),code_natural,groups8242544230860333062m_list(code_natural),aa(list(product_prod(code_natural,A)),list(code_natural),map(product_prod(code_natural,A),code_natural,product_fst(code_natural,A)),Xs)))
     => member(A,aa(product_prod(A,product_prod(code_natural,code_natural)),A,product_fst(A,product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),select_weight(A,Xs),S)),aa(list(A),set(A),set2(A),aa(list(product_prod(code_natural,A)),list(A),map(product_prod(code_natural,A),A,product_snd(code_natural,A)),Xs))) ) ).

% select_weight_member
tff(fact_7931_select,axiom,
    ! [A: $tType,Xs: list(A),S: product_prod(code_natural,code_natural)] :
      ( ( Xs != nil(A) )
     => member(A,aa(product_prod(A,product_prod(code_natural,code_natural)),A,product_fst(A,product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),select(A,Xs),S)),aa(list(A),set(A),set2(A),Xs)) ) ).

% select
tff(fact_7932_select__weight__select,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( select_weight(A,aa(list(A),list(product_prod(code_natural,A)),map(A,product_prod(code_natural,A),product_Pair(code_natural,A,one_one(code_natural))),Xs)) = select(A,Xs) ) ) ).

% select_weight_select
tff(fact_7933_pick__member,axiom,
    ! [A: $tType,I: code_natural,Xs: list(product_prod(code_natural,A))] :
      ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),I),aa(list(code_natural),code_natural,groups8242544230860333062m_list(code_natural),aa(list(product_prod(code_natural,A)),list(code_natural),map(product_prod(code_natural,A),code_natural,product_fst(code_natural,A)),Xs)))
     => member(A,pick(A,Xs,I),aa(list(A),set(A),set2(A),aa(list(product_prod(code_natural,A)),list(A),map(product_prod(code_natural,A),A,product_snd(code_natural,A)),Xs))) ) ).

% pick_member
tff(fact_7934_iterate_Opelims,axiom,
    ! [A: $tType,B: $tType,Xb: code_natural,Xaa: fun(B,fun(A,product_prod(B,A))),Xba: B,Y: fun(A,product_prod(B,A))] :
      ( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,Xb,Xaa),Xba) = Y )
     => ( accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B),Xb),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),product_Pair(fun(B,fun(A,product_prod(B,A))),B,Xaa),Xba)))
       => ~ ( ( Y = $ite(Xb = zero_zero(code_natural),product_Pair(B,A,Xba),product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xaa,Xba),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),Xb),one_one(code_natural)),Xaa))) )
           => ~ accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B),Xb),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),product_Pair(fun(B,fun(A,product_prod(B,A))),B,Xaa),Xba))) ) ) ) ).

% iterate.pelims
tff(fact_7935_pick__same,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),L),aa(list(A),nat,size_size(list(A)),Xs))
     => ( pick(A,aa(list(A),list(product_prod(code_natural,A)),map(A,product_prod(code_natural,A),product_Pair(code_natural,A,one_one(code_natural))),Xs),aa(nat,code_natural,code_natural_of_nat,L)) = aa(nat,A,nth(A,Xs),L) ) ) ).

% pick_same
tff(fact_7936_integer__of__natural__def,axiom,
    code_i5400310926305786745atural = aa(fun(nat,int),fun(code_natural,code_integer),map_fun(code_natural,nat,int,code_integer,code_nat_of_natural,code_integer_of_int),semiring_1_of_nat(int)) ).

% integer_of_natural_def
tff(fact_7937_zero__natural__def,axiom,
    zero_zero(code_natural) = aa(nat,code_natural,code_natural_of_nat,zero_zero(nat)) ).

% zero_natural_def
tff(fact_7938_less__natural_Oabs__eq,axiom,
    ! [Xaa: nat,Xb: nat] :
      ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),aa(nat,code_natural,code_natural_of_nat,Xaa)),aa(nat,code_natural,code_natural_of_nat,Xb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Xb) ) ).

% less_natural.abs_eq
tff(fact_7939_plus__natural_Oabs__eq,axiom,
    ! [Xaa: nat,Xb: nat] : ( aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),aa(nat,code_natural,code_natural_of_nat,Xaa)),aa(nat,code_natural,code_natural_of_nat,Xb)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xaa),Xb)) ) ).

% plus_natural.abs_eq
tff(fact_7940_minus__natural_Oabs__eq,axiom,
    ! [Xaa: nat,Xb: nat] : ( aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),aa(nat,code_natural,code_natural_of_nat,Xaa)),aa(nat,code_natural,code_natural_of_nat,Xb)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xaa),Xb)) ) ).

% minus_natural.abs_eq
tff(fact_7941_one__natural__def,axiom,
    one_one(code_natural) = aa(nat,code_natural,code_natural_of_nat,one_one(nat)) ).

% one_natural_def
tff(fact_7942_integer__of__natural_Oabs__eq,axiom,
    ! [Xb: nat] : ( aa(code_natural,code_integer,code_i5400310926305786745atural,aa(nat,code_natural,code_natural_of_nat,Xb)) = aa(int,code_integer,code_integer_of_int,aa(nat,int,semiring_1_of_nat(int),Xb)) ) ).

% integer_of_natural.abs_eq
tff(fact_7943_natural__of__integer_Oabs__eq,axiom,
    ! [Xb: int] : ( aa(code_integer,code_natural,code_n4118661773612635043nteger,aa(int,code_integer,code_integer_of_int,Xb)) = aa(nat,code_natural,code_natural_of_nat,aa(int,nat,nat2,Xb)) ) ).

% natural_of_integer.abs_eq
tff(fact_7944_Code__Numeral_ONat_Oabs__eq,axiom,
    ! [Xb: int] : ( aa(code_integer,code_natural,code_Nat,aa(int,code_integer,code_integer_of_int,Xb)) = aa(nat,code_natural,code_natural_of_nat,aa(int,nat,nat2,Xb)) ) ).

% Code_Numeral.Nat.abs_eq
tff(fact_7945_Code__Numeral_ONat__def,axiom,
    code_Nat = aa(fun(int,nat),fun(code_integer,code_natural),map_fun(code_integer,int,nat,code_natural,code_int_of_integer,code_natural_of_nat),nat2) ).

% Code_Numeral.Nat_def
tff(fact_7946_natural__of__integer__def,axiom,
    code_n4118661773612635043nteger = aa(fun(int,nat),fun(code_integer,code_natural),map_fun(code_integer,int,nat,code_natural,code_int_of_integer,code_natural_of_nat),nat2) ).

% natural_of_integer_def
tff(fact_7947_Suc_Orep__eq,axiom,
    ! [Xb: code_natural] : ( aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,code_Suc,Xb)) = aa(nat,nat,suc,aa(code_natural,nat,code_nat_of_natural,Xb)) ) ).

% Suc.rep_eq
tff(fact_7948_Code__Numeral_ONat_Otransfer,axiom,
    aa(fun(code_integer,code_natural),$o,aa(fun(int,nat),fun(fun(code_integer,code_natural),$o),bNF_rel_fun(int,code_integer,nat,code_natural,code_pcr_integer,code_pcr_natural),nat2),code_Nat) ).

% Code_Numeral.Nat.transfer
tff(fact_7949_zero__natural_Otransfer,axiom,
    aa(code_natural,$o,aa(nat,fun(code_natural,$o),code_pcr_natural,zero_zero(nat)),zero_zero(code_natural)) ).

% zero_natural.transfer
tff(fact_7950_Suc_Otransfer,axiom,
    aa(fun(code_natural,code_natural),$o,aa(fun(nat,nat),fun(fun(code_natural,code_natural),$o),bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural),suc),code_Suc) ).

% Suc.transfer
tff(fact_7951_less__natural_Otransfer,axiom,
    aa(fun(code_natural,fun(code_natural,$o)),$o,aa(fun(nat,fun(nat,$o)),fun(fun(code_natural,fun(code_natural,$o)),$o),bNF_rel_fun(nat,code_natural,fun(nat,$o),fun(code_natural,$o),code_pcr_natural,bNF_rel_fun(nat,code_natural,$o,$o,code_pcr_natural,fequal($o))),ord_less(nat)),ord_less(code_natural)) ).

% less_natural.transfer
tff(fact_7952_minus__natural_Otransfer,axiom,
    aa(fun(code_natural,fun(code_natural,code_natural)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(code_natural,fun(code_natural,code_natural)),$o),bNF_rel_fun(nat,code_natural,fun(nat,nat),fun(code_natural,code_natural),code_pcr_natural,bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),minus_minus(nat)),minus_minus(code_natural)) ).

% minus_natural.transfer
tff(fact_7953_plus__natural_Otransfer,axiom,
    aa(fun(code_natural,fun(code_natural,code_natural)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(code_natural,fun(code_natural,code_natural)),$o),bNF_rel_fun(nat,code_natural,fun(nat,nat),fun(code_natural,code_natural),code_pcr_natural,bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),plus_plus(nat)),plus_plus(code_natural)) ).

% plus_natural.transfer
tff(fact_7954_one__natural_Otransfer,axiom,
    aa(code_natural,$o,aa(nat,fun(code_natural,$o),code_pcr_natural,one_one(nat)),one_one(code_natural)) ).

% one_natural.transfer
tff(fact_7955_Suc_Oabs__eq,axiom,
    ! [Xb: nat] : ( aa(code_natural,code_natural,code_Suc,aa(nat,code_natural,code_natural_of_nat,Xb)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,suc,Xb)) ) ).

% Suc.abs_eq
tff(fact_7956_integer__of__natural_Otransfer,axiom,
    aa(fun(code_natural,code_integer),$o,aa(fun(nat,int),fun(fun(code_natural,code_integer),$o),bNF_rel_fun(nat,code_natural,int,code_integer,code_pcr_natural,code_pcr_integer),semiring_1_of_nat(int)),code_i5400310926305786745atural) ).

% integer_of_natural.transfer
tff(fact_7957_natural__of__integer_Otransfer,axiom,
    aa(fun(code_integer,code_natural),$o,aa(fun(int,nat),fun(fun(code_integer,code_natural),$o),bNF_rel_fun(int,code_integer,nat,code_natural,code_pcr_integer,code_pcr_natural),nat2),code_n4118661773612635043nteger) ).

% natural_of_integer.transfer
tff(fact_7958_Suc__natural__minus__one,axiom,
    ! [Nb: code_natural] : ( aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),aa(code_natural,code_natural,code_Suc,Nb)),one_one(code_natural)) = Nb ) ).

% Suc_natural_minus_one
tff(fact_7959_Code__Numeral_OSuc__def,axiom,
    code_Suc = aa(fun(nat,nat),fun(code_natural,code_natural),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat),suc) ).

% Code_Numeral.Suc_def
tff(fact_7960_Code__Numeral_Osub__def,axiom,
    code_sub = aa(fun(num,fun(num,int)),fun(num,fun(num,code_integer)),map_fun(num,num,fun(num,int),fun(num,code_integer),id(num),map_fun(num,num,int,code_integer,id(num),code_integer_of_int)),aTP_Lamp_aaj(num,fun(num,int))) ).

% Code_Numeral.sub_def
tff(fact_7961_minus__natural__def,axiom,
    minus_minus(code_natural) = aa(fun(nat,fun(nat,nat)),fun(code_natural,fun(code_natural,code_natural)),map_fun(code_natural,nat,fun(nat,nat),fun(code_natural,code_natural),code_nat_of_natural,map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),minus_minus(nat)) ).

% minus_natural_def
tff(fact_7962_plus__natural__def,axiom,
    plus_plus(code_natural) = aa(fun(nat,fun(nat,nat)),fun(code_natural,fun(code_natural,code_natural)),map_fun(code_natural,nat,fun(nat,nat),fun(code_natural,code_natural),code_nat_of_natural,map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),plus_plus(nat)) ).

% plus_natural_def
tff(fact_7963_Quotient3__int,axiom,
    quotient3(product_prod(nat,nat),int,intrel,abs_Integ,rep_Integ) ).

% Quotient3_int
tff(fact_7964_Field__natLeq__on,axiom,
    ! [Nb: nat] : ( field2(nat,collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_aez(nat,fun(nat,fun(nat,$o)),Nb)))) = collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aa(nat,fun(nat,$o)),Nb)) ) ).

% Field_natLeq_on
tff(fact_7965_natLeq__on__Well__order,axiom,
    ! [Nb: nat] : order_well_order_on(nat,field2(nat,collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_aez(nat,fun(nat,fun(nat,$o)),Nb)))),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_aez(nat,fun(nat,fun(nat,$o)),Nb)))) ).

% natLeq_on_Well_order
tff(fact_7966_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xb)
      <=> (Y) )
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv2)) )
               => ( ~ (Y)
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Y)
                   => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
                   => ( (Y)
                     => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) )
                     => ( ~ (Y)
                       => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_7967_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xb)
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,Xb)
       => ( ! [Uv2: $o] :
              ( ( Xb = vEBT_Leaf($true,(Uv2)) )
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($true,(Uv2))) )
         => ( ! [Uu2: $o] :
                ( ( Xb = vEBT_Leaf((Uu2),$true) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf((Uu2),$true)) )
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) )
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_7968_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xb)
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($false,$false)) )
         => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_7969_natLeq__on__well__order__on,axiom,
    ! [Nb: nat] : order_well_order_on(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aa(nat,fun(nat,$o)),Nb)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_aez(nat,fun(nat,fun(nat,$o)),Nb)))) ).

% natLeq_on_well_order_on
tff(fact_7970_list__all__length,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( list_all(A,P,Xs)
    <=> ! [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),N4)) ) ) ).

% list_all_length
tff(fact_7971_linear__injective__on__subspace__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),S: set(A)] :
          ( real_Vector_linear(A,B,F2)
         => ( real_Vector_subspace(A,S)
           => ( inj_on(A,B,F2,S)
            <=> ! [X4: A] :
                  ( member(A,X4,S)
                 => ( ( aa(A,B,F2,X4) = zero_zero(B) )
                   => ( X4 = zero_zero(A) ) ) ) ) ) ) ) ).

% linear_injective_on_subspace_0
tff(fact_7972_subspace__diff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A),Xb: A,Y: A] :
          ( real_Vector_subspace(A,S3)
         => ( member(A,Xb,S3)
           => ( member(A,Y,S3)
             => member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y),S3) ) ) ) ) ).

% subspace_diff
tff(fact_7973_subspace__add,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A),Xb: A,Y: A] :
          ( real_Vector_subspace(A,S3)
         => ( member(A,Xb,S3)
           => ( member(A,Y,S3)
             => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y),S3) ) ) ) ) ).

% subspace_add
tff(fact_7974_subspace__sums,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A),T4: set(A)] :
          ( real_Vector_subspace(A,S3)
         => ( real_Vector_subspace(A,T4)
           => real_Vector_subspace(A,collect(A,aa(set(A),fun(A,$o),aTP_Lamp_agf(set(A),fun(set(A),fun(A,$o)),S3),T4))) ) ) ) ).

% subspace_sums
tff(fact_7975_Ball__set__list__all,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X4) )
    <=> list_all(A,P,Xs) ) ).

% Ball_set_list_all
tff(fact_7976_list__all__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: list(A)] :
      ( list_all(A,P,Xb)
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xb))
         => aa(A,$o,P,X4) ) ) ).

% list_all_iff
tff(fact_7977_list_Opred__set,axiom,
    ! [A: $tType,P: fun(A,$o),X3: list(A)] :
      ( list_all(A,P,X3)
    <=> ! [Xa3: A] :
          ( member(A,Xa3,aa(list(A),set(A),set2(A),X3))
         => aa(A,$o,P,Xa3) ) ) ).

% list.pred_set
tff(fact_7978_subspace__neg,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A),Xb: A] :
          ( real_Vector_subspace(A,S3)
         => ( member(A,Xb,S3)
           => member(A,aa(A,A,uminus_uminus(A),Xb),S3) ) ) ) ).

% subspace_neg
tff(fact_7979_list__all__cong,axiom,
    ! [A: $tType,Xb: list(A),Ya: list(A),P: fun(A,$o),Pa: fun(A,$o)] :
      ( ( Xb = Ya )
     => ( ! [Z2: A] :
            ( member(A,Z2,aa(list(A),set(A),set2(A),Ya))
           => ( aa(A,$o,P,Z2)
            <=> aa(A,$o,Pa,Z2) ) )
       => ( list_all(A,P,Xb)
        <=> list_all(A,Pa,Ya) ) ) ) ).

% list_all_cong
tff(fact_7980_list_Opred__mono__strong,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: list(A),Pa: fun(A,$o)] :
      ( list_all(A,P,Xb)
     => ( ! [Z2: A] :
            ( member(A,Z2,aa(list(A),set(A),set2(A),Xb))
           => ( aa(A,$o,P,Z2)
             => aa(A,$o,Pa,Z2) ) )
       => list_all(A,Pa,Xb) ) ) ).

% list.pred_mono_strong
tff(fact_7981_subspace__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] :
          ( real_Vector_subspace(A,S3)
         => member(A,zero_zero(A),S3) ) ) ).

% subspace_0
tff(fact_7982_linear__subspace__kernel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_Vector_linear(A,B,F2)
         => real_Vector_subspace(A,collect(A,aTP_Lamp_agg(fun(A,B),fun(A,$o),F2))) ) ) ).

% linear_subspace_kernel
tff(fact_7983_subspace__single__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => real_Vector_subspace(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A)))) ) ).

% subspace_single_0
tff(fact_7984_subspaceI,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] :
          ( member(A,zero_zero(A),S3)
         => ( ! [X: A,Y3: A] :
                ( member(A,X,S3)
               => ( member(A,Y3,S3)
                 => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3),S3) ) )
           => ( ! [C3: real,X: A] :
                  ( member(A,X,S3)
                 => member(A,real_V8093663219630862766scaleR(A,C3,X),S3) )
             => real_Vector_subspace(A,S3) ) ) ) ) ).

% subspaceI
tff(fact_7985_subspace__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] :
          ( real_Vector_subspace(A,S3)
        <=> ( member(A,zero_zero(A),S3)
            & ! [X4: A] :
                ( member(A,X4,S3)
               => ! [Xa3: A] :
                    ( member(A,Xa3,S3)
                   => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Xa3),S3) ) )
            & ! [C4: real,X4: A] :
                ( member(A,X4,S3)
               => member(A,real_V8093663219630862766scaleR(A,C4,X4),S3) ) ) ) ) ).

% subspace_def
tff(fact_7986_eventually__filtercomap__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(B)
        & no_bot(B) )
     => ! [P: fun(A,$o),F2: fun(A,B)] :
          ( eventually(A,P,filtercomap(A,B,F2,at_bot(B)))
        <=> ? [N6: B] :
            ! [X4: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),N6)
             => aa(A,$o,P,X4) ) ) ) ).

% eventually_filtercomap_at_bot_dense
tff(fact_7987_of__int__code_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : ( aa(int,A,ring_1_of_int(A),neg(K)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K)) ) ) ).

% of_int_code(1)
tff(fact_7988_less__eq__int__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),neg(K)),neg(L))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),L),K) ) ).

% less_eq_int_code(9)
tff(fact_7989_less__eq__int__code_I7_J,axiom,
    ! [K: num] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),neg(K)),zero_zero(int)) ).

% less_eq_int_code(7)
tff(fact_7990_less__eq__int__code_I3_J,axiom,
    ! [L: num] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),neg(L)) ).

% less_eq_int_code(3)
tff(fact_7991_nat__code_I1_J,axiom,
    ! [K: num] : ( aa(int,nat,nat2,neg(K)) = zero_zero(nat) ) ).

% nat_code(1)
tff(fact_7992_less__int__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),neg(K)),neg(L))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),L),K) ) ).

% less_int_code(9)
tff(fact_7993_plus__int__code_I6_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),plus_plus(int),neg(Ma)),neg(Nb)) = neg(aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),Nb)) ) ).

% plus_int_code(6)
tff(fact_7994_less__int__code_I3_J,axiom,
    ! [L: num] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),neg(L)) ).

% less_int_code(3)
tff(fact_7995_less__int__code_I7_J,axiom,
    ! [K: num] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),neg(K)),zero_zero(int)) ).

% less_int_code(7)
tff(fact_7996_eventually__filtercomap__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(B)
        & no_top(B) )
     => ! [P: fun(A,$o),F2: fun(A,B)] :
          ( eventually(A,P,filtercomap(A,B,F2,at_top(B)))
        <=> ? [N6: B] :
            ! [X4: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),N6),aa(A,B,F2,X4))
             => aa(A,$o,P,X4) ) ) ) ).

% eventually_filtercomap_at_top_dense
tff(fact_7997_Int_Osub__code_I4_J,axiom,
    ! [Nb: num] : ( sub(one2,bit0(Nb)) = neg(bitM(Nb)) ) ).

% Int.sub_code(4)
tff(fact_7998_Int_Osub__code_I5_J,axiom,
    ! [Nb: num] : ( sub(one2,aa(num,num,bit1,Nb)) = neg(bit0(Nb)) ) ).

% Int.sub_code(5)
tff(fact_7999_Int_Osub__code_I1_J,axiom,
    sub(one2,one2) = zero_zero(int) ).

% Int.sub_code(1)
tff(fact_8000_Int_Osub__def,axiom,
    ! [Ma: num,Nb: num] : ( sub(Ma,Nb) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Ma)),aa(num,int,numeral_numeral(int),Nb)) ) ).

% Int.sub_def
tff(fact_8001_minus__int__code_I6_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),minus_minus(int),neg(Ma)),neg(Nb)) = sub(Nb,Ma) ) ).

% minus_int_code(6)
tff(fact_8002_Int_Osub__code_I9_J,axiom,
    ! [Ma: num,Nb: num] : ( sub(bit0(Ma),aa(num,num,bit1,Nb)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),dup(sub(Ma,Nb))),one_one(int)) ) ).

% Int.sub_code(9)
tff(fact_8003_Int_Osub__code_I8_J,axiom,
    ! [Ma: num,Nb: num] : ( sub(aa(num,num,bit1,Ma),bit0(Nb)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),dup(sub(Ma,Nb))),one_one(int)) ) ).

% Int.sub_code(8)
tff(fact_8004_Int_Odup__code_I1_J,axiom,
    dup(zero_zero(int)) = zero_zero(int) ).

% Int.dup_code(1)
tff(fact_8005_Int_Odup__def,axiom,
    ! [K: int] : ( dup(K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),K) ) ).

% Int.dup_def
tff(fact_8006_Int_Odup__code_I3_J,axiom,
    ! [Nb: num] : ( dup(neg(Nb)) = neg(bit0(Nb)) ) ).

% Int.dup_code(3)
tff(fact_8007_Int_Osub__code_I6_J,axiom,
    ! [Ma: num,Nb: num] : ( sub(bit0(Ma),bit0(Nb)) = dup(sub(Ma,Nb)) ) ).

% Int.sub_code(6)
tff(fact_8008_Int_Osub__code_I7_J,axiom,
    ! [Ma: num,Nb: num] : ( sub(aa(num,num,bit1,Ma),aa(num,num,bit1,Nb)) = dup(sub(Ma,Nb)) ) ).

% Int.sub_code(7)
tff(fact_8009_Int_Osub__code_I2_J,axiom,
    ! [Ma: num] : ( sub(bit0(Ma),one2) = aa(num,int,pos,bitM(Ma)) ) ).

% Int.sub_code(2)
tff(fact_8010_Int_Osub__code_I3_J,axiom,
    ! [Ma: num] : ( sub(aa(num,num,bit1,Ma),one2) = aa(num,int,pos,bit0(Ma)) ) ).

% Int.sub_code(3)
tff(fact_8011_Int_Odup__code_I2_J,axiom,
    ! [Nb: num] : ( dup(aa(num,int,pos,Nb)) = aa(num,int,pos,bit0(Nb)) ) ).

% Int.dup_code(2)
tff(fact_8012_less__eq__int__code_I8_J,axiom,
    ! [K: num,L: num] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),neg(K)),aa(num,int,pos,L)) ).

% less_eq_int_code(8)
tff(fact_8013_less__eq__int__code_I6_J,axiom,
    ! [K: num,L: num] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,pos,K)),neg(L)) ).

% less_eq_int_code(6)
tff(fact_8014_uminus__int__code_I3_J,axiom,
    ! [Ma: num] : ( aa(int,int,uminus_uminus(int),neg(Ma)) = aa(num,int,pos,Ma) ) ).

% uminus_int_code(3)
tff(fact_8015_uminus__int__code_I2_J,axiom,
    ! [Ma: num] : ( aa(int,int,uminus_uminus(int),aa(num,int,pos,Ma)) = neg(Ma) ) ).

% uminus_int_code(2)
tff(fact_8016_Int_ONeg__def,axiom,
    ! [Nb: num] : ( neg(Nb) = aa(int,int,uminus_uminus(int),aa(num,int,pos,Nb)) ) ).

% Int.Neg_def
tff(fact_8017_less__int__code_I6_J,axiom,
    ! [K: num,L: num] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,pos,K)),neg(L)) ).

% less_int_code(6)
tff(fact_8018_less__int__code_I8_J,axiom,
    ! [K: num,L: num] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),neg(K)),aa(num,int,pos,L)) ).

% less_int_code(8)
tff(fact_8019_less__int__code_I4_J,axiom,
    ! [K: num] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,pos,K)),zero_zero(int)) ).

% less_int_code(4)
tff(fact_8020_less__int__code_I2_J,axiom,
    ! [L: num] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(num,int,pos,L)) ).

% less_int_code(2)
tff(fact_8021_one__int__code,axiom,
    one_one(int) = aa(num,int,pos,one2) ).

% one_int_code
tff(fact_8022_plus__int__code_I3_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,pos,Ma)),aa(num,int,pos,Nb)) = aa(num,int,pos,aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),Nb)) ) ).

% plus_int_code(3)
tff(fact_8023_less__int__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,pos,K)),aa(num,int,pos,L))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),K),L) ) ).

% less_int_code(5)
tff(fact_8024_nat__code_I3_J,axiom,
    ! [K: num] : ( aa(int,nat,nat2,aa(num,int,pos,K)) = aa(num,nat,nat_of_num,K) ) ).

% nat_code(3)
tff(fact_8025_less__eq__int__code_I2_J,axiom,
    ! [L: num] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(num,int,pos,L)) ).

% less_eq_int_code(2)
tff(fact_8026_less__eq__int__code_I4_J,axiom,
    ! [K: num] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,pos,K)),zero_zero(int)) ).

% less_eq_int_code(4)
tff(fact_8027_less__eq__int__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,pos,K)),aa(num,int,pos,L))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),K),L) ) ).

% less_eq_int_code(5)
tff(fact_8028_Int_OPos__def,axiom,
    pos = numeral_numeral(int) ).

% Int.Pos_def
tff(fact_8029_of__int__code_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : ( aa(int,A,ring_1_of_int(A),aa(num,int,pos,K)) = aa(num,A,numeral_numeral(A),K) ) ) ).

% of_int_code(3)
tff(fact_8030_times__int__code_I3_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,pos,Ma)),aa(num,int,pos,Nb)) = aa(num,int,pos,aa(num,num,aa(num,fun(num,num),times_times(num),Ma),Nb)) ) ).

% times_int_code(3)
tff(fact_8031_minus__int__code_I3_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,pos,Ma)),aa(num,int,pos,Nb)) = sub(Ma,Nb) ) ).

% minus_int_code(3)
tff(fact_8032_minus__int__code_I5_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),minus_minus(int),neg(Ma)),aa(num,int,pos,Nb)) = neg(aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),Nb)) ) ).

% minus_int_code(5)
tff(fact_8033_minus__int__code_I4_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,pos,Ma)),neg(Nb)) = aa(num,int,pos,aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),Nb)) ) ).

% minus_int_code(4)
tff(fact_8034_times__int__code_I4_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,pos,Ma)),neg(Nb)) = neg(aa(num,num,aa(num,fun(num,num),times_times(num),Ma),Nb)) ) ).

% times_int_code(4)
tff(fact_8035_times__int__code_I5_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),times_times(int),neg(Ma)),aa(num,int,pos,Nb)) = neg(aa(num,num,aa(num,fun(num,num),times_times(num),Ma),Nb)) ) ).

% times_int_code(5)
tff(fact_8036_times__int__code_I6_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),times_times(int),neg(Ma)),neg(Nb)) = aa(num,int,pos,aa(num,num,aa(num,fun(num,num),times_times(num),Ma),Nb)) ) ).

% times_int_code(6)
tff(fact_8037_plus__int__code_I4_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,pos,Ma)),neg(Nb)) = sub(Ma,Nb) ) ).

% plus_int_code(4)
tff(fact_8038_plus__int__code_I5_J,axiom,
    ! [Ma: num,Nb: num] : ( aa(int,int,aa(int,fun(int,int),plus_plus(int),neg(Ma)),aa(num,int,pos,Nb)) = sub(Nb,Ma) ) ).

% plus_int_code(5)
tff(fact_8039_is__arg__max__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),P: fun(A,$o),Xb: A] :
          ( lattic501386751176901750rg_max(A,B,F2,P,Xb)
        <=> ( aa(A,$o,P,Xb)
            & ~ ? [Y5: A] :
                  ( aa(A,$o,P,Y5)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y5)) ) ) ) ) ).

% is_arg_max_def
tff(fact_8040_prod__list_Ocomm__monoid__list__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => groups1828464146339083142d_list(A,times_times(A),one_one(A)) ) ).

% prod_list.comm_monoid_list_axioms
tff(fact_8041_sum__list_Ocomm__monoid__list__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => groups1828464146339083142d_list(A,plus_plus(A),zero_zero(A)) ) ).

% sum_list.comm_monoid_list_axioms
tff(fact_8042_VEBT_Orec__transfer,axiom,
    ! [A: $tType,B: $tType,S3: fun(A,fun(B,$o))] : aa(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(B,B))))),fun(fun($o,fun($o,B)),fun(vEBT_VEBT,B))),$o,aa(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,A))),fun(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(B,B))))),fun(fun($o,fun($o,B)),fun(vEBT_VEBT,B))),$o),bNF_rel_fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(B,B))))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,A)),fun(fun($o,fun($o,B)),fun(vEBT_VEBT,B)),bNF_rel_fun(option(product_prod(nat,nat)),option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A)))),fun(nat,fun(list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(B,B)))),fequal(option(product_prod(nat,nat))),bNF_rel_fun(nat,nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))),fun(list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(B,B))),fequal(nat),bNF_rel_fun(list(product_prod(vEBT_VEBT,A)),list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(A,A)),fun(vEBT_VEBT,fun(B,B)),list_all2(product_prod(vEBT_VEBT,A),product_prod(vEBT_VEBT,B),basic_rel_prod(vEBT_VEBT,vEBT_VEBT,A,B,fequal(vEBT_VEBT),S3)),bNF_rel_fun(vEBT_VEBT,vEBT_VEBT,fun(A,A),fun(B,B),fequal(vEBT_VEBT),bNF_rel_fun(A,B,A,B,S3,S3))))),bNF_rel_fun(fun($o,fun($o,A)),fun($o,fun($o,B)),fun(vEBT_VEBT,A),fun(vEBT_VEBT,B),bNF_rel_fun($o,$o,fun($o,A),fun($o,B),fequal($o),bNF_rel_fun($o,$o,A,B,fequal($o),S3)),bNF_rel_fun(vEBT_VEBT,vEBT_VEBT,A,B,fequal(vEBT_VEBT),S3))),vEBT_rec_VEBT(A)),vEBT_rec_VEBT(B)) ).

% VEBT.rec_transfer
tff(fact_8043_group_Oaxioms_I2_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( group(A,F2,Z,Inverse)
     => group_axioms(A,F2,Z,Inverse) ) ).

% group.axioms(2)
tff(fact_8044_group__axioms__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( group_axioms(A,F2,Z,Inverse)
    <=> ( ! [A5: A] : ( aa(A,A,aa(A,fun(A,A),F2,Z),A5) = A5 )
        & ! [A5: A] : ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,A5)),A5) = Z ) ) ) ).

% group_axioms_def
tff(fact_8045_group__axioms_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( ! [A4: A] : ( aa(A,A,aa(A,fun(A,A),F2,Z),A4) = A4 )
     => ( ! [A4: A] : ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,A4)),A4) = Z )
       => group_axioms(A,F2,Z,Inverse) ) ) ).

% group_axioms.intro
tff(fact_8046_VEBT_Ocase__transfer,axiom,
    ! [A: $tType,B: $tType,S3: fun(A,fun(B,$o))] : aa(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(fun($o,fun($o,B)),fun(vEBT_VEBT,B))),$o,aa(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,A))),fun(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(fun($o,fun($o,B)),fun(vEBT_VEBT,B))),$o),bNF_rel_fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,A)),fun(fun($o,fun($o,B)),fun(vEBT_VEBT,B)),bNF_rel_fun(option(product_prod(nat,nat)),option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A))),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B))),fequal(option(product_prod(nat,nat))),bNF_rel_fun(nat,nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)),fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)),fequal(nat),bNF_rel_fun(list(vEBT_VEBT),list(vEBT_VEBT),fun(vEBT_VEBT,A),fun(vEBT_VEBT,B),fequal(list(vEBT_VEBT)),bNF_rel_fun(vEBT_VEBT,vEBT_VEBT,A,B,fequal(vEBT_VEBT),S3)))),bNF_rel_fun(fun($o,fun($o,A)),fun($o,fun($o,B)),fun(vEBT_VEBT,A),fun(vEBT_VEBT,B),bNF_rel_fun($o,$o,fun($o,A),fun($o,B),fequal($o),bNF_rel_fun($o,$o,A,B,fequal($o),S3)),bNF_rel_fun(vEBT_VEBT,vEBT_VEBT,A,B,fequal(vEBT_VEBT),S3))),vEBT_case_VEBT(A)),vEBT_case_VEBT(B)) ).

% VEBT.case_transfer
tff(fact_8047_group_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( semigroup(A,F2)
     => ( group_axioms(A,F2,Z,Inverse)
       => group(A,F2,Z,Inverse) ) ) ).

% group.intro
tff(fact_8048_mult_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => semigroup(A,times_times(A)) ) ).

% mult.semigroup_axioms
tff(fact_8049_lcm_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => semigroup(A,gcd_lcm(A)) ) ).

% lcm.semigroup_axioms
tff(fact_8050_group_Oaxioms_I1_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( group(A,F2,Z,Inverse)
     => semigroup(A,F2) ) ).

% group.axioms(1)
tff(fact_8051_inf_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => semigroup(A,inf_inf(A)) ) ).

% inf.semigroup_axioms
tff(fact_8052_gcd_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => semigroup(A,gcd_gcd(A)) ) ).

% gcd.semigroup_axioms
tff(fact_8053_max_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semigroup(A,ord_max(A)) ) ).

% max.semigroup_axioms
tff(fact_8054_VEBT_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),F22: fun($o,fun($o,B)),VEBT: vEBT_VEBT] : ( aa(B,A,H,aa(vEBT_VEBT,B,aa(fun($o,fun($o,B)),fun(vEBT_VEBT,B),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(fun($o,fun($o,B)),fun(vEBT_VEBT,B)),vEBT_case_VEBT(B),F1),F22),VEBT)) = aa(vEBT_VEBT,A,aa(fun($o,fun($o,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,A)),vEBT_case_VEBT(A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),aTP_Lamp_agh(fun(B,A),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A))))),H),F1)),aa(fun($o,fun($o,B)),fun($o,fun($o,A)),aTP_Lamp_agi(fun(B,A),fun(fun($o,fun($o,B)),fun($o,fun($o,A))),H),F22)),VEBT) ) ).

% VEBT.case_distrib
tff(fact_8055_semigroup_Oassoc,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A2: A,B2: A,C2: A] :
      ( semigroup(A,F2)
     => ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),F2,A2),aa(A,A,aa(A,fun(A,A),F2,B2),C2)) ) ) ).

% semigroup.assoc
tff(fact_8056_semigroup_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A))] :
      ( ! [A4: A,B4: A,C3: A] : ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A4),B4)),C3) = aa(A,A,aa(A,fun(A,A),F2,A4),aa(A,A,aa(A,fun(A,A),F2,B4),C3)) )
     => semigroup(A,F2) ) ).

% semigroup.intro
tff(fact_8057_semigroup__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A))] :
      ( semigroup(A,F2)
    <=> ! [A5: A,B5: A,C4: A] : ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A5),B5)),C4) = aa(A,A,aa(A,fun(A,A),F2,A5),aa(A,A,aa(A,fun(A,A),F2,B5),C4)) ) ) ).

% semigroup_def
tff(fact_8058_min_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semigroup(A,ord_min(A)) ) ).

% min.semigroup_axioms
tff(fact_8059_VEBT_Osimps_I5_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),F22: fun($o,fun($o,A)),X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : ( aa(vEBT_VEBT,A,aa(fun($o,fun($o,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,A)),vEBT_case_VEBT(A),F1),F22),vEBT_Node(X11,X12,X13,X14)) = aa(vEBT_VEBT,A,aa(list(vEBT_VEBT),fun(vEBT_VEBT,A),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A))),F1,X11),X12),X13),X14) ) ).

% VEBT.simps(5)
tff(fact_8060_VEBT_Osimps_I6_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),F22: fun($o,fun($o,A)),X21: $o,X22: $o] : ( aa(vEBT_VEBT,A,aa(fun($o,fun($o,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,A)),vEBT_case_VEBT(A),F1),F22),vEBT_Leaf((X21),(X22))) = aa($o,A,aa($o,fun($o,A),F22,(X21)),(X22)) ) ).

% VEBT.simps(6)
tff(fact_8061_sup_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => semigroup(A,sup_sup(A)) ) ).

% sup.semigroup_axioms
tff(fact_8062_monoid_Oaxioms_I1_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A] :
      ( monoid(A,F2,Z)
     => semigroup(A,F2) ) ).

% monoid.axioms(1)
tff(fact_8063_add_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => semigroup(A,plus_plus(A)) ) ).

% add.semigroup_axioms
tff(fact_8064_group__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( group(A,F2,Z,Inverse)
    <=> ( semigroup(A,F2)
        & group_axioms(A,F2,Z,Inverse) ) ) ).

% group_def
tff(fact_8065_monoid_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A] :
      ( semigroup(A,F2)
     => ( monoid_axioms(A,F2,Z)
       => monoid(A,F2,Z) ) ) ).

% monoid.intro
tff(fact_8066_monoid__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A] :
      ( monoid(A,F2,Z)
    <=> ( semigroup(A,F2)
        & monoid_axioms(A,F2,Z) ) ) ).

% monoid_def
tff(fact_8067_monoid_Oaxioms_I2_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A] :
      ( monoid(A,F2,Z)
     => monoid_axioms(A,F2,Z) ) ).

% monoid.axioms(2)
tff(fact_8068_monoid__axioms__def,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A] :
      ( monoid_axioms(A,F2,Z)
    <=> ( ! [A5: A] : ( aa(A,A,aa(A,fun(A,A),F2,Z),A5) = A5 )
        & ! [A5: A] : ( aa(A,A,aa(A,fun(A,A),F2,A5),Z) = A5 ) ) ) ).

% monoid_axioms_def
tff(fact_8069_monoid__axioms_Ointro,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A] :
      ( ! [A4: A] : ( aa(A,A,aa(A,fun(A,A),F2,Z),A4) = A4 )
     => ( ! [A4: A] : ( aa(A,A,aa(A,fun(A,A),F2,A4),Z) = A4 )
       => monoid_axioms(A,F2,Z) ) ) ).

% monoid_axioms.intro
tff(fact_8070_Quotient3__real,axiom,
    quotient3(fun(nat,rat),real,realrel,real2,rep_real) ).

% Quotient3_real
tff(fact_8071_sum__encode__def,axiom,
    ! [Xb: sum_sum(nat,nat)] : ( aa(sum_sum(nat,nat),nat,nat_sum_encode,Xb) = sum_case_sum(nat,nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aTP_Lamp_agj(nat,nat),Xb) ) ).

% sum_encode_def
tff(fact_8072_inj__sum__encode,axiom,
    ! [A3: set(sum_sum(nat,nat))] : inj_on(sum_sum(nat,nat),nat,nat_sum_encode,A3) ).

% inj_sum_encode
tff(fact_8073_bij__sum__encode,axiom,
    bij_betw(sum_sum(nat,nat),nat,nat_sum_encode,top_top(set(sum_sum(nat,nat))),top_top(set(nat))) ).

% bij_sum_encode
tff(fact_8074_sum__encode__eq,axiom,
    ! [Xb: sum_sum(nat,nat),Y: sum_sum(nat,nat)] :
      ( ( aa(sum_sum(nat,nat),nat,nat_sum_encode,Xb) = aa(sum_sum(nat,nat),nat,nat_sum_encode,Y) )
    <=> ( Xb = Y ) ) ).

% sum_encode_eq
tff(fact_8075_surj__sum__encode,axiom,
    image(sum_sum(nat,nat),nat,nat_sum_encode,top_top(set(sum_sum(nat,nat)))) = top_top(set(nat)) ).

% surj_sum_encode
tff(fact_8076_int__encode__def,axiom,
    ! [I: int] :
      ( aa(int,nat,nat_int_encode,I) = aa(sum_sum(nat,nat),nat,nat_sum_encode,
          $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I),aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),aa(int,nat,nat2,I)),aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),I)),one_one(int)))))) ) ).

% int_encode_def
tff(fact_8077_nths__nths,axiom,
    ! [A: $tType,Xs: list(A),A3: set(nat),B3: set(nat)] : ( nths(A,nths(A,Xs,A3),B3) = nths(A,Xs,collect(nat,aa(set(nat),fun(nat,$o),aTP_Lamp_agl(set(nat),fun(set(nat),fun(nat,$o)),A3),B3))) ) ).

% nths_nths
tff(fact_8078_sum_OPlus,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A3: set(A),B3: set(B),G: fun(sum_sum(A,B),C)] :
          ( aa(set(A),$o,finite_finite(A),A3)
         => ( aa(set(B),$o,finite_finite(B),B3)
           => ( aa(set(sum_sum(A,B)),C,aa(fun(sum_sum(A,B),C),fun(set(sum_sum(A,B)),C),groups7311177749621191930dd_sum(sum_sum(A,B),C),G),sum_Plus(A,B,A3,B3)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(A,sum_sum(A,B)),fun(A,C),comp(sum_sum(A,B),C,A,G),sum_Inl(A,B))),A3)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,sum_sum(A,B)),fun(B,C),comp(sum_sum(A,B),C,B,G),sum_Inr(B,A))),B3)) ) ) ) ) ).

% sum.Plus
tff(fact_8079_Bex__set__list__ex,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X4) )
    <=> list_ex(A,P,Xs) ) ).

% Bex_set_list_ex
tff(fact_8080_ndepth__Push__Node__aux,axiom,
    ! [A: $tType,I: nat,F2: fun(nat,sum_sum(A,nat)),K: nat] :
      ( ( case_nat(sum_sum(A,nat),aa(nat,sum_sum(A,nat),sum_Inr(nat,A),aa(nat,nat,suc,I)),F2,K) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_agm(fun(nat,sum_sum(A,nat)),fun(nat,$o),F2)))),K) ) ).

% ndepth_Push_Node_aux
tff(fact_8081_sum_Osize_I3_J,axiom,
    ! [A: $tType,B: $tType,X15: A] : ( aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),aa(A,sum_sum(A,B),sum_Inl(A,B),X15)) = aa(nat,nat,suc,zero_zero(nat)) ) ).

% sum.size(3)
tff(fact_8082_prod_Osize__neq,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(A,B)] : ( aa(product_prod(A,B),nat,size_size(product_prod(A,B)),Xb) != zero_zero(nat) ) ).

% prod.size_neq
tff(fact_8083_sum_Osize__neq,axiom,
    ! [A: $tType,B: $tType,Xb: sum_sum(A,B)] : ( aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),Xb) != zero_zero(nat) ) ).

% sum.size_neq
tff(fact_8084_sum_Osize_I4_J,axiom,
    ! [B: $tType,A: $tType,X2: B] : ( aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),aa(B,sum_sum(A,B),sum_Inr(B,A),X2)) = aa(nat,nat,suc,zero_zero(nat)) ) ).

% sum.size(4)
tff(fact_8085_sum_Osize__gen_I1_J,axiom,
    ! [B: $tType,A: $tType,Xaa: fun(A,nat),Xb: fun(B,nat),X15: A] : ( basic_BNF_size_sum(A,B,Xaa,Xb,aa(A,sum_sum(A,B),sum_Inl(A,B),X15)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xaa,X15)),aa(nat,nat,suc,zero_zero(nat))) ) ).

% sum.size_gen(1)
tff(fact_8086_sum_Osize__gen_I2_J,axiom,
    ! [A: $tType,B: $tType,Xaa: fun(A,nat),Xb: fun(B,nat),X2: B] : ( basic_BNF_size_sum(A,B,Xaa,Xb,aa(B,sum_sum(A,B),sum_Inr(B,A),X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(B,nat,Xb,X2)),aa(nat,nat,suc,zero_zero(nat))) ) ).

% sum.size_gen(2)
tff(fact_8087_sum__decode__def,axiom,
    ! [Nb: nat] :
      ( aa(nat,sum_sum(nat,nat),nat_sum_decode,Nb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb),aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% sum_decode_def
tff(fact_8088_Node__def,axiom,
    ! [B: $tType,A: $tType] : ( old_Node(A,B) = collect(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),aTP_Lamp_agn(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),$o)) ) ).

% Node_def
tff(fact_8089_sum__encode__inverse,axiom,
    ! [Xb: sum_sum(nat,nat)] : ( aa(nat,sum_sum(nat,nat),nat_sum_decode,aa(sum_sum(nat,nat),nat,nat_sum_encode,Xb)) = Xb ) ).

% sum_encode_inverse
tff(fact_8090_sum__decode__inverse,axiom,
    ! [Nb: nat] : ( aa(sum_sum(nat,nat),nat,nat_sum_encode,aa(nat,sum_sum(nat,nat),nat_sum_decode,Nb)) = Nb ) ).

% sum_decode_inverse
tff(fact_8091_surj__sum__decode,axiom,
    image(nat,sum_sum(nat,nat),nat_sum_decode,top_top(set(nat))) = top_top(set(sum_sum(nat,nat))) ).

% surj_sum_decode
tff(fact_8092_bij__sum__decode,axiom,
    bij_betw(nat,sum_sum(nat,nat),nat_sum_decode,top_top(set(nat)),top_top(set(sum_sum(nat,nat)))) ).

% bij_sum_decode
tff(fact_8093_inj__sum__decode,axiom,
    ! [A3: set(nat)] : inj_on(nat,sum_sum(nat,nat),nat_sum_decode,A3) ).

% inj_sum_decode
tff(fact_8094_sum__decode__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,Xb) = aa(nat,sum_sum(nat,nat),nat_sum_decode,Y) )
    <=> ( Xb = Y ) ) ).

% sum_decode_eq
tff(fact_8095_Node__K0__I,axiom,
    ! [B: $tType,A: $tType,A2: sum_sum(B,nat)] : member(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),aa(sum_sum(B,nat),product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),product_Pair(fun(nat,sum_sum(A,nat)),sum_sum(B,nat),aTP_Lamp_ago(nat,sum_sum(A,nat))),A2),old_Node(A,B)) ).

% Node_K0_I
tff(fact_8096_nth__item_Opinduct,axiom,
    ! [A0: nat,P: fun(nat,$o)] :
      ( accp(nat,nth_item_rel,A0)
     => ( ( accp(nat,nth_item_rel,zero_zero(nat))
         => aa(nat,$o,P,zero_zero(nat)) )
       => ( ! [N: nat] :
              ( accp(nat,nth_item_rel,aa(nat,nat,suc,N))
             => ( ! [A8: nat,Aa2: nat] :
                    ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,N) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),A8) )
                   => ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,A8) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),Aa2) )
                     => aa(nat,$o,P,Aa2) ) )
               => ( ! [A8: nat,B14: nat] :
                      ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,N) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),A8) )
                     => ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,A8) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B14) )
                       => aa(nat,$o,P,B14) ) )
                 => ( ! [B14: nat,Ba: nat,X3: nat,Y4: nat] :
                        ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,N) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B14) )
                       => ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,B14) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),Ba) )
                         => ( ( aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X3),Y4) = aa(nat,product_prod(nat,nat),nat_prod_decode,Ba) )
                           => aa(nat,$o,P,X3) ) ) )
                   => ( ! [B14: nat,Ba: nat,X3: nat,Y4: nat] :
                          ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,N) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B14) )
                         => ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,B14) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),Ba) )
                           => ( ( aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X3),Y4) = aa(nat,product_prod(nat,nat),nat_prod_decode,Ba) )
                             => aa(nat,$o,P,Y4) ) ) )
                     => aa(nat,$o,P,aa(nat,nat,suc,N)) ) ) ) ) )
         => aa(nat,$o,P,A0) ) ) ) ).

% nth_item.pinduct
tff(fact_8097_int__decode__def,axiom,
    ! [Nb: nat] : ( aa(nat,int,nat_int_decode,Nb) = sum_case_sum(nat,int,nat,semiring_1_of_nat(int),aTP_Lamp_agp(nat,int),aa(nat,sum_sum(nat,nat),nat_sum_decode,Nb)) ) ).

% int_decode_def
tff(fact_8098_Push__neq__K0,axiom,
    ! [A: $tType,K: nat,F2: fun(nat,sum_sum(A,nat))] :
      ~ ! [X: nat] : ( old_Push(A,aa(nat,sum_sum(A,nat),sum_Inr(nat,A),aa(nat,nat,suc,K)),F2,X) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ) ).

% Push_neq_K0
tff(fact_8099_ndepth__def,axiom,
    ! [B: $tType,A: $tType,Nb: old_node(A,B)] : ( old_ndepth(A,B,Nb) = aa(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),nat,product_case_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat),nat,aTP_Lamp_agr(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat))),old_Rep_Node(A,B,Nb)) ) ).

% ndepth_def
tff(fact_8100_ndepth__K0,axiom,
    ! [A: $tType,B: $tType,Xb: sum_sum(A,nat)] : ( old_ndepth(A,B,old_Abs_Node(B,A,aa(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),product_Pair(fun(nat,sum_sum(B,nat)),sum_sum(A,nat),aTP_Lamp_ags(nat,sum_sum(B,nat))),Xb))) = zero_zero(nat) ) ).

% ndepth_K0
tff(fact_8101_Atom__def,axiom,
    ! [B: $tType,A: $tType,X3: sum_sum(A,nat)] : ( old_Atom(A,B,X3) = aa(set(old_node(A,B)),set(old_node(A,B)),aa(old_node(A,B),fun(set(old_node(A,B)),set(old_node(A,B))),insert2(old_node(A,B)),old_Abs_Node(B,A,aa(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),product_Pair(fun(nat,sum_sum(B,nat)),sum_sum(A,nat),aTP_Lamp_ags(nat,sum_sum(B,nat))),X3))),bot_bot(set(old_node(A,B)))) ) ).

% Atom_def
tff(fact_8102_ndepth__Push__Node,axiom,
    ! [B: $tType,A: $tType,I: nat,Nb: old_node(A,B)] : ( old_ndepth(A,B,aa(old_node(A,B),old_node(A,B),old_Push_Node(B,A,aa(nat,sum_sum(B,nat),sum_Inr(nat,B),aa(nat,nat,suc,I))),Nb)) = aa(nat,nat,suc,old_ndepth(A,B,Nb)) ) ).

% ndepth_Push_Node
tff(fact_8103_Scons__def,axiom,
    ! [B: $tType,A: $tType,M9: set(old_node(A,B)),N5: set(old_node(A,B))] : ( old_Scons(A,B,M9,N5) = aa(set(old_node(A,B)),set(old_node(A,B)),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(old_node(A,B))),sup_sup(set(old_node(A,B))),image(old_node(A,B),old_node(A,B),old_Push_Node(B,A,aa(nat,sum_sum(B,nat),sum_Inr(nat,B),one_one(nat))),M9)),image(old_node(A,B),old_node(A,B),old_Push_Node(B,A,aa(nat,sum_sum(B,nat),sum_Inr(nat,B),aa(nat,nat,suc,one_one(nat)))),N5)) ) ).

% Scons_def
tff(fact_8104_ntrunc__0,axiom,
    ! [B: $tType,A: $tType,M9: set(old_node(A,B))] : ( old_ntrunc(A,B,zero_zero(nat),M9) = bot_bot(set(old_node(A,B))) ) ).

% ntrunc_0
tff(fact_8105_ntrunc__Scons,axiom,
    ! [B: $tType,A: $tType,K: nat,M9: set(old_node(A,B)),N5: set(old_node(A,B))] : ( old_ntrunc(A,B,aa(nat,nat,suc,K),old_Scons(A,B,M9,N5)) = old_Scons(A,B,old_ntrunc(A,B,K,M9),old_ntrunc(A,B,K,N5)) ) ).

% ntrunc_Scons
tff(fact_8106_ntrunc__Atom,axiom,
    ! [B: $tType,A: $tType,K: nat,A2: sum_sum(A,nat)] : ( old_ntrunc(A,B,aa(nat,nat,suc,K),old_Atom(A,B,A2)) = old_Atom(A,B,A2) ) ).

% ntrunc_Atom
tff(fact_8107_ntrunc__def,axiom,
    ! [B: $tType,A: $tType,K: nat,N5: set(old_node(A,B))] : ( old_ntrunc(A,B,K,N5) = collect(old_node(A,B),aa(set(old_node(A,B)),fun(old_node(A,B),$o),aTP_Lamp_agt(nat,fun(set(old_node(A,B)),fun(old_node(A,B),$o)),K),N5)) ) ).

% ntrunc_def
tff(fact_8108_ntrunc__one__In1,axiom,
    ! [B: $tType,A: $tType,M9: set(old_node(A,B))] : ( old_ntrunc(A,B,aa(nat,nat,suc,zero_zero(nat)),old_In1(A,B,M9)) = bot_bot(set(old_node(A,B))) ) ).

% ntrunc_one_In1
tff(fact_8109_ntrunc__one__In0,axiom,
    ! [B: $tType,A: $tType,M9: set(old_node(A,B))] : ( old_ntrunc(A,B,aa(nat,nat,suc,zero_zero(nat)),old_In0(A,B,M9)) = bot_bot(set(old_node(A,B))) ) ).

% ntrunc_one_In0
tff(fact_8110_ntrunc__In1,axiom,
    ! [B: $tType,A: $tType,K: nat,M9: set(old_node(A,B))] : ( old_ntrunc(A,B,aa(nat,nat,suc,aa(nat,nat,suc,K)),old_In1(A,B,M9)) = old_In1(A,B,old_ntrunc(A,B,aa(nat,nat,suc,K),M9)) ) ).

% ntrunc_In1
tff(fact_8111_ntrunc__In0,axiom,
    ! [B: $tType,A: $tType,K: nat,M9: set(old_node(A,B))] : ( old_ntrunc(A,B,aa(nat,nat,suc,aa(nat,nat,suc,K)),old_In0(A,B,M9)) = old_In0(A,B,old_ntrunc(A,B,aa(nat,nat,suc,K),M9)) ) ).

% ntrunc_In0
tff(fact_8112_ntrunc__Leaf,axiom,
    ! [B: $tType,A: $tType,K: nat,A2: A] : ( old_ntrunc(A,B,aa(nat,nat,suc,K),old_Leaf(A,B,A2)) = old_Leaf(A,B,A2) ) ).

% ntrunc_Leaf
tff(fact_8113_In1__def,axiom,
    ! [B: $tType,A: $tType,M9: set(old_node(A,B))] : ( old_In1(A,B,M9) = old_Scons(A,B,old_Numb(A,B,one_one(nat)),M9) ) ).

% In1_def
tff(fact_8114_ntrunc__Numb,axiom,
    ! [A: $tType,B: $tType,K: nat,I: nat] : ( old_ntrunc(A,B,aa(nat,nat,suc,K),old_Numb(A,B,I)) = old_Numb(A,B,I) ) ).

% ntrunc_Numb
tff(fact_8115_In0__def,axiom,
    ! [B: $tType,A: $tType,M9: set(old_node(A,B))] : ( old_In0(A,B,M9) = old_Scons(A,B,old_Numb(A,B,zero_zero(nat)),M9) ) ).

% In0_def
tff(fact_8116_ATP_Olambda__1,axiom,
    ! [Uu: product_prod(int,int)] :
      ( aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_abb(product_prod(int,int),product_prod(int,int)),Uu) = $ite(aa(product_prod(int,int),int,product_fst(int,int),Uu) = zero_zero(int),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_fst(int,int),Uu))) ) ).

% ATP.lambda_1
tff(fact_8117_ATP_Olambda__2,axiom,
    ! [Uu: fun(nat,rat)] :
      ( aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aah(fun(nat,rat),fun(nat,rat)),Uu) = $ite(vanishes(Uu),aTP_Lamp_aac(nat,rat),aTP_Lamp_aaa(fun(nat,rat),fun(nat,rat),Uu)) ) ).

% ATP.lambda_2
tff(fact_8118_ATP_Olambda__3,axiom,
    ! [Uu: nat] : ( aa(nat,real,aTP_Lamp_bf(nat,real),Uu) = 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_3
tff(fact_8119_ATP_Olambda__4,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : ( aa(A,A,aTP_Lamp_uq(A,A),Uu) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),exp(A,Uu)),one_one(A)),Uu) ) ) ).

% ATP.lambda_4
tff(fact_8120_ATP_Olambda__5,axiom,
    ! [A: $tType,Uu: set(set(A))] : ( aa(set(set(A)),int,aTP_Lamp_pe(set(set(A)),int),Uu) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(set(A)),nat,finite_card(set(A)),Uu)),one_one(nat)))),aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Uu)))) ) ).

% ATP.lambda_5
tff(fact_8121_ATP_Olambda__6,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A] :
          ( aa(A,$o,aTP_Lamp_oo(A,$o),Uu)
        <=> ( member(A,Uu,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Uu) ) ) ) ).

% ATP.lambda_6
tff(fact_8122_ATP_Olambda__7,axiom,
    ! [Uu: nat] : ( aa(nat,real,aTP_Lamp_da(nat,real),Uu) = aa(nat,real,aa(real,fun(nat,real),power_power(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,nat,suc,Uu)) ) ).

% ATP.lambda_7
tff(fact_8123_ATP_Olambda__8,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: product_prod(int,int)] : ( aa(product_prod(int,int),A,aTP_Lamp_abg(product_prod(int,int),A),Uu) = divide_divide(A,aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_snd(int,int),Uu))) ) ) ).

% ATP.lambda_8
tff(fact_8124_ATP_Olambda__9,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : ( aa(nat,A,aTP_Lamp_vu(nat,A),Uu) = 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_9
tff(fact_8125_ATP_Olambda__10,axiom,
    ! [Uu: product_prod(int,int)] : ( aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_abf(product_prod(int,int),product_prod(int,int)),Uu) = aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,uminus_uminus(int),aa(product_prod(int,int),int,product_fst(int,int),Uu))),aa(product_prod(int,int),int,product_snd(int,int),Uu)) ) ).

% ATP.lambda_10
tff(fact_8126_ATP_Olambda__11,axiom,
    ! [Uu: nat] : ( aa(nat,int,aTP_Lamp_agp(nat,int),Uu) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Uu))),one_one(int)) ) ).

% ATP.lambda_11
tff(fact_8127_ATP_Olambda__12,axiom,
    ! [Uu: nat] : ( aa(nat,real,aTP_Lamp_du(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_12
tff(fact_8128_ATP_Olambda__13,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : ( aa(nat,A,aTP_Lamp_vt(nat,A),Uu) = 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_13
tff(fact_8129_ATP_Olambda__14,axiom,
    ! [A: $tType] :
      ( topolo6943815403480290642id_add(A)
     => ! [Uu: product_prod(A,A)] : ( aa(product_prod(A,A),A,aTP_Lamp_adl(product_prod(A,A),A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(product_prod(A,A),A,product_fst(A,A),Uu)),aa(product_prod(A,A),A,product_snd(A,A),Uu)) ) ) ).

% ATP.lambda_14
tff(fact_8130_ATP_Olambda__15,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat] :
          ( aa(nat,$o,aTP_Lamp_lk(nat,$o),Uu)
        <=> ( aa(nat,A,semiring_1_of_nat(A),Uu) = zero_zero(A) ) ) ) ).

% ATP.lambda_15
tff(fact_8131_ATP_Olambda__16,axiom,
    ! [Uu: nat] : ( aa(nat,real,aTP_Lamp_vf(nat,real),Uu) = aa(real,real,root(Uu),aa(nat,real,semiring_1_of_nat(real),Uu)) ) ).

% ATP.lambda_16
tff(fact_8132_ATP_Olambda__17,axiom,
    ! [Uu: nat] : ( aa(nat,nat,aTP_Lamp_kv(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ) ).

% ATP.lambda_17
tff(fact_8133_ATP_Olambda__18,axiom,
    ! [Uu: int] : ( aa(int,int,aTP_Lamp_aai(int,int),Uu) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),Uu) ) ).

% ATP.lambda_18
tff(fact_8134_ATP_Olambda__19,axiom,
    ! [A: $tType,Uu: A] : ( aa(A,product_prod(A,A),aTP_Lamp_ok(A,product_prod(A,A)),Uu) = aa(A,product_prod(A,A),product_Pair(A,A,Uu),Uu) ) ).

% ATP.lambda_19
tff(fact_8135_ATP_Olambda__20,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : ( aa(A,A,aTP_Lamp_hc(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ) ).

% ATP.lambda_20
tff(fact_8136_ATP_Olambda__21,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: A] :
          ( aa(A,$o,aTP_Lamp_afi(A,$o),Uu)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Uu),one_one(A)) ) ) ).

% ATP.lambda_21
tff(fact_8137_ATP_Olambda__22,axiom,
    ! [Uu: nat] : ( aa(nat,product_prod(nat,nat),aTP_Lamp_aak(nat,product_prod(nat,nat)),Uu) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Uu),zero_zero(nat)) ) ).

% ATP.lambda_22
tff(fact_8138_ATP_Olambda__23,axiom,
    ! [Uu: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aTP_Lamp_aay(product_prod(int,int),$o),Uu)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uu))) ) ).

% ATP.lambda_23
tff(fact_8139_ATP_Olambda__24,axiom,
    ! [Uu: code_natural] : ( aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_agd(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,aTP_Lamp_agc(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu)) ) ).

% ATP.lambda_24
tff(fact_8140_ATP_Olambda__25,axiom,
    ! [Uu: nat] : ( aa(nat,real,aTP_Lamp_vn(nat,real),Uu) = divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),Uu)) ) ).

% ATP.lambda_25
tff(fact_8141_ATP_Olambda__26,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : ( aa(nat,A,aTP_Lamp_vs(nat,A),Uu) = divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ) ).

% ATP.lambda_26
tff(fact_8142_ATP_Olambda__27,axiom,
    ! [Uu: nat] : ( aa(nat,nat,aTP_Lamp_agj(nat,nat),Uu) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu)) ) ).

% ATP.lambda_27
tff(fact_8143_ATP_Olambda__28,axiom,
    ! [B: $tType,Uu: list(B)] : ( aa(list(B),fun(nat,nat),aTP_Lamp_oa(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Uu)),aa(nat,nat,suc,zero_zero(nat)))) ) ).

% ATP.lambda_28
tff(fact_8144_ATP_Olambda__29,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_ob(list(B),$o),Uu)
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_29
tff(fact_8145_ATP_Olambda__30,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_oc(list(A),$o),Uu)
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_30
tff(fact_8146_ATP_Olambda__31,axiom,
    ! [Uu: real] : ( aa(real,real,aTP_Lamp_qk(real,real),Uu) = suminf(real,aTP_Lamp_as(real,fun(nat,real),Uu)) ) ).

% ATP.lambda_31
tff(fact_8147_ATP_Olambda__32,axiom,
    ! [Uu: nat] : ( aa(nat,set(nat),aTP_Lamp_zw(nat,set(nat)),Uu) = collect(nat,aTP_Lamp_iy(nat,fun(nat,$o),Uu)) ) ).

% ATP.lambda_32
tff(fact_8148_ATP_Olambda__33,axiom,
    ! [Uu: nat] : ( aa(nat,real,aTP_Lamp_vp(nat,real),Uu) = aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uu))) ) ).

% ATP.lambda_33
tff(fact_8149_ATP_Olambda__34,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: nat] : ( aa(nat,A,aTP_Lamp_es(nat,A),Uu) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uu)) ) ) ).

% ATP.lambda_34
tff(fact_8150_ATP_Olambda__35,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : ( aa(nat,A,aTP_Lamp_vh(nat,A),Uu) = aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ) ).

% ATP.lambda_35
tff(fact_8151_ATP_Olambda__36,axiom,
    ! [B: $tType,Uu: list(B)] : ( aa(list(B),fun(nat,nat),aTP_Lamp_nz(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(B),nat,size_size(list(B)),Uu)) ) ).

% ATP.lambda_36
tff(fact_8152_ATP_Olambda__37,axiom,
    ! [A: $tType,Uu: list(A)] : ( aa(list(A),fun(nat,nat),aTP_Lamp_nq(list(A),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Uu)) ) ).

% ATP.lambda_37
tff(fact_8153_ATP_Olambda__38,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A] : ( aa(A,B,aTP_Lamp_zj(A,B),Uu) = aa(int,B,ring_1_of_int(B),aa(A,int,archim6421214686448440834_floor(A),Uu)) ) ) ).

% ATP.lambda_38
tff(fact_8154_ATP_Olambda__39,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A] : ( aa(A,B,aTP_Lamp_zk(A,B),Uu) = aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,Uu)) ) ) ).

% ATP.lambda_39
tff(fact_8155_ATP_Olambda__40,axiom,
    ! [Uu: num] : ( aa(num,option(num),aTP_Lamp_mo(num,option(num)),Uu) = aa(num,option(num),some(num),bit0(Uu)) ) ).

% ATP.lambda_40
tff(fact_8156_ATP_Olambda__41,axiom,
    ! [Uu: int] : ( aa(int,fun(int,product_prod(int,int)),aTP_Lamp_km(int,fun(int,product_prod(int,int))),Uu) = product_Pair(int,int,aa(int,int,uminus_uminus(int),Uu)) ) ).

% ATP.lambda_41
tff(fact_8157_ATP_Olambda__42,axiom,
    ! [Uu: nat] : ( aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_hl(nat,fun(nat,product_prod(nat,nat))),Uu) = product_Pair(nat,nat,aa(nat,nat,suc,Uu)) ) ).

% ATP.lambda_42
tff(fact_8158_ATP_Olambda__43,axiom,
    ! [Uu: nat] : ( aa(nat,extended_enat,aTP_Lamp_adv(nat,extended_enat),Uu) = extended_enat2(aa(nat,nat,suc,Uu)) ) ).

% ATP.lambda_43
tff(fact_8159_ATP_Olambda__44,axiom,
    ! [Uu: int] : ( aa(int,nat,aTP_Lamp_ml(int,nat),Uu) = aa(int,nat,nat2,aa(int,int,abs_abs(int),Uu)) ) ).

% ATP.lambda_44
tff(fact_8160_ATP_Olambda__45,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: A] :
          ( aa(A,$o,aTP_Lamp_oj(A,$o),Uu)
        <=> ? [N4: int] :
              ( ( Uu = aa(int,A,ring_1_of_int(A),N4) )
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N4) ) ) ) ).

% ATP.lambda_45
tff(fact_8161_ATP_Olambda__46,axiom,
    ! [Uu: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aTP_Lamp_aag(fun(nat,rat),$o),Uu)
    <=> ? [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
          & ? [K3: nat] :
            ! [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N4)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,Uu,N4)) ) ) ) ).

% ATP.lambda_46
tff(fact_8162_ATP_Olambda__47,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_of(real,$o),Uu)
    <=> ? [I3: int,N4: nat] :
          ( ( Uu = divide_divide(real,aa(int,real,ring_1_of_int(real),I3),aa(nat,real,semiring_1_of_nat(real),N4)) )
          & ( N4 != zero_zero(nat) ) ) ) ).

% ATP.lambda_47
tff(fact_8163_ATP_Olambda__48,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_og(real,$o),Uu)
    <=> ? [I3: int,J3: int] :
          ( ( Uu = divide_divide(real,aa(int,real,ring_1_of_int(real),I3),aa(int,real,ring_1_of_int(real),J3)) )
          & ( J3 != zero_zero(int) ) ) ) ).

% ATP.lambda_48
tff(fact_8164_ATP_Olambda__49,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_ye(product_prod(A,A),$o),Uu)
        <=> ? [X4: A,Y5: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X4),Y5) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y5) ) ) ) ).

% ATP.lambda_49
tff(fact_8165_ATP_Olambda__50,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_yf(product_prod(A,A),$o),Uu)
        <=> ? [X4: A,Y5: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X4),Y5) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X4) ) ) ) ).

% ATP.lambda_50
tff(fact_8166_ATP_Olambda__51,axiom,
    ! [B: $tType,A: $tType,Uu: product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))] :
      ( aa(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),$o,aTP_Lamp_agn(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),$o),Uu)
    <=> ? [F5: fun(nat,sum_sum(A,nat)),X4: sum_sum(B,nat),K3: nat] :
          ( ( Uu = aa(sum_sum(B,nat),product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),product_Pair(fun(nat,sum_sum(A,nat)),sum_sum(B,nat),F5),X4) )
          & ( aa(nat,sum_sum(A,nat),F5,K3) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ) ) ) ).

% ATP.lambda_51
tff(fact_8167_ATP_Olambda__52,axiom,
    ! [B: $tType,Uu: nat] : ( aa(nat,sum_sum(B,nat),aTP_Lamp_ags(nat,sum_sum(B,nat)),Uu) = aa(nat,sum_sum(B,nat),sum_Inr(nat,B),zero_zero(nat)) ) ).

% ATP.lambda_52
tff(fact_8168_ATP_Olambda__53,axiom,
    ! [A: $tType,Uu: nat] : ( aa(nat,sum_sum(A,nat),aTP_Lamp_ago(nat,sum_sum(A,nat)),Uu) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ) ).

% ATP.lambda_53
tff(fact_8169_ATP_Olambda__54,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          ( aa(nat,A,aTP_Lamp_ha(A,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),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_54
tff(fact_8170_ATP_Olambda__55,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          ( aa(nat,A,aTP_Lamp_eq(nat,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ) ).

% ATP.lambda_55
tff(fact_8171_ATP_Olambda__56,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          ( aa(nat,A,aTP_Lamp_hb(A,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),zero_zero(A),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_56
tff(fact_8172_ATP_Olambda__57,axiom,
    ! [Uu: fun(nat,real),Uua: nat] :
      ( aa(nat,real,aTP_Lamp_dc(fun(nat,real),fun(nat,real),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),zero_zero(real),aa(nat,real,Uu,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_57
tff(fact_8173_ATP_Olambda__58,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: product_prod(A,C),Uua: product_prod(C,B)] :
      ( aa(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_abi(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu),Uua) = $ite(aa(product_prod(A,C),C,product_snd(A,C),Uu) = aa(product_prod(C,B),C,product_fst(C,B),Uua),cons(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,aa(product_prod(A,C),A,product_fst(A,C),Uu)),aa(product_prod(C,B),B,product_snd(C,B),Uua)),nil(product_prod(A,B))),nil(product_prod(A,B))) ) ).

% ATP.lambda_58
tff(fact_8174_ATP_Olambda__59,axiom,
    ! [Uu: int,Uua: int] :
      ( aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_kh(int,fun(int,product_prod(int,int))),Uu),Uua) = $ite(Uu = zero_zero(int),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),Uu)),Uua)),aa(int,int,abs_abs(int),Uu))) ) ).

% ATP.lambda_59
tff(fact_8175_ATP_Olambda__60,axiom,
    ! [Uu: int,Uua: int] :
      ( aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_abe(int,fun(int,product_prod(int,int))),Uu),Uua) = $ite(Uua = zero_zero(int),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),product_Pair(int,int,Uu),Uua)) ) ).

% ATP.lambda_60
tff(fact_8176_ATP_Olambda__61,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( aa(nat,extended_enat,aTP_Lamp_adu(extended_enat,fun(nat,extended_enat),Uu),Uua) = extended_case_enat(extended_enat,aTP_Lamp_adt(nat,fun(nat,extended_enat),Uua),
          $ite(Uua = zero_zero(nat),zero_zero(extended_enat),extend4730790105801354508finity(extended_enat)),
          Uu) ) ).

% ATP.lambda_61
tff(fact_8177_ATP_Olambda__62,axiom,
    ! [Uu: extended_enat,Uua: nat] : ( aa(nat,extended_enat,aTP_Lamp_ads(extended_enat,fun(nat,extended_enat),Uu),Uua) = extended_case_enat(extended_enat,aTP_Lamp_adr(nat,fun(nat,extended_enat),Uua),extend4730790105801354508finity(extended_enat),Uu) ) ).

% ATP.lambda_62
tff(fact_8178_ATP_Olambda__63,axiom,
    ! [Uu: extended_enat,Uua: nat] : ( aa(nat,extended_enat,aTP_Lamp_adq(extended_enat,fun(nat,extended_enat),Uu),Uua) = extended_case_enat(extended_enat,aTP_Lamp_adp(nat,fun(nat,extended_enat),Uua),zero_zero(extended_enat),Uu) ) ).

% ATP.lambda_63
tff(fact_8179_ATP_Olambda__64,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          ( aa(nat,A,aTP_Lamp_gl(nat,fun(nat,A),Uu),Uua) = $ite(~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ) ).

% ATP.lambda_64
tff(fact_8180_ATP_Olambda__65,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_acv(extended_enat,fun(nat,$o),Uu),Uua)
    <=> extended_case_enat($o,aa(nat,fun(nat,$o),ord_less(nat),Uua),$true,Uu) ) ).

% ATP.lambda_65
tff(fact_8181_ATP_Olambda__66,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Uu: fun(A,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_yp(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),top_top(A)) ) ) ).

% ATP.lambda_66
tff(fact_8182_ATP_Olambda__67,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu: fun(A,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_ou(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),bot_bot(A)) ) ) ).

% ATP.lambda_67
tff(fact_8183_ATP_Olambda__68,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_abw(list(list(A)),fun(list(A),$o),Uu),Uua)
    <=> aa(list(list(A)),$o,aa(list(A),fun(list(list(A)),$o),list_all2(A,list(A),aTP_Lamp_abv(A,fun(list(A),$o))),Uua),Uu) ) ).

% ATP.lambda_68
tff(fact_8184_ATP_Olambda__69,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_aat(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),Uu),Uua)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),B,product_snd(A,B),Uua)) ) ).

% ATP.lambda_69
tff(fact_8185_ATP_Olambda__70,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : ( aa(nat,list(A),aTP_Lamp_no(list(list(A)),fun(nat,list(A)),Uu),Uua) = aa(list(nat),list(A),map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_nn(list(list(A)),fun(nat,fun(nat,A)),Uu),Uua)),upt(zero_zero(nat),aa(list(list(A)),nat,size_size(list(list(A))),Uu))) ) ).

% ATP.lambda_70
tff(fact_8186_ATP_Olambda__71,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : ( aa(nat,A,aTP_Lamp_ht(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hs(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ) ).

% ATP.lambda_71
tff(fact_8187_ATP_Olambda__72,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : ( aa(nat,A,aTP_Lamp_hr(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hq(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ) ).

% ATP.lambda_72
tff(fact_8188_ATP_Olambda__73,axiom,
    ! [Uu: product_prod(int,int),Uua: product_prod(int,int)] : ( aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aTP_Lamp_aba(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu),Uua) = aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uua)),aa(product_prod(int,int),int,product_snd(int,int),Uu)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))) ) ).

% ATP.lambda_73
tff(fact_8189_ATP_Olambda__74,axiom,
    ! [Uu: real,Uua: nat] : ( aa(nat,real,aTP_Lamp_ar(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)),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_74
tff(fact_8190_ATP_Olambda__75,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),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_75
tff(fact_8191_ATP_Olambda__76,axiom,
    ! [Uu: real,Uua: nat] : ( aa(nat,real,aTP_Lamp_di(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(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_76
tff(fact_8192_ATP_Olambda__77,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : ( aa(nat,A,aTP_Lamp_gc(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_77
tff(fact_8193_ATP_Olambda__78,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : ( aa(nat,A,aTP_Lamp_gd(nat,fun(nat,A),Uu),Uua) = 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_78
tff(fact_8194_ATP_Olambda__79,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_ho(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(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),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( sin(real,Uua) = Uu ) ) ) ).

% ATP.lambda_79
tff(fact_8195_ATP_Olambda__80,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_hn(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(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),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( aa(real,real,tan(real),Uua) = Uu ) ) ) ).

% ATP.lambda_80
tff(fact_8196_ATP_Olambda__81,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_kx(code_integer,fun(code_integer,int)),Uu),Uua) = $let(
          l2: int,
          l2:= aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(code_integer,int,code_int_of_integer,Uu)),
          $ite(Uua = zero_zero(code_integer),l2,aa(int,int,aa(int,fun(int,int),plus_plus(int),l2),one_one(int))) ) ) ).

% ATP.lambda_81
tff(fact_8197_ATP_Olambda__82,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,a,aa(nat,fun(nat,a),aTP_Lamp_hv(nat,fun(nat,a)),Uu),Uua) = $let(
          m3: a,
          m3:= 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),m3,aa(a,a,aa(a,fun(a,a),plus_plus(a),m3),one_one(a))) ) ) ).

% ATP.lambda_82
tff(fact_8198_ATP_Olambda__83,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa(real,$o,aTP_Lamp_he(complex,fun(real,$o),Uu),Uua)
    <=> ( ( aa(complex,complex,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_83
tff(fact_8199_ATP_Olambda__84,axiom,
    ! [Uu: real,Uua: int] :
      ( aa(int,$o,aTP_Lamp_kd(real,fun(int,$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_84
tff(fact_8200_ATP_Olambda__85,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_ke(rat,fun(int,$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_85
tff(fact_8201_ATP_Olambda__86,axiom,
    ! [Uu: real,Uua: nat] : ( aa(nat,real,aTP_Lamp_as(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),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_86
tff(fact_8202_ATP_Olambda__87,axiom,
    ! [Uu: real,Uua: nat] : ( aa(nat,real,aTP_Lamp_ql(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_8203_ATP_Olambda__88,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : ( aa(nat,real,aTP_Lamp_uw(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_8204_ATP_Olambda__89,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : ( aa(nat,A,aTP_Lamp_gi(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_8205_ATP_Olambda__90,axiom,
    ! [Uu: rat,Uua: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aTP_Lamp_abl(rat,fun(product_prod(int,int),$o),Uu),Uua)
    <=> ( ( Uu = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),Uua)),aa(product_prod(int,int),int,product_snd(int,int),Uua)) )
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),Uua))
        & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Uua),aa(product_prod(int,int),int,product_snd(int,int),Uua)) ) ) ).

% ATP.lambda_90
tff(fact_8206_ATP_Olambda__91,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_fe(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_91
tff(fact_8207_ATP_Olambda__92,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_dj(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),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_8208_ATP_Olambda__93,axiom,
    ! [A: $tType] :
      ( field_char_0(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,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_8209_ATP_Olambda__94,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_abc(nat,fun(nat,$o)),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_94
tff(fact_8210_ATP_Olambda__95,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( aa(set(set(A)),$o,aTP_Lamp_pf(set(set(A)),fun(set(set(A)),$o),Uu),Uua)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),Uua),Uu)
        & ( Uua != bot_bot(set(set(A))) ) ) ) ).

% ATP.lambda_95
tff(fact_8211_ATP_Olambda__96,axiom,
    ! [Uu: real,Uua: nat] : ( aa(nat,real,aTP_Lamp_ek(real,fun(nat,real),Uu),Uua) = divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua),semiring_char_0_fact(real,Uua)) ) ).

% ATP.lambda_96
tff(fact_8212_ATP_Olambda__97,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,set(nat),aTP_Lamp_abr(nat,fun(nat,set(nat)),Uu),Uua) = set_or3652927894154168847AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uua),Uu) ) ).

% ATP.lambda_97
tff(fact_8213_ATP_Olambda__98,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_ov(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu),Uua)),Uua) ) ).

% ATP.lambda_98
tff(fact_8214_ATP_Olambda__99,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_fb(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_99
tff(fact_8215_ATP_Olambda__100,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_fa(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_100
tff(fact_8216_ATP_Olambda__101,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa(complex,$o,aTP_Lamp_ak(nat,fun(complex,$o),Uu),Uua)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_101
tff(fact_8217_ATP_Olambda__102,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( aa(A,$o,aTP_Lamp_al(nat,fun(A,$o),Uu),Uua)
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_102
tff(fact_8218_ATP_Olambda__103,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_iz(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_103
tff(fact_8219_ATP_Olambda__104,axiom,
    ! [Uu: real,Uua: nat] : ( aa(nat,real,aTP_Lamp_wb(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)),divide_divide(real,Uu,aa(nat,real,semiring_1_of_nat(real),Uua)))),Uua) ) ).

% ATP.lambda_104
tff(fact_8220_ATP_Olambda__105,axiom,
    ! [Uu: real,Uua: real] : ( aa(real,real,aTP_Lamp_yb(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)),divide_divide(real,one_one(real),Uua)) ) ).

% ATP.lambda_105
tff(fact_8221_ATP_Olambda__106,axiom,
    ! [Uu: real,Uua: real] : ( aa(real,real,aTP_Lamp_xy(real,fun(real,real),Uu),Uua) = powr(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),divide_divide(real,Uu,Uua)),Uua) ) ).

% ATP.lambda_106
tff(fact_8222_ATP_Olambda__107,axiom,
    ! [Uu: real,Uua: nat] : ( aa(nat,real,aTP_Lamp_ap(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(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_107
tff(fact_8223_ATP_Olambda__108,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: set(A),Uua: list(A)] :
          ( aa(list(A),$o,aTP_Lamp_aej(set(A),fun(list(A),$o),Uu),Uua)
        <=> ( sorted_wrt(A,ord_less(A),Uua)
            & ( aa(list(A),set(A),set2(A),Uua) = Uu ) ) ) ) ).

% ATP.lambda_108
tff(fact_8224_ATP_Olambda__109,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aeh(set(product_prod(A,A)),fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu)) ) ) ).

% ATP.lambda_109
tff(fact_8225_ATP_Olambda__110,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aei(nat,fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(nat,nat,suc,Uu)) ) ) ).

% ATP.lambda_110
tff(fact_8226_ATP_Olambda__111,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_bd(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_111
tff(fact_8227_ATP_Olambda__112,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_db(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_112
tff(fact_8228_ATP_Olambda__113,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_vj(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_113
tff(fact_8229_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_cv(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_114
tff(fact_8230_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : ( aa(nat,A,aTP_Lamp_jm(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ) ).

% ATP.lambda_115
tff(fact_8231_ATP_Olambda__116,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : ( aa(nat,A,aTP_Lamp_jk(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ) ).

% ATP.lambda_116
tff(fact_8232_ATP_Olambda__117,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_bu(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_117
tff(fact_8233_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_av(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_118
tff(fact_8234_ATP_Olambda__119,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_ij(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_119
tff(fact_8235_ATP_Olambda__120,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_bv(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_120
tff(fact_8236_ATP_Olambda__121,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_cz(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_121
tff(fact_8237_ATP_Olambda__122,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_cy(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_122
tff(fact_8238_ATP_Olambda__123,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_vk(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_123
tff(fact_8239_ATP_Olambda__124,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_ea(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_124
tff(fact_8240_ATP_Olambda__125,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : ( aa(nat,A,aTP_Lamp_ex(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_ord_lessThan(nat,Uua)) ) ) ).

% ATP.lambda_125
tff(fact_8241_ATP_Olambda__126,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : ( aa(nat,A,aTP_Lamp_eu(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_ord_lessThan(nat,Uua)) ) ) ).

% ATP.lambda_126
tff(fact_8242_ATP_Olambda__127,axiom,
    ! [B: $tType,Uu: fun(nat,sum_sum(B,nat)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_agq(fun(nat,sum_sum(B,nat)),fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,sum_sum(B,nat),Uu,Uua) = aa(nat,sum_sum(B,nat),sum_Inr(nat,B),zero_zero(nat)) ) ) ).

% ATP.lambda_127
tff(fact_8243_ATP_Olambda__128,axiom,
    ! [A: $tType,Uu: fun(nat,sum_sum(A,nat)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_agm(fun(nat,sum_sum(A,nat)),fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,sum_sum(A,nat),Uu,Uua) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ) ) ).

% ATP.lambda_128
tff(fact_8244_ATP_Olambda__129,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(A,real),Uua: A] : ( aa(A,A,aTP_Lamp_aar(fun(A,real),fun(A,A),Uu),Uua) = real_V8093663219630862766scaleR(A,aa(A,real,Uu,Uua),Uua) ) ) ).

% ATP.lambda_129
tff(fact_8245_ATP_Olambda__130,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_yo(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,Uu,Uua)),zero_zero(real)) ) ).

% ATP.lambda_130
tff(fact_8246_ATP_Olambda__131,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(B,real),Uua: B] : ( aa(B,real,aTP_Lamp_sm(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_131
tff(fact_8247_ATP_Olambda__132,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_agg(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_132
tff(fact_8248_ATP_Olambda__133,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_iw(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_133
tff(fact_8249_ATP_Olambda__134,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ix(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_134
tff(fact_8250_ATP_Olambda__135,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : ( aa(nat,real,aTP_Lamp_ux(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_uw(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_135
tff(fact_8251_ATP_Olambda__136,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : ( aa(nat,real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_uw(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_136
tff(fact_8252_ATP_Olambda__137,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : ( aa(list(A),list(list(A)),aTP_Lamp_om(list(A),fun(list(A),list(list(A))),Uu),Uua) = aa(list(A),list(list(A)),map(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_ol(list(A),fun(A,list(A))),Uua)),Uu) ) ).

% ATP.lambda_137
tff(fact_8253_ATP_Olambda__138,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_lm(nat,fun(nat,nat)),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uu) ) ).

% ATP.lambda_138
tff(fact_8254_ATP_Olambda__139,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,real,aTP_Lamp_xk(fun(A,B),fun(A,real),Uu),Uua) = divide_divide(real,real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)),real_V7770717601297561774m_norm(A,Uua)) ) ) ).

% ATP.lambda_139
tff(fact_8255_ATP_Olambda__140,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_gs(A,fun(nat,A),Uu),Uua) = 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_140
tff(fact_8256_ATP_Olambda__141,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_gy(A,fun(nat,A),Uu),Uua) = 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_141
tff(fact_8257_ATP_Olambda__142,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_aem(set(A),fun(set(A),$o),Uu),Uua)
        <=> ( ~ real_V358717886546972837endent(A,Uua)
            & ( real_Vector_span(A,Uua) = real_Vector_span(A,Uu) ) ) ) ) ).

% ATP.lambda_142
tff(fact_8258_ATP_Olambda__143,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_gt(A,fun(nat,A),Uu),Uua) = 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_143
tff(fact_8259_ATP_Olambda__144,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_cb(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_144
tff(fact_8260_ATP_Olambda__145,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_dk(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_145
tff(fact_8261_ATP_Olambda__146,axiom,
    ! [Uu: num,Uua: num] : ( aa(num,int,aTP_Lamp_mn(num,fun(num,int),Uu),Uua) = bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),Uu),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,int,numeral_numeral(int),Uua))) ) ).

% ATP.lambda_146
tff(fact_8262_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_gw(A,fun(nat,A),Uu),Uua) = 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_147
tff(fact_8263_ATP_Olambda__148,axiom,
    ! [Uu: nat,Uua: real] : ( aa(real,real,aTP_Lamp_pg(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Uua)),Uu)) ) ).

% ATP.lambda_148
tff(fact_8264_ATP_Olambda__149,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_wg(A,fun(nat,A),Uu),Uua) = 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_149
tff(fact_8265_ATP_Olambda__150,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_wf(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_150
tff(fact_8266_ATP_Olambda__151,axiom,
    ! [Uu: real,Uua: nat] : ( aa(nat,real,aTP_Lamp_dt(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sin_coeff(Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ) ).

% ATP.lambda_151
tff(fact_8267_ATP_Olambda__152,axiom,
    ! [Uu: real,Uua: nat] : ( aa(nat,real,aTP_Lamp_ds(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_152
tff(fact_8268_ATP_Olambda__153,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_aer(set(A),fun(set(A),$o),Uu),Uua)
    <=> ( aa(set(A),$o,finite_finite(A),Uua)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu) ) ) ).

% ATP.lambda_153
tff(fact_8269_ATP_Olambda__154,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: list(product_prod(A,B))] :
      ( aa(list(product_prod(A,B)),$o,aTP_Lamp_abu(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),Uu),Uua)
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Uua)),collect(product_prod(A,B),product_case_prod(A,B,$o,Uu))) ) ).

% ATP.lambda_154
tff(fact_8270_ATP_Olambda__155,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: nat,Uua: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_lb(nat,fun(nat,A)),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ) ).

% ATP.lambda_155
tff(fact_8271_ATP_Olambda__156,axiom,
    ! [Uu: num,Uua: num] : ( aa(num,int,aa(num,fun(num,int),aTP_Lamp_aaj(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_156
tff(fact_8272_ATP_Olambda__157,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_ni(list(A),fun(list(A),$o)),Uu),Uua)
    <=> ( aa(list(A),nat,size_size(list(A)),Uu) = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_157
tff(fact_8273_ATP_Olambda__158,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_ln(code_integer,fun(code_integer,num)),Uu),Uua) = $let(
          l2: num,
          l2:= aa(code_integer,num,code_num_of_integer,Uu),
          $let(
            l3: num,
            l3:= aa(num,num,aa(num,fun(num,num),plus_plus(num),l2),l2),
            $ite(Uua = zero_zero(code_integer),l3,aa(num,num,aa(num,fun(num,num),plus_plus(num),l3),one2)) ) ) ) ).

% ATP.lambda_158
tff(fact_8274_ATP_Olambda__159,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_kw(code_integer,fun(code_integer,nat)),Uu),Uua) = $let(
          l2: nat,
          l2:= aa(code_integer,nat,code_nat_of_integer,Uu),
          $let(
            l3: nat,
            l3:= aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l2),l2),
            $ite(Uua = zero_zero(code_integer),l3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l3),one_one(nat))) ) ) ) ).

% ATP.lambda_159
tff(fact_8275_ATP_Olambda__160,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_aes(set(A),fun(list(A),$o),Uu),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu) ) ).

% ATP.lambda_160
tff(fact_8276_ATP_Olambda__161,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : ( aa(nat,A,aTP_Lamp_de(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ) ).

% ATP.lambda_161
tff(fact_8277_ATP_Olambda__162,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : ( aa(A,list(product_prod(A,B)),aTP_Lamp_nj(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),product_Pair(A,B,Uua)),Uu) ) ).

% ATP.lambda_162
tff(fact_8278_ATP_Olambda__163,axiom,
    ! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_aaw(set(nat),fun(product_prod(A,nat),$o),Uu),Uua)
    <=> member(nat,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua),Uu) ) ).

% ATP.lambda_163
tff(fact_8279_ATP_Olambda__164,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_oq(set(nat),fun(nat,$o),Uu),Uua)
    <=> member(nat,aa(nat,nat,suc,Uua),Uu) ) ).

% ATP.lambda_164
tff(fact_8280_ATP_Olambda__165,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_hf(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(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_165
tff(fact_8281_ATP_Olambda__166,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_aq(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uua),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% ATP.lambda_166
tff(fact_8282_ATP_Olambda__167,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_gh(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_167
tff(fact_8283_ATP_Olambda__168,axiom,
    ! [Uu: real,Uua: nat] : ( aa(nat,real,aTP_Lamp_wd(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_168
tff(fact_8284_ATP_Olambda__169,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_vx(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or1337092689740270186AtMost(nat,zero_zero(nat),Uua)) ) ) ).

% ATP.lambda_169
tff(fact_8285_ATP_Olambda__170,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : ( aa(A,product_prod(A,B),aTP_Lamp_abk(fun(A,B),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),product_Pair(A,B,Uua),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_170
tff(fact_8286_ATP_Olambda__171,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : ( aa(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_adn(nat,fun(vEBT_VEBT,vEBT_VEBT),Uu),Uua) = vEBT_VEBT_elim_dead(Uua,extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Uu,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% ATP.lambda_171
tff(fact_8287_ATP_Olambda__172,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_jq(A,fun(nat,A),Uu),Uua) = aa(A,A,bit_se4730199178511100633sh_bit(A,Uua),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ) ).

% ATP.lambda_172
tff(fact_8288_ATP_Olambda__173,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A] : ( aa(A,product_prod(A,B),aTP_Lamp_abx(fun(A,option(B)),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),product_Pair(A,B,Uua),the2(B,aa(A,option(B),Uu,Uua))) ) ).

% ATP.lambda_173
tff(fact_8289_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,real,aTP_Lamp_vy(fun(nat,A),fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua))) ) ) ).

% ATP.lambda_174
tff(fact_8290_ATP_Olambda__175,axiom,
    ! [Uu: int,Uua: int] : ( aa(int,int,aa(int,fun(int,int),aTP_Lamp_adb(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_175
tff(fact_8291_ATP_Olambda__176,axiom,
    ! [Uu: real,Uua: nat] : ( aa(nat,real,aTP_Lamp_vz(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_176
tff(fact_8292_ATP_Olambda__177,axiom,
    ! [Uu: real,Uua: nat] : ( aa(nat,real,aTP_Lamp_vq(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_177
tff(fact_8293_ATP_Olambda__178,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_vg(A,fun(nat,A),Uu),Uua) = divide_divide(A,Uu,aa(nat,A,semiring_1_of_nat(A),Uua)) ) ) ).

% ATP.lambda_178
tff(fact_8294_ATP_Olambda__179,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_nr(nat,fun(list(A),$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uu),aa(list(A),nat,size_size(list(A)),Uua)) ) ).

% ATP.lambda_179
tff(fact_8295_ATP_Olambda__180,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_be(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_180
tff(fact_8296_ATP_Olambda__181,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_jt(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_181
tff(fact_8297_ATP_Olambda__182,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_bb(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_182
tff(fact_8298_ATP_Olambda__183,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,list(nat),aa(nat,fun(nat,list(nat)),aTP_Lamp_adi(nat,fun(nat,list(nat))),Uu),Uua) = cons(nat,Uu,aa(nat,list(nat),nat_list_decode,Uua)) ) ).

% ATP.lambda_183
tff(fact_8299_ATP_Olambda__184,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_abv(A,fun(list(A),$o)),Uu),Uua)
    <=> member(A,Uu,aa(list(A),set(A),set2(A),Uua)) ) ).

% ATP.lambda_184
tff(fact_8300_ATP_Olambda__185,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_nx(list(A),fun(A,$o),Uu),Uua)
    <=> member(A,Uua,aa(list(A),set(A),set2(A),Uu)) ) ).

% ATP.lambda_185
tff(fact_8301_ATP_Olambda__186,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_acu(nat,fun(nat,$o)),Uu),Uua)
    <=> ( Uua = aa(nat,nat,suc,Uu) ) ) ).

% ATP.lambda_186
tff(fact_8302_ATP_Olambda__187,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ab(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uu) ) ).

% ATP.lambda_187
tff(fact_8303_ATP_Olambda__188,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_zu(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_188
tff(fact_8304_ATP_Olambda__189,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_adg(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_189
tff(fact_8305_ATP_Olambda__190,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_acl(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_190
tff(fact_8306_ATP_Olambda__191,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_afz(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_191
tff(fact_8307_ATP_Olambda__192,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ade(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_192
tff(fact_8308_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_ot(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ) ).

% ATP.lambda_193
tff(fact_8309_ATP_Olambda__194,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_lu(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ) ).

% ATP.lambda_194
tff(fact_8310_ATP_Olambda__195,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_ys(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ) ).

% ATP.lambda_195
tff(fact_8311_ATP_Olambda__196,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aa(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu) ) ).

% ATP.lambda_196
tff(fact_8312_ATP_Olambda__197,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_zv(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_197
tff(fact_8313_ATP_Olambda__198,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_yk(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_198
tff(fact_8314_ATP_Olambda__199,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_adh(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_199
tff(fact_8315_ATP_Olambda__200,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_acm(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_200
tff(fact_8316_ATP_Olambda__201,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_act(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_201
tff(fact_8317_ATP_Olambda__202,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_adf(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_202
tff(fact_8318_ATP_Olambda__203,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_dv(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_203
tff(fact_8319_ATP_Olambda__204,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_ve(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu) ) ).

% ATP.lambda_204
tff(fact_8320_ATP_Olambda__205,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_md(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ) ).

% ATP.lambda_205
tff(fact_8321_ATP_Olambda__206,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_lr(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ) ).

% ATP.lambda_206
tff(fact_8322_ATP_Olambda__207,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_adm(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ) ).

% ATP.lambda_207
tff(fact_8323_ATP_Olambda__208,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_lt(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ) ).

% ATP.lambda_208
tff(fact_8324_ATP_Olambda__209,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_lv(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ) ).

% ATP.lambda_209
tff(fact_8325_ATP_Olambda__210,axiom,
    ! [Uu: nat,Uua: real] : ( aa(real,real,aTP_Lamp_qo(nat,fun(real,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu) ) ).

% ATP.lambda_210
tff(fact_8326_ATP_Olambda__211,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_zr(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),Uu) ) ) ).

% ATP.lambda_211
tff(fact_8327_ATP_Olambda__212,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_oz(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),Uu) ) ) ).

% ATP.lambda_212
tff(fact_8328_ATP_Olambda__213,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_nk(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ) ).

% ATP.lambda_213
tff(fact_8329_ATP_Olambda__214,axiom,
    ! [Uu: int,Uua: int] : ( aa(int,int,aTP_Lamp_mm(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ) ).

% ATP.lambda_214
tff(fact_8330_ATP_Olambda__215,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_ado(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ) ).

% ATP.lambda_215
tff(fact_8331_ATP_Olambda__216,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_yw(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ) ).

% ATP.lambda_216
tff(fact_8332_ATP_Olambda__217,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_ls(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ) ).

% ATP.lambda_217
tff(fact_8333_ATP_Olambda__218,axiom,
    ! [Uu: real,Uua: real] : ( aa(real,real,aTP_Lamp_px(real,fun(real,real),Uu),Uua) = powr(real,Uua,Uu) ) ).

% ATP.lambda_218
tff(fact_8334_ATP_Olambda__219,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_iy(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uua),Uu) ) ).

% ATP.lambda_219
tff(fact_8335_ATP_Olambda__220,axiom,
    ! [Uu: int,Uua: int] :
      ( aa(int,$o,aTP_Lamp_iv(int,fun(int,$o),Uu),Uua)
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Uua),Uu) ) ).

% ATP.lambda_220
tff(fact_8336_ATP_Olambda__221,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_af(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Uua),Uu) ) ) ).

% ATP.lambda_221
tff(fact_8337_ATP_Olambda__222,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_la(nat,fun(nat,product_prod(nat,nat))),Uu),Uua) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Uua),Uu) ) ).

% ATP.lambda_222
tff(fact_8338_ATP_Olambda__223,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : ( aa(nat,product_prod(nat,A),aTP_Lamp_nm(A,fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),product_Pair(nat,A,Uua),Uu) ) ).

% ATP.lambda_223
tff(fact_8339_ATP_Olambda__224,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_et(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ) ).

% ATP.lambda_224
tff(fact_8340_ATP_Olambda__225,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : ( aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_ol(list(A),fun(A,list(A))),Uu),Uua) = cons(A,Uua,Uu) ) ).

% ATP.lambda_225
tff(fact_8341_ATP_Olambda__226,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: int,Uua: A] : ( aa(A,A,aTP_Lamp_sd(int,fun(A,A),Uu),Uua) = power_int(A,Uua,Uu) ) ) ).

% ATP.lambda_226
tff(fact_8342_ATP_Olambda__227,axiom,
    ! [Uu: real,Uua: nat] : ( aa(nat,real,aTP_Lamp_vo(real,fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),Uu) ) ).

% ATP.lambda_227
tff(fact_8343_ATP_Olambda__228,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aap(set(A),fun(A,$o),Uu),Uua)
        <=> member(A,Uua,Uu) ) ) ).

% ATP.lambda_228
tff(fact_8344_ATP_Olambda__229,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_229
tff(fact_8345_ATP_Olambda__230,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : ( aa(nat,set(product_prod(A,A)),aTP_Lamp_aeg(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),Uu),Uua) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Uua),Uu) ) ).

% ATP.lambda_230
tff(fact_8346_ATP_Olambda__231,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : ( aa(list(A),A,aTP_Lamp_ns(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ) ).

% ATP.lambda_231
tff(fact_8347_ATP_Olambda__232,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_hg(A,fun(A,$o),Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_232
tff(fact_8348_ATP_Olambda__233,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( aa(A,$o,aTP_Lamp_xb(fun(A,real),fun(A,$o),Uu),Uua)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_233
tff(fact_8349_ATP_Olambda__234,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_xd(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_234
tff(fact_8350_ATP_Olambda__235,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_xx(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_235
tff(fact_8351_ATP_Olambda__236,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,set(nat),aTP_Lamp_afa(nat,fun(nat,set(nat)),Uu),Uua) = order_underS(nat,bNF_Ca8665028551170535155natLeq,Uu) ) ).

% ATP.lambda_236
tff(fact_8352_ATP_Olambda__237,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : ( aa(nat,real,aTP_Lamp_ej(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_237
tff(fact_8353_ATP_Olambda__238,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : ( aa(nat,real,aTP_Lamp_ei(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_238
tff(fact_8354_ATP_Olambda__239,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : ( aa(A,A,aTP_Lamp_ya(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,Uu,divide_divide(A,one_one(A),Uua)) ) ) ).

% ATP.lambda_239
tff(fact_8355_ATP_Olambda__240,axiom,
    ! [A: $tType,Uu: fun(nat,$o),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_aau(fun(nat,$o),fun(product_prod(A,nat),$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua))) ) ).

% ATP.lambda_240
tff(fact_8356_ATP_Olambda__241,axiom,
    ! [Uu: fun(real,$o),Uua: real] :
      ( aa(real,$o,aTP_Lamp_yg(fun(real,$o),fun(real,$o),Uu),Uua)
    <=> aa(real,$o,Uu,aa(real,real,inverse_inverse(real),Uua)) ) ).

% ATP.lambda_241
tff(fact_8357_ATP_Olambda__242,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real] : ( aa(real,A,aTP_Lamp_yh(fun(real,A),fun(real,A),Uu),Uua) = aa(real,A,Uu,aa(real,real,inverse_inverse(real),Uua)) ) ).

% ATP.lambda_242
tff(fact_8358_ATP_Olambda__243,axiom,
    ! [A: $tType,Uu: fun(int,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_xs(fun(int,A),fun(nat,A),Uu),Uua) = aa(int,A,Uu,aa(nat,int,semiring_1_of_nat(int),Uua)) ) ).

% ATP.lambda_243
tff(fact_8359_ATP_Olambda__244,axiom,
    ! [Uu: fun(real,real),Uua: real] : ( aa(real,real,aTP_Lamp_qx(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,Uu,aa(real,real,uminus_uminus(real),Uua)) ) ).

% ATP.lambda_244
tff(fact_8360_ATP_Olambda__245,axiom,
    ! [Uu: fun(real,$o),Uua: real] :
      ( aa(real,$o,aTP_Lamp_yd(fun(real,$o),fun(real,$o),Uu),Uua)
    <=> aa(real,$o,Uu,aa(real,real,uminus_uminus(real),Uua)) ) ).

% ATP.lambda_245
tff(fact_8361_ATP_Olambda__246,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real] : ( aa(real,A,aTP_Lamp_yc(fun(real,A),fun(real,A),Uu),Uua) = aa(real,A,Uu,aa(real,real,uminus_uminus(real),Uua)) ) ).

% ATP.lambda_246
tff(fact_8362_ATP_Olambda__247,axiom,
    ! [A: $tType,Uu: fun(nat,$o),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_aav(fun(nat,$o),fun(product_prod(A,nat),$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)) ) ).

% ATP.lambda_247
tff(fact_8363_ATP_Olambda__248,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_wn(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_248
tff(fact_8364_ATP_Olambda__249,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(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_249
tff(fact_8365_ATP_Olambda__250,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_vb(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_250
tff(fact_8366_ATP_Olambda__251,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_cq(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_251
tff(fact_8367_ATP_Olambda__252,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_aw(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_252
tff(fact_8368_ATP_Olambda__253,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_co(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_253
tff(fact_8369_ATP_Olambda__254,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_on(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_254
tff(fact_8370_ATP_Olambda__255,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_aan(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_aam(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ) ).

% ATP.lambda_255
tff(fact_8371_ATP_Olambda__256,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_lj(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = product_case_prod(nat,nat,product_prod(nat,nat),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_li(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ) ).

% ATP.lambda_256
tff(fact_8372_ATP_Olambda__257,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_lh(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = product_case_prod(nat,nat,product_prod(nat,nat),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_lg(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ) ).

% ATP.lambda_257
tff(fact_8373_ATP_Olambda__258,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_lf(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_le(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ) ).

% ATP.lambda_258
tff(fact_8374_ATP_Olambda__259,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_ld(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_lc(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ) ).

% ATP.lambda_259
tff(fact_8375_ATP_Olambda__260,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = product_case_prod(nat,nat,product_prod(nat,nat),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_ky(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ) ).

% ATP.lambda_260
tff(fact_8376_ATP_Olambda__261,axiom,
    ! [Uu: fun(nat,real),Uua: real] : ( aa(real,real,aTP_Lamp_qc(fun(nat,real),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_qb(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua)) ) ).

% ATP.lambda_261
tff(fact_8377_ATP_Olambda__262,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C)] : ( aa(fun(B,C),A,aTP_Lamp_pq(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_pp(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua),top_top(set(B)))) ) ) ).

% ATP.lambda_262
tff(fact_8378_ATP_Olambda__263,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : ( aa(B,A,aTP_Lamp_pr(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_pn(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),top_top(set(C)))) ) ) ).

% ATP.lambda_263
tff(fact_8379_ATP_Olambda__264,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C)] : ( aa(fun(B,C),A,aTP_Lamp_ps(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_pp(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua),top_top(set(B)))) ) ) ).

% ATP.lambda_264
tff(fact_8380_ATP_Olambda__265,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : ( aa(B,A,aTP_Lamp_po(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_pn(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),top_top(set(C)))) ) ) ).

% ATP.lambda_265
tff(fact_8381_ATP_Olambda__266,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,complex,aTP_Lamp_ep(nat,fun(nat,complex),Uu),Uua) = cis(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_266
tff(fact_8382_ATP_Olambda__267,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_ru(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_267
tff(fact_8383_ATP_Olambda__268,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aas(fun(A,real),fun(A,$o),Uu),Uua)
        <=> ( aa(A,real,Uu,Uua) != zero_zero(real) ) ) ) ).

% ATP.lambda_268
tff(fact_8384_ATP_Olambda__269,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: list(product_prod(C,B)),Uua: product_prod(A,C)] : ( aa(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_abj(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Uu),Uua) = concat(product_prod(A,B),aa(list(product_prod(C,B)),list(list(product_prod(A,B))),map(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_abi(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua)),Uu)) ) ).

% ATP.lambda_269
tff(fact_8385_ATP_Olambda__270,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : ( aa(nat,real,aTP_Lamp_gu(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(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_270
tff(fact_8386_ATP_Olambda__271,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_am(nat,fun(nat,$o),Uu),Uua)
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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_271
tff(fact_8387_ATP_Olambda__272,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_gv(A,fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,sin_coeff(Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uu)),Uua))) ) ) ).

% ATP.lambda_272
tff(fact_8388_ATP_Olambda__273,axiom,
    ! [Uu: code_natural,Uua: code_natural] : ( aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_agc(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu),Uua) = product_Pair(code_natural,product_prod(code_natural,code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),Uua),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),Uu),aa(num,code_natural,numeral_numeral(code_natural),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,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,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,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))))))))))))))))))))))))))))))) ) ).

% ATP.lambda_273
tff(fact_8389_ATP_Olambda__274,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : ( aa(nat,set(A),aTP_Lamp_pb(fun(nat,set(A)),fun(nat,set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),Uu,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ) ).

% ATP.lambda_274
tff(fact_8390_ATP_Olambda__275,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,fun(A,A),aTP_Lamp_ao(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_275
tff(fact_8391_ATP_Olambda__276,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,fun(A,A),aTP_Lamp_an(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_276
tff(fact_8392_ATP_Olambda__277,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_nw(list(A),fun(A,$o),Uu),Uua)
    <=> ~ member(A,Uua,aa(list(A),set(A),set2(A),Uu)) ) ).

% ATP.lambda_277
tff(fact_8393_ATP_Olambda__278,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: set(B)] : ( aa(set(B),A,aTP_Lamp_pk(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,Uu,Uua)) ) ) ).

% ATP.lambda_278
tff(fact_8394_ATP_Olambda__279,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: set(B)] : ( aa(set(B),A,aTP_Lamp_pj(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,Uu,Uua)) ) ) ).

% ATP.lambda_279
tff(fact_8395_ATP_Olambda__280,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,set(nat),aTP_Lamp_abq(nat,fun(nat,set(nat)),Uu),Uua) = set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uua)) ) ).

% ATP.lambda_280
tff(fact_8396_ATP_Olambda__281,axiom,
    ! [Uu: real,Uua: nat] : ( aa(nat,real,aTP_Lamp_vw(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_281
tff(fact_8397_ATP_Olambda__282,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_pv(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_282
tff(fact_8398_ATP_Olambda__283,axiom,
    ! [Uu: code_natural,Uua: code_natural] : ( aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_age(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu),Uua) = product_Pair(code_natural,product_prod(code_natural,code_natural),modulo_modulo(code_natural,Uua,Uu)) ) ).

% ATP.lambda_283
tff(fact_8399_ATP_Olambda__284,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,extended_enat,aTP_Lamp_adt(nat,fun(nat,extended_enat),Uu),Uua) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uua)) ) ).

% ATP.lambda_284
tff(fact_8400_ATP_Olambda__285,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,extended_enat,aTP_Lamp_adp(nat,fun(nat,extended_enat),Uu),Uua) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uua)) ) ).

% ATP.lambda_285
tff(fact_8401_ATP_Olambda__286,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,extended_enat,aTP_Lamp_adr(nat,fun(nat,extended_enat),Uu),Uua) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)) ) ).

% ATP.lambda_286
tff(fact_8402_ATP_Olambda__287,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : ( aa(A,filter(A),aTP_Lamp_aet(A,fun(A,filter(A)),Uu),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uu,Uua)) ) ) ).

% ATP.lambda_287
tff(fact_8403_ATP_Olambda__288,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : ( aa(A,filter(A),aTP_Lamp_aeu(A,fun(A,filter(A)),Uu),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uua,Uu)) ) ) ).

% ATP.lambda_288
tff(fact_8404_ATP_Olambda__289,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_mq(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_289
tff(fact_8405_ATP_Olambda__290,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_mp(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_290
tff(fact_8406_ATP_Olambda__291,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_kg(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_291
tff(fact_8407_ATP_Olambda__292,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,nat,aTP_Lamp_kf(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_292
tff(fact_8408_ATP_Olambda__293,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ad(set(A),fun(A,$o),Uu),Uua)
    <=> ~ member(A,Uua,Uu) ) ).

% ATP.lambda_293
tff(fact_8409_ATP_Olambda__294,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_nv(A,fun(A,$o),Uu),Uua)
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_294
tff(fact_8410_ATP_Olambda__295,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,real,aTP_Lamp_fj(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ) ).

% ATP.lambda_295
tff(fact_8411_ATP_Olambda__296,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,real,aTP_Lamp_sz(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_296
tff(fact_8412_ATP_Olambda__297,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,$o),Uua: A] : ( aa(A,B,aTP_Lamp_lq(fun(A,$o),fun(A,B),Uu),Uua) = aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu,Uua)) ) ) ).

% ATP.lambda_297
tff(fact_8413_ATP_Olambda__298,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : ( aa(nat,real,aTP_Lamp_vm(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_298
tff(fact_8414_ATP_Olambda__299,axiom,
    ! [Uu: fun(nat,rat),Uua: nat] : ( aa(nat,rat,aTP_Lamp_aaa(fun(nat,rat),fun(nat,rat),Uu),Uua) = aa(rat,rat,inverse_inverse(rat),aa(nat,rat,Uu,Uua)) ) ).

% ATP.lambda_299
tff(fact_8415_ATP_Olambda__300,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B] : ( aa(B,A,aTP_Lamp_rs(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)) ) ) ).

% ATP.lambda_300
tff(fact_8416_ATP_Olambda__301,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,B,aTP_Lamp_acb(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_301
tff(fact_8417_ATP_Olambda__302,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,B,aTP_Lamp_za(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_302
tff(fact_8418_ATP_Olambda__303,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : ( aa(A,A,aTP_Lamp_qm(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu,Uua)) ) ) ).

% ATP.lambda_303
tff(fact_8419_ATP_Olambda__304,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,B,aTP_Lamp_um(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_304
tff(fact_8420_ATP_Olambda__305,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : ( aa(A,real,aTP_Lamp_xr(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_305
tff(fact_8421_ATP_Olambda__306,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,B,aTP_Lamp_tb(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_306
tff(fact_8422_ATP_Olambda__307,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : ( aa(B,A,aTP_Lamp_at(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ) ).

% ATP.lambda_307
tff(fact_8423_ATP_Olambda__308,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : ( aa(B,A,aTP_Lamp_cf(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ) ).

% ATP.lambda_308
tff(fact_8424_ATP_Olambda__309,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : ( aa(A,int,aTP_Lamp_bi(fun(A,nat),fun(A,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(A,nat,Uu,Uua)) ) ).

% ATP.lambda_309
tff(fact_8425_ATP_Olambda__310,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : ( aa(A,real,aTP_Lamp_rz(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_310
tff(fact_8426_ATP_Olambda__311,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(real,A),Uua: real] : ( aa(real,A,aTP_Lamp_adx(fun(real,A),fun(real,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(real,A,Uu,Uua)) ) ) ).

% ATP.lambda_311
tff(fact_8427_ATP_Olambda__312,axiom,
    ! [Uu: fun(nat,rat),Uua: nat] : ( aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_mu(fun(nat,rat),fun(nat,rat)),Uu),Uua) = aa(rat,rat,uminus_uminus(rat),aa(nat,rat,Uu,Uua)) ) ).

% ATP.lambda_312
tff(fact_8428_ATP_Olambda__313,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(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ) ).

% ATP.lambda_313
tff(fact_8429_ATP_Olambda__314,axiom,
    ! [A: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_acf(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ) ).

% ATP.lambda_314
tff(fact_8430_ATP_Olambda__315,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_hd(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ) ).

% ATP.lambda_315
tff(fact_8431_ATP_Olambda__316,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,A,aTP_Lamp_er(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ) ).

% ATP.lambda_316
tff(fact_8432_ATP_Olambda__317,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : ( aa(B,set(A),aTP_Lamp_ph(fun(B,set(A)),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),aa(B,set(A),Uu,Uua)) ) ).

% ATP.lambda_317
tff(fact_8433_ATP_Olambda__318,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [Uu: fun(B,A),Uua: B] : ( aa(B,A,aTP_Lamp_pm(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ) ).

% ATP.lambda_318
tff(fact_8434_ATP_Olambda__319,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: B] : ( aa(B,A,aTP_Lamp_cl(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ) ).

% ATP.lambda_319
tff(fact_8435_ATP_Olambda__320,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,B,aTP_Lamp_rl(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_320
tff(fact_8436_ATP_Olambda__321,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,B,aTP_Lamp_yx(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_321
tff(fact_8437_ATP_Olambda__322,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : ( aa(A,A,aTP_Lamp_qt(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,Uu,Uua)) ) ) ).

% ATP.lambda_322
tff(fact_8438_ATP_Olambda__323,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,B,aTP_Lamp_aeq(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_323
tff(fact_8439_ATP_Olambda__324,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,B,aTP_Lamp_ul(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_324
tff(fact_8440_ATP_Olambda__325,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,B,aTP_Lamp_tc(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_325
tff(fact_8441_ATP_Olambda__326,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : ( aa(A,real,aTP_Lamp_yl(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,uminus_uminus(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_326
tff(fact_8442_ATP_Olambda__327,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,B,aTP_Lamp_aga(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_327
tff(fact_8443_ATP_Olambda__328,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,B,aTP_Lamp_td(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_328
tff(fact_8444_ATP_Olambda__329,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,fun(A,A),aTP_Lamp_ba(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_329
tff(fact_8445_ATP_Olambda__330,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : ( aa(B,A,aTP_Lamp_au(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ) ).

% ATP.lambda_330
tff(fact_8446_ATP_Olambda__331,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : ( aa(B,A,aTP_Lamp_cg(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ) ).

% ATP.lambda_331
tff(fact_8447_ATP_Olambda__332,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : ( aa(nat,fun(A,A),aTP_Lamp_cw(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_332
tff(fact_8448_ATP_Olambda__333,axiom,
    ! [Uu: fun(real,real),Uua: real] : ( aa(real,real,aTP_Lamp_zi(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,artanh(real),aa(real,real,Uu,Uua)) ) ).

% ATP.lambda_333
tff(fact_8449_ATP_Olambda__334,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : ( aa(A,real,aTP_Lamp_xz(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_334
tff(fact_8450_ATP_Olambda__335,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : ( aa(A,real,aTP_Lamp_uf(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_335
tff(fact_8451_ATP_Olambda__336,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : ( aa(A,real,aTP_Lamp_sj(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_336
tff(fact_8452_ATP_Olambda__337,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : ( aa(A,real,aTP_Lamp_rb(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_337
tff(fact_8453_ATP_Olambda__338,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : ( aa(A,real,aTP_Lamp_zh(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_338
tff(fact_8454_ATP_Olambda__339,axiom,
    ! [Uu: fun(real,real),Uua: real] : ( aa(real,real,aTP_Lamp_ze(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,arcosh(real),aa(real,real,Uu,Uua)) ) ).

% ATP.lambda_339
tff(fact_8455_ATP_Olambda__340,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : ( aa(A,real,aTP_Lamp_xc(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_340
tff(fact_8456_ATP_Olambda__341,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : ( aa(A,real,aTP_Lamp_tv(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_341
tff(fact_8457_ATP_Olambda__342,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : ( aa(A,real,aTP_Lamp_rd(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_342
tff(fact_8458_ATP_Olambda__343,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : ( aa(A,real,aTP_Lamp_zg(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_343
tff(fact_8459_ATP_Olambda__344,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,B,aTP_Lamp_yt(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_344
tff(fact_8460_ATP_Olambda__345,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,B,aTP_Lamp_un(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_345
tff(fact_8461_ATP_Olambda__346,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,B,aTP_Lamp_sy(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_346
tff(fact_8462_ATP_Olambda__347,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu: fun(B,A),Uua: B] : ( aa(B,A,aTP_Lamp_ch(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ) ).

% ATP.lambda_347
tff(fact_8463_ATP_Olambda__348,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,B,aTP_Lamp_zd(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_348
tff(fact_8464_ATP_Olambda__349,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_us(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_349
tff(fact_8465_ATP_Olambda__350,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : ( aa(A,A,aTP_Lamp_pz(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tanh(A),aa(A,A,Uu,Uua)) ) ) ).

% ATP.lambda_350
tff(fact_8466_ATP_Olambda__351,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : ( aa(A,B,aTP_Lamp_sx(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ) ).

% ATP.lambda_351
tff(fact_8467_ATP_Olambda__352,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : ( aa(A,real,aTP_Lamp_rf(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,tan(real),aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_352
tff(fact_8468_ATP_Olambda__353,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : ( aa(A,A,aTP_Lamp_te(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tan(A),aa(A,A,Uu,Uua)) ) ) ).

% ATP.lambda_353
tff(fact_8469_ATP_Olambda__354,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : ( aa(A,A,aTP_Lamp_sw(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cot(A),aa(A,A,Uu,Uua)) ) ) ).

% ATP.lambda_354
tff(fact_8470_ATP_Olambda__355,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : ( aa(nat,real,aTP_Lamp_wj(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,cos(real),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_355
tff(fact_8471_ATP_Olambda__356,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : ( aa(A,real,aTP_Lamp_rv(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,cos(real),aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_356
tff(fact_8472_ATP_Olambda__357,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : ( aa(A,A,aTP_Lamp_pu(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cos(A),aa(A,A,Uu,Uua)) ) ) ).

% ATP.lambda_357
tff(fact_8473_ATP_Olambda__358,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_uu(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ) ).

% ATP.lambda_358
tff(fact_8474_ATP_Olambda__359,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,list(A)),Uua: B] : ( aa(B,set(A),aTP_Lamp_zm(fun(B,list(A)),fun(B,set(A)),Uu),Uua) = aa(list(A),set(A),set2(A),aa(B,list(A),Uu,Uua)) ) ).

% ATP.lambda_359
tff(fact_8475_ATP_Olambda__360,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : ( aa(A,real,aTP_Lamp_sh(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_360
tff(fact_8476_ATP_Olambda__361,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : ( aa(A,nat,aTP_Lamp_dn(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ) ).

% ATP.lambda_361
tff(fact_8477_ATP_Olambda__362,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ae(fun(A,$o),fun(A,$o),Uu),Uua)
    <=> ~ aa(A,$o,Uu,Uua) ) ).

% ATP.lambda_362
tff(fact_8478_ATP_Olambda__363,axiom,
    ! [A: $tType,B: $tType,Uu: fun(nat,sum_sum(B,nat)),Uua: sum_sum(A,nat)] : ( aa(sum_sum(A,nat),nat,aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat),aTP_Lamp_agr(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat)),Uu),Uua) = ord_Least(nat,aTP_Lamp_agq(fun(nat,sum_sum(B,nat)),fun(nat,$o),Uu)) ) ).

% ATP.lambda_363
tff(fact_8479_ATP_Olambda__364,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,set(nat),aTP_Lamp_aey(nat,fun(nat,set(nat)),Uu),Uua) = collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aa(nat,fun(nat,$o)),Uu)) ) ).

% ATP.lambda_364
tff(fact_8480_ATP_Olambda__365,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu: fun(real,A),Uua: real] : ( aa(real,real,aTP_Lamp_uv(fun(real,A),fun(real,real),Uu),Uua) = aa(int,real,ring_1_of_int(real),aa(A,int,archim6421214686448440834_floor(A),aa(real,A,Uu,Uua))) ) ) ).

% ATP.lambda_365
tff(fact_8481_ATP_Olambda__366,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_tu(fun(A,B),fun(A,C),Uu),Uua) = aa(int,C,ring_1_of_int(C),aa(B,int,archim6421214686448440834_floor(B),aa(A,B,Uu,Uua))) ) ) ).

% ATP.lambda_366
tff(fact_8482_ATP_Olambda__367,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_tt(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_367
tff(fact_8483_ATP_Olambda__368,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ne(list(A),fun(A,$o),Uu),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu),I3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Uu)) ) ) ).

% ATP.lambda_368
tff(fact_8484_ATP_Olambda__369,axiom,
    ! [B: $tType,Uu: set(set(B)),Uua: set(B)] :
      ( aa(set(B),$o,aTP_Lamp_pl(set(set(B)),fun(set(B),$o),Uu),Uua)
    <=> ? [F5: fun(set(B),B)] :
          ( ( Uua = image(set(B),B,F5,Uu) )
          & ! [X4: set(B)] :
              ( member(set(B),X4,Uu)
             => member(B,aa(set(B),B,F5,X4),X4) ) ) ) ).

% ATP.lambda_369
tff(fact_8485_ATP_Olambda__370,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_pi(set(set(A)),fun(set(A),$o),Uu),Uua)
        <=> ? [F5: fun(set(A),A)] :
              ( ( Uua = image(set(A),A,F5,Uu) )
              & ! [X4: set(A)] :
                  ( member(set(A),X4,Uu)
                 => member(A,aa(set(A),A,F5,X4),X4) ) ) ) ) ).

% ATP.lambda_370
tff(fact_8486_ATP_Olambda__371,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_aco(set(A),fun(set(A),$o),Uu),Uua)
    <=> ? [B9: set(A)] :
          ( ( Uua = aa(set(A),set(A),uminus_uminus(set(A)),B9) )
          & member(set(A),Uu,pow2(A,B9)) ) ) ).

% ATP.lambda_371
tff(fact_8487_ATP_Olambda__372,axiom,
    ! [Uu: set(real),Uua: real] :
      ( aa(real,$o,aTP_Lamp_aaq(set(real),fun(real,$o),Uu),Uua)
    <=> ! [X4: real] :
          ( member(real,X4,Uu)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),Uua) ) ) ).

% ATP.lambda_372
tff(fact_8488_ATP_Olambda__373,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_afh(set(nat),fun(nat,$o),Uu),Uua)
    <=> ! [X4: nat] :
          ( member(nat,X4,Uu)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uua),X4) ) ) ).

% ATP.lambda_373
tff(fact_8489_ATP_Olambda__374,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_afg(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X4: A] :
              ( member(A,X4,Uu)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Uua),X4) ) ) ) ).

% ATP.lambda_374
tff(fact_8490_ATP_Olambda__375,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aff(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X4: A] :
              ( member(A,X4,Uu)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X4),Uua) ) ) ) ).

% ATP.lambda_375
tff(fact_8491_ATP_Olambda__376,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,set(nat),aTP_Lamp_abp(nat,fun(nat,set(nat)),Uu),Uua) = set_ord_atMost(nat,Uu) ) ).

% ATP.lambda_376
tff(fact_8492_ATP_Olambda__377,axiom,
    ! [Uu: nat,Uua: nat] : ( aa(nat,rat,aTP_Lamp_aaf(nat,fun(nat,rat),Uu),Uua) = aa(nat,rat,semiring_1_of_nat(rat),Uu) ) ).

% ATP.lambda_377
tff(fact_8493_ATP_Olambda__378,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : ( aa(A,set(B),aTP_Lamp_abs(set(B),fun(A,set(B)),Uu),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),Uu) ) ).

% ATP.lambda_378
tff(fact_8494_ATP_Olambda__379,axiom,
    ! [Uu: int,Uua: nat] : ( aa(nat,rat,aTP_Lamp_aae(int,fun(nat,rat),Uu),Uua) = aa(int,rat,ring_1_of_int(rat),Uu) ) ).

% ATP.lambda_379
tff(fact_8495_ATP_Olambda__380,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : ( aa(A,set(B),aTP_Lamp_abo(list(B),fun(A,set(B)),Uu),Uua) = aa(list(B),set(B),set2(B),Uu) ) ).

% ATP.lambda_380
tff(fact_8496_ATP_Olambda__381,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : ( aa(list(A),fun(set(A),set(A)),aa(A,fun(list(A),fun(set(A),set(A))),aTP_Lamp_aby(A,fun(list(A),fun(set(A),set(A)))),Uu),Uua) = aa(A,fun(set(A),set(A)),insert2(A),Uu) ) ).

% ATP.lambda_381
tff(fact_8497_ATP_Olambda__382,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      ( aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_dd(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(nat,real,Uua,divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,Uu,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_382
tff(fact_8498_ATP_Olambda__383,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      ( aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_eh(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(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_383
tff(fact_8499_ATP_Olambda__384,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_jz(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),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),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_384
tff(fact_8500_ATP_Olambda__385,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_hi(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),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),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_385
tff(fact_8501_ATP_Olambda__386,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      ( aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_hj(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),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),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_386
tff(fact_8502_ATP_Olambda__387,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_hk(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),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),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_387
tff(fact_8503_ATP_Olambda__388,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_ia(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_388
tff(fact_8504_ATP_Olambda__389,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_lw(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_389
tff(fact_8505_ATP_Olambda__390,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_lx(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_390
tff(fact_8506_ATP_Olambda__391,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_if(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),zero_zero(B)) ) ) ).

% ATP.lambda_391
tff(fact_8507_ATP_Olambda__392,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_ii(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ) ).

% ATP.lambda_392
tff(fact_8508_ATP_Olambda__393,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_bk(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(nat,A,Uua,Uub),zero_zero(A)) ) ) ).

% ATP.lambda_393
tff(fact_8509_ATP_Olambda__394,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_ig(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),zero_zero(B)) ) ) ).

% ATP.lambda_394
tff(fact_8510_ATP_Olambda__395,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_ih(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),one_one(B)) ) ) ).

% ATP.lambda_395
tff(fact_8511_ATP_Olambda__396,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      ( aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_kr(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uu)),Uub))) ) ).

% ATP.lambda_396
tff(fact_8512_ATP_Olambda__397,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      ( aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_ks(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),Uu)),Uub))) ) ).

% ATP.lambda_397
tff(fact_8513_ATP_Olambda__398,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      ( aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_kq(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uu),Uub))) ) ).

% ATP.lambda_398
tff(fact_8514_ATP_Olambda__399,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_ib(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_399
tff(fact_8515_ATP_Olambda__400,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,$o),Uub: B] :
          ( aa(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_nu(fun(B,A),fun(fun(B,$o),fun(B,A)),Uu),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ) ).

% ATP.lambda_400
tff(fact_8516_ATP_Olambda__401,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,$o),fun(A,B),aTP_Lamp_is(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),zero_zero(B)) ) ) ).

% ATP.lambda_401
tff(fact_8517_ATP_Olambda__402,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,$o),fun(A,B),aTP_Lamp_it(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),one_one(B)) ) ) ).

% ATP.lambda_402
tff(fact_8518_ATP_Olambda__403,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : ( aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_zn(fun(A,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uub),Uu),Uua) ) ).

% ATP.lambda_403
tff(fact_8519_ATP_Olambda__404,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: fun(nat,A),Uub: nat] : ( aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_afy(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),Uu,aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub))),aa(nat,A,Uua,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)))) ) ).

% ATP.lambda_404
tff(fact_8520_ATP_Olambda__405,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C),Uub: B] : ( aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_pp(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,aa(B,C,Uua,Uub)),Uub) ) ) ).

% ATP.lambda_405
tff(fact_8521_ATP_Olambda__406,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_hs(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_406
tff(fact_8522_ATP_Olambda__407,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_hq(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_407
tff(fact_8523_ATP_Olambda__408,axiom,
    ! [A: $tType,Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afk(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_408
tff(fact_8524_ATP_Olambda__409,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_ey(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_409
tff(fact_8525_ATP_Olambda__410,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_ev(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_410
tff(fact_8526_ATP_Olambda__411,axiom,
    ! [A: $tType,Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aft(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_411
tff(fact_8527_ATP_Olambda__412,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : ( aa(C,A,aa(B,fun(C,A),aTP_Lamp_pn(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ) ) ).

% ATP.lambda_412
tff(fact_8528_ATP_Olambda__413,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_st(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_413
tff(fact_8529_ATP_Olambda__414,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_tx(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_414
tff(fact_8530_ATP_Olambda__415,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_gr(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gq(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ) ).

% ATP.lambda_415
tff(fact_8531_ATP_Olambda__416,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_gp(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_go(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ) ).

% ATP.lambda_416
tff(fact_8532_ATP_Olambda__417,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_gn(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gm(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ) ).

% ATP.lambda_417
tff(fact_8533_ATP_Olambda__418,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_fl(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_fk(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ) ).

% ATP.lambda_418
tff(fact_8534_ATP_Olambda__419,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_cu(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_419
tff(fact_8535_ATP_Olambda__420,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      ( aa(code_integer,product_prod(code_integer,$o),aa(code_integer,fun(code_integer,product_prod(code_integer,$o)),aTP_Lamp_kp(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),Uu),Uua),Uub) = aa($o,product_prod(code_integer,$o),
          product_Pair(code_integer,$o,
            $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),Uu),Uua,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),Uub))),
          Uub = one_one(code_integer)) ) ).

% ATP.lambda_420
tff(fact_8536_ATP_Olambda__421,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_ez(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ey(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ) ).

% ATP.lambda_421
tff(fact_8537_ATP_Olambda__422,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_ew(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ev(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ) ).

% ATP.lambda_422
tff(fact_8538_ATP_Olambda__423,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ko(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_kn(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_423
tff(fact_8539_ATP_Olambda__424,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : ( aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_kl(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_kk(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_424
tff(fact_8540_ATP_Olambda__425,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : ( aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_kj(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_ki(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_425
tff(fact_8541_ATP_Olambda__426,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: nat] : ( aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_qd(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(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_426
tff(fact_8542_ATP_Olambda__427,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_qe(real,fun(fun(nat,fun(real,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(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_427
tff(fact_8543_ATP_Olambda__428,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_eg(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(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_428
tff(fact_8544_ATP_Olambda__429,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_fh(fun(nat,A),fun(nat,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_429
tff(fact_8545_ATP_Olambda__430,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_up(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = 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_430
tff(fact_8546_ATP_Olambda__431,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_uh(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = 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_431
tff(fact_8547_ATP_Olambda__432,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_ik(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = 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_432
tff(fact_8548_ATP_Olambda__433,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_tz(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = 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_433
tff(fact_8549_ATP_Olambda__434,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_ui(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = 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_434
tff(fact_8550_ATP_Olambda__435,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : ( aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_qa(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_435
tff(fact_8551_ATP_Olambda__436,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_mz(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),one_one(nat)) )
        & ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_436
tff(fact_8552_ATP_Olambda__437,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_eo(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_437
tff(fact_8553_ATP_Olambda__438,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_mr(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub))
        | ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Uua),Uub),lex(A,Uu)) ) ) ) ).

% ATP.lambda_438
tff(fact_8554_ATP_Olambda__439,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_na(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_439
tff(fact_8555_ATP_Olambda__440,axiom,
    ! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(A),fun(list(A),$o),aTP_Lamp_hm(nat,fun(set(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua) ) ) ).

% ATP.lambda_440
tff(fact_8556_ATP_Olambda__441,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_hw(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu) ) ) ).

% ATP.lambda_441
tff(fact_8557_ATP_Olambda__442,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_acq(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uua) ) ) ) ).

% ATP.lambda_442
tff(fact_8558_ATP_Olambda__443,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_ll(nat,fun(list(A),fun(list(A),$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)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua)) ) ) ).

% ATP.lambda_443
tff(fact_8559_ATP_Olambda__444,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_hx(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),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_444
tff(fact_8560_ATP_Olambda__445,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_in(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_445
tff(fact_8561_ATP_Olambda__446,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_my(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_446
tff(fact_8562_ATP_Olambda__447,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_sn(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_447
tff(fact_8563_ATP_Olambda__448,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ah(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,aa(nat,nat,suc,Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_448
tff(fact_8564_ATP_Olambda__449,axiom,
    ! [A: $tType,Uu: set(nat),Uua: nat,Uub: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aa(nat,fun(product_prod(A,nat),$o),aTP_Lamp_aax(set(nat),fun(nat,fun(product_prod(A,nat),$o)),Uu),Uua),Uub)
    <=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(product_prod(A,nat),nat,product_snd(A,nat),Uub)),Uua),Uu) ) ).

% ATP.lambda_449
tff(fact_8565_ATP_Olambda__450,axiom,
    ! [Uu: nat,Uua: nat,Uub: set(nat)] :
      ( aa(set(nat),$o,aa(nat,fun(set(nat),$o),aTP_Lamp_acp(nat,fun(nat,fun(set(nat),$o)),Uu),Uua),Uub)
    <=> ( member(set(nat),Uub,pow2(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uu)))
        & ( aa(set(nat),nat,finite_card(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_450
tff(fact_8566_ATP_Olambda__451,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_oi(list(A),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu))
        & member(nat,Uub,Uua) ) ) ).

% ATP.lambda_451
tff(fact_8567_ATP_Olambda__452,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_ny(fun(A,$o),fun(list(A),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uua))
        & aa(A,$o,Uu,aa(nat,A,nth(A,Uua),Uub)) ) ) ).

% ATP.lambda_452
tff(fact_8568_ATP_Olambda__453,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_zz(list(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,aa(list(A),set(A),set2(A),Uu))
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_453
tff(fact_8569_ATP_Olambda__454,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_nt(fun(A,$o),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,aa(list(A),set(A),set2(A),Uua))
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_454
tff(fact_8570_ATP_Olambda__455,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_js(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = 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_455
tff(fact_8571_ATP_Olambda__456,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_op(list(A),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu)),Uua) ) ).

% ATP.lambda_456
tff(fact_8572_ATP_Olambda__457,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aez(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uub) ) ) ).

% ATP.lambda_457
tff(fact_8573_ATP_Olambda__458,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_aao(set(A),fun(nat,fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),infini527867602293511546merate(A,Uu,Uua)),Uub) ) ) ) ).

% ATP.lambda_458
tff(fact_8574_ATP_Olambda__459,axiom,
    ! [Uu: set(nat),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_agl(set(nat),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & member(nat,aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_agk(set(nat),fun(nat,fun(nat,$o)),Uu),Uub))),Uua) ) ) ).

% ATP.lambda_459
tff(fact_8575_ATP_Olambda__460,axiom,
    ! [A: $tType,B: $tType,Uu: nat,Uua: set(old_node(A,B)),Uub: old_node(A,B)] :
      ( aa(old_node(A,B),$o,aa(set(old_node(A,B)),fun(old_node(A,B),$o),aTP_Lamp_agt(nat,fun(set(old_node(A,B)),fun(old_node(A,B),$o)),Uu),Uua),Uub)
    <=> ( member(old_node(A,B),Uub,Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),old_ndepth(A,B,Uub)),Uu) ) ) ).

% ATP.lambda_460
tff(fact_8576_ATP_Olambda__461,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ai(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_461
tff(fact_8577_ATP_Olambda__462,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_en(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_462
tff(fact_8578_ATP_Olambda__463,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_ff(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_463
tff(fact_8579_ATP_Olambda__464,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_jy(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_464
tff(fact_8580_ATP_Olambda__465,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_hy(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_465
tff(fact_8581_ATP_Olambda__466,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_jx(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_466
tff(fact_8582_ATP_Olambda__467,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ie(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_467
tff(fact_8583_ATP_Olambda__468,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_id(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_468
tff(fact_8584_ATP_Olambda__469,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_hz(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_469
tff(fact_8585_ATP_Olambda__470,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_abd(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uub),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uub),Uu) ) ) ).

% ATP.lambda_470
tff(fact_8586_ATP_Olambda__471,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_aaz(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Uub),Uua)
        & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Uub),Uu) ) ) ).

% ATP.lambda_471
tff(fact_8587_ATP_Olambda__472,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_agk(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_472
tff(fact_8588_ATP_Olambda__473,axiom,
    ! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : ( aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_nc(list(A),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uu),Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_473
tff(fact_8589_ATP_Olambda__474,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: list(A),Uub: A] : ( aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_nf(fun(A,nat),fun(list(A),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uua),Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_474
tff(fact_8590_ATP_Olambda__475,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_hh(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_475
tff(fact_8591_ATP_Olambda__476,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,$o),aTP_Lamp_io(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_476
tff(fact_8592_ATP_Olambda__477,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,$o),aTP_Lamp_jg(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_477
tff(fact_8593_ATP_Olambda__478,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,$o),aTP_Lamp_iq(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_478
tff(fact_8594_ATP_Olambda__479,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,$o),aTP_Lamp_jh(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_479
tff(fact_8595_ATP_Olambda__480,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_lo(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_480
tff(fact_8596_ATP_Olambda__481,axiom,
    ! [A: $tType,B: $tType,Uu: list(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_abh(list(product_prod(A,B)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(A,option(B),map_of(A,B,Uu),Uua) = aa(B,option(B),some(B),Uub) ) ) ).

% ATP.lambda_481
tff(fact_8597_ATP_Olambda__482,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_zx(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_482
tff(fact_8598_ATP_Olambda__483,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_hp(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_483
tff(fact_8599_ATP_Olambda__484,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_hu(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_484
tff(fact_8600_ATP_Olambda__485,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : ( aa(A,A,aa(A,fun(A,A),aTP_Lamp_mh(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Uub,Uu)),Uua) ) ) ).

% ATP.lambda_485
tff(fact_8601_ATP_Olambda__486,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : ( aa(A,A,aa(A,fun(A,A),aTP_Lamp_mf(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_486
tff(fact_8602_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_mg(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Uub,Uu)),Uua) ) ) ).

% ATP.lambda_487
tff(fact_8603_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_me(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_488
tff(fact_8604_ATP_Olambda__489,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_np(set(product_prod(A,B)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Uua),Uub),Uu) ) ).

% ATP.lambda_489
tff(fact_8605_ATP_Olambda__490,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_nn(list(list(A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_490
tff(fact_8606_ATP_Olambda__491,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_dq(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_491
tff(fact_8607_ATP_Olambda__492,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa(complex,$o,aa(nat,fun(complex,$o),aTP_Lamp_aj(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_492
tff(fact_8608_ATP_Olambda__493,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_iu(A,fun(A,fun(A,$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_493
tff(fact_8609_ATP_Olambda__494,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : ( aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bw(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_494
tff(fact_8610_ATP_Olambda__495,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_bz(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ) ).

% ATP.lambda_495
tff(fact_8611_ATP_Olambda__496,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,int),Uub: nat] : ( aa(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_wi(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_496
tff(fact_8612_ATP_Olambda__497,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : ( aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_qb(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_497
tff(fact_8613_ATP_Olambda__498,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : ( aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_cc(fun(nat,real),fun(real,fun(nat,real)),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_498
tff(fact_8614_ATP_Olambda__499,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fs(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_499
tff(fact_8615_ATP_Olambda__500,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_fd(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ) ).

% ATP.lambda_500
tff(fact_8616_ATP_Olambda__501,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : ( aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bx(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ) ).

% ATP.lambda_501
tff(fact_8617_ATP_Olambda__502,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : ( aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_by(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ) ).

% ATP.lambda_502
tff(fact_8618_ATP_Olambda__503,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_ut(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_8619_ATP_Olambda__504,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_fc(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_8620_ATP_Olambda__505,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : ( aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fi(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_8621_ATP_Olambda__506,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_im(fun(nat,$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_506
tff(fact_8622_ATP_Olambda__507,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(real,real),Uua: fun(real,A),Uub: real] : ( aa(real,A,aa(fun(real,A),fun(real,A),aTP_Lamp_adw(fun(real,real),fun(fun(real,A),fun(real,A)),Uu),Uua),Uub) = real_V8093663219630862766scaleR(A,aa(real,real,Uu,Uub),aa(real,A,Uua,Uub)) ) ) ).

% ATP.lambda_507
tff(fact_8623_ATP_Olambda__508,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_rh(fun(A,real),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_508
tff(fact_8624_ATP_Olambda__509,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_sb(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_509
tff(fact_8625_ATP_Olambda__510,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_rq(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_510
tff(fact_8626_ATP_Olambda__511,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : ( aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yv(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_511
tff(fact_8627_ATP_Olambda__512,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_qn(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,Uu,Uub),aa(A,A,Uua,Uub)) ) ) ).

% ATP.lambda_512
tff(fact_8628_ATP_Olambda__513,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_uj(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_513
tff(fact_8629_ATP_Olambda__514,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : ( aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xu(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_514
tff(fact_8630_ATP_Olambda__515,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_tl(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_515
tff(fact_8631_ATP_Olambda__516,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_zb(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_516
tff(fact_8632_ATP_Olambda__517,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(real,A),Uua: fun(real,A),Uub: real] : ( aa(real,A,aa(fun(real,A),fun(real,A),aTP_Lamp_aee(fun(real,A),fun(fun(real,A),fun(real,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(real,A,Uu,Uub)),aa(real,A,Uua,Uub)) ) ) ).

% ATP.lambda_517
tff(fact_8633_ATP_Olambda__518,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : ( aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_ms(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),Uu),Uua),Uub) = aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),aa(nat,rat,Uu,Uub)),aa(nat,rat,Uua,Uub)) ) ).

% ATP.lambda_518
tff(fact_8634_ATP_Olambda__519,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : ( aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_jw(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_519
tff(fact_8635_ATP_Olambda__520,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_jv(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_520
tff(fact_8636_ATP_Olambda__521,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_ro(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_521
tff(fact_8637_ATP_Olambda__522,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_qu(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_522
tff(fact_8638_ATP_Olambda__523,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_tr(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_523
tff(fact_8639_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_xt(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_524
tff(fact_8640_ATP_Olambda__525,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_ts(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_525
tff(fact_8641_ATP_Olambda__526,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : ( aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_lp(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_526
tff(fact_8642_ATP_Olambda__527,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : ( aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_cx(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_527
tff(fact_8643_ATP_Olambda__528,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(real,A),Uua: fun(real,A),Uub: real] : ( aa(real,A,aa(fun(real,A),fun(real,A),aTP_Lamp_aed(fun(real,A),fun(fun(real,A),fun(real,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(real,A,Uu,Uub)),aa(real,A,Uua,Uub)) ) ) ).

% ATP.lambda_528
tff(fact_8644_ATP_Olambda__529,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : ( aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_vl(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_529
tff(fact_8645_ATP_Olambda__530,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : ( aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_mw(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),Uu),Uua),Uub) = aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,Uu,Uub)),aa(nat,rat,Uua,Uub)) ) ).

% ATP.lambda_530
tff(fact_8646_ATP_Olambda__531,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_bq(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_531
tff(fact_8647_ATP_Olambda__532,axiom,
    ! [A: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : ( aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ack(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_532
tff(fact_8648_ATP_Olambda__533,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_ck(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_533
tff(fact_8649_ATP_Olambda__534,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_rn(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_534
tff(fact_8650_ATP_Olambda__535,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : ( aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yz(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_535
tff(fact_8651_ATP_Olambda__536,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_qs(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_536
tff(fact_8652_ATP_Olambda__537,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : ( aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aeo(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_537
tff(fact_8653_ATP_Olambda__538,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_uk(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_538
tff(fact_8654_ATP_Olambda__539,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_ti(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_539
tff(fact_8655_ATP_Olambda__540,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_to(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_540
tff(fact_8656_ATP_Olambda__541,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_tj(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_541
tff(fact_8657_ATP_Olambda__542,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_jb(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_542
tff(fact_8658_ATP_Olambda__543,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_dy(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_543
tff(fact_8659_ATP_Olambda__544,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : ( aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_dl(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_544
tff(fact_8660_ATP_Olambda__545,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : ( aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tn(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_545
tff(fact_8661_ATP_Olambda__546,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(real,A),Uua: fun(real,A),Uub: real] : ( aa(real,A,aa(fun(real,A),fun(real,A),aTP_Lamp_aea(fun(real,A),fun(fun(real,A),fun(real,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,Uu,Uub)),aa(real,A,Uua,Uub)) ) ) ).

% ATP.lambda_546
tff(fact_8662_ATP_Olambda__547,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : ( aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_mv(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),Uu),Uua),Uub) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(nat,rat,Uu,Uub)),aa(nat,rat,Uua,Uub)) ) ).

% ATP.lambda_547
tff(fact_8663_ATP_Olambda__548,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_bp(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_548
tff(fact_8664_ATP_Olambda__549,axiom,
    ! [A: $tType] :
      ( topolo6943815403480290642id_add(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : ( aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aci(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_549
tff(fact_8665_ATP_Olambda__550,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_cj(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_550
tff(fact_8666_ATP_Olambda__551,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_rm(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_551
tff(fact_8667_ATP_Olambda__552,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : ( aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yy(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_552
tff(fact_8668_ATP_Olambda__553,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_qr(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_553
tff(fact_8669_ATP_Olambda__554,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : ( aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aen(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_554
tff(fact_8670_ATP_Olambda__555,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_th(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_555
tff(fact_8671_ATP_Olambda__556,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_xp(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_556
tff(fact_8672_ATP_Olambda__557,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_tg(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_557
tff(fact_8673_ATP_Olambda__558,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_ji(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_558
tff(fact_8674_ATP_Olambda__559,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : ( aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zf(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log2(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ) ).

% ATP.lambda_559
tff(fact_8675_ATP_Olambda__560,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_vr(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log2(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ) ).

% ATP.lambda_560
tff(fact_8676_ATP_Olambda__561,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_tw(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log2(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_561
tff(fact_8677_ATP_Olambda__562,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_zt(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_562
tff(fact_8678_ATP_Olambda__563,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_zc(fun(A,B),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).

% ATP.lambda_563
tff(fact_8679_ATP_Olambda__564,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_or(A,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uua,Uu) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_564
tff(fact_8680_ATP_Olambda__565,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_xg(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,Uua))) ) ) ).

% ATP.lambda_565
tff(fact_8681_ATP_Olambda__566,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_qy(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_566
tff(fact_8682_ATP_Olambda__567,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_su(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),Uua,Uub)),Uu) ) ) ).

% ATP.lambda_567
tff(fact_8683_ATP_Olambda__568,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_ty(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(B,fun(A,C),Uua,Uub)),Uu) ) ) ).

% ATP.lambda_568
tff(fact_8684_ATP_Olambda__569,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_yn(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_569
tff(fact_8685_ATP_Olambda__570,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wy(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_570
tff(fact_8686_ATP_Olambda__571,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_bo(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,Uu,Uub),Uua) ) ) ).

% ATP.lambda_571
tff(fact_8687_ATP_Olambda__572,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_ma(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),Uua) ) ) ).

% ATP.lambda_572
tff(fact_8688_ATP_Olambda__573,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_tk(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),Uua) ) ) ).

% ATP.lambda_573
tff(fact_8689_ATP_Olambda__574,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_va(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,Uua,Uub),Uu) ) ) ).

% ATP.lambda_574
tff(fact_8690_ATP_Olambda__575,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_wu(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_575
tff(fact_8691_ATP_Olambda__576,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ws(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_576
tff(fact_8692_ATP_Olambda__577,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,$o),aTP_Lamp_ym(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_577
tff(fact_8693_ATP_Olambda__578,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : ( aa(A,B,aa(B,fun(A,B),aTP_Lamp_abz(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_578
tff(fact_8694_ATP_Olambda__579,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_tq(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_579
tff(fact_8695_ATP_Olambda__580,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_cn(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_580
tff(fact_8696_ATP_Olambda__581,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_uz(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_581
tff(fact_8697_ATP_Olambda__582,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_sq(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_582
tff(fact_8698_ATP_Olambda__583,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(real,A),Uua: A,Uub: real] : ( aa(real,A,aa(A,fun(real,A),aTP_Lamp_aec(fun(real,A),fun(A,fun(real,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(real,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_583
tff(fact_8699_ATP_Olambda__584,axiom,
    ! [A: $tType] :
      ( topolo1287966508704411220up_add(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : ( aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_acj(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_584
tff(fact_8700_ATP_Olambda__585,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : ( aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dz(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_585
tff(fact_8701_ATP_Olambda__586,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_tm(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_586
tff(fact_8702_ATP_Olambda__587,axiom,
    ! [Uu: fun(real,real),Uua: nat,Uub: real] : ( aa(real,real,aa(nat,fun(real,real),aTP_Lamp_pw(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_587
tff(fact_8703_ATP_Olambda__588,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_rx(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_588
tff(fact_8704_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_qz(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_589
tff(fact_8705_ATP_Olambda__590,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : ( aa(A,B,aa(nat,fun(A,B),aTP_Lamp_xv(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_590
tff(fact_8706_ATP_Olambda__591,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_sv(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_591
tff(fact_8707_ATP_Olambda__592,axiom,
    ! [Uu: nat,Uua: fun(real,real),Uub: real] : ( aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yj(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_592
tff(fact_8708_ATP_Olambda__593,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,real),Uub: A] : ( aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ss(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_593
tff(fact_8709_ATP_Olambda__594,axiom,
    ! [B: $tType,A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : ( aa(B,A,aa(A,fun(B,A),aTP_Lamp_zs(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_594
tff(fact_8710_ATP_Olambda__595,axiom,
    ! [B: $tType,A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : ( aa(B,A,aa(A,fun(B,A),aTP_Lamp_pa(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_595
tff(fact_8711_ATP_Olambda__596,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(real,A),Uua: A,Uub: real] : ( aa(real,A,aa(A,fun(real,A),aTP_Lamp_aeb(fun(real,A),fun(A,fun(real,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_596
tff(fact_8712_ATP_Olambda__597,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : ( aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_yq(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_597
tff(fact_8713_ATP_Olambda__598,axiom,
    ! [A: $tType] :
      ( topolo1287966508704411220up_add(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : ( aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_acg(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_598
tff(fact_8714_ATP_Olambda__599,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_zy(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_599
tff(fact_8715_ATP_Olambda__600,axiom,
    ! [Uu: fun(real,real),Uua: real,Uub: real] : ( aa(real,real,aa(real,fun(real,real),aTP_Lamp_py(fun(real,real),fun(real,fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),Uua) ) ).

% ATP.lambda_600
tff(fact_8716_ATP_Olambda__601,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(B,A),Uua: int,Uub: B] : ( aa(B,A,aa(int,fun(B,A),aTP_Lamp_sf(fun(B,A),fun(int,fun(B,A)),Uu),Uua),Uub) = power_int(A,aa(B,A,Uu,Uub),Uua) ) ) ).

% ATP.lambda_601
tff(fact_8717_ATP_Olambda__602,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: int,Uub: A] : ( aa(A,B,aa(int,fun(A,B),aTP_Lamp_acc(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ) ).

% ATP.lambda_602
tff(fact_8718_ATP_Olambda__603,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: int,Uub: A] : ( aa(A,B,aa(int,fun(A,B),aTP_Lamp_yu(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ) ).

% ATP.lambda_603
tff(fact_8719_ATP_Olambda__604,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: int,Uub: A] : ( aa(A,A,aa(int,fun(A,A),aTP_Lamp_ra(fun(A,A),fun(int,fun(A,A)),Uu),Uua),Uub) = power_int(A,aa(A,A,Uu,Uub),Uua) ) ) ).

% ATP.lambda_604
tff(fact_8720_ATP_Olambda__605,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: int,Uub: A] : ( aa(A,B,aa(int,fun(A,B),aTP_Lamp_uo(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ) ).

% ATP.lambda_605
tff(fact_8721_ATP_Olambda__606,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: int,Uub: A] : ( aa(A,B,aa(int,fun(A,B),aTP_Lamp_ta(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ) ).

% ATP.lambda_606
tff(fact_8722_ATP_Olambda__607,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_afd(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,A,Uu,Uub) = Uua ) ) ).

% ATP.lambda_607
tff(fact_8723_ATP_Olambda__608,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jr(nat,fun(nat,fun(nat,A)),Uu),Uua),Uub) = 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_608
tff(fact_8724_ATP_Olambda__609,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : ( aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aab(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),Uu),Uua),Uub) = aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(rat,rat,inverse_inverse(rat),aa(nat,rat,Uu,Uub))),aa(rat,rat,inverse_inverse(rat),aa(nat,rat,Uua,Uub))) ) ).

% ATP.lambda_609
tff(fact_8725_ATP_Olambda__610,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_xa(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( archimedean_ceiling(B,aa(A,B,Uu,Uub)) = archimedean_ceiling(B,Uua) ) ) ) ).

% ATP.lambda_610
tff(fact_8726_ATP_Olambda__611,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_gz(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = 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_611
tff(fact_8727_ATP_Olambda__612,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_xf(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(int,B,ring_1_of_int(B),aa(B,int,archim6421214686448440834_floor(B),Uua))),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_612
tff(fact_8728_ATP_Olambda__613,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(list(A),fun(list(A),$o)),Uua: list(A),Uub: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_pd(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Uu),Uua),Uub)
        <=> ( ? [Y5: A,Ys4: list(A)] :
                ( ( Uua = nil(A) )
                & ( Uub = cons(A,Y5,Ys4) ) )
            | ? [X4: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = cons(A,X4,Xs3) )
                & ( Uub = cons(A,Y5,Ys4) )
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y5) )
            | ? [X4: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = cons(A,X4,Xs3) )
                & ( Uub = cons(A,Y5,Ys4) )
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y5)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X4)
                & aa(list(A),$o,aa(list(A),fun(list(A),$o),Uu,Xs3),Ys4) ) ) ) ) ).

% ATP.lambda_613
tff(fact_8729_ATP_Olambda__614,axiom,
    ! [A: $tType,Uu: fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),Uua: fun($o,fun($o,A)),Uub: vEBT_VEBT] : ( aa(vEBT_VEBT,product_prod(vEBT_VEBT,A),aa(fun($o,fun($o,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A)),aTP_Lamp_acd(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A))),Uu),Uua),Uub) = aa(A,product_prod(vEBT_VEBT,A),product_Pair(vEBT_VEBT,A,Uub),aa(vEBT_VEBT,A,aa(fun($o,fun($o,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun($o,fun($o,A)),fun(vEBT_VEBT,A)),vEBT_rec_VEBT(A),Uu),Uua),Uub)) ) ).

% ATP.lambda_614
tff(fact_8730_ATP_Olambda__615,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_jo(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ) ).

% ATP.lambda_615
tff(fact_8731_ATP_Olambda__616,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_jp(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ) ).

% ATP.lambda_616
tff(fact_8732_ATP_Olambda__617,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_jn(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ) ).

% ATP.lambda_617
tff(fact_8733_ATP_Olambda__618,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: int,Uub: A] : ( aa(A,A,aa(int,fun(A,A),aTP_Lamp_se(A,fun(int,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Uua)),power_int(A,Uu,aa(int,int,aa(int,fun(int,int),minus_minus(int),Uua),one_one(int))))) ) ) ).

% ATP.lambda_618
tff(fact_8734_ATP_Olambda__619,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : ( aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dh(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_619
tff(fact_8735_ATP_Olambda__620,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : ( aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_df(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_620
tff(fact_8736_ATP_Olambda__621,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_ef(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_621
tff(fact_8737_ATP_Olambda__622,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : ( aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_vv(real,fun(real,fun(nat,real)),Uu),Uua),Uub) = divide_divide(real,Uua,aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ) ).

% ATP.lambda_622
tff(fact_8738_ATP_Olambda__623,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_cs(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_623
tff(fact_8739_ATP_Olambda__624,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : ( aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_dg(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_624
tff(fact_8740_ATP_Olambda__625,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_wz(B,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uu),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_625
tff(fact_8741_ATP_Olambda__626,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wq(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_626
tff(fact_8742_ATP_Olambda__627,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_ku(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,Uua,aa(nat,A,Uu,Uub)) ) ) ).

% ATP.lambda_627
tff(fact_8743_ATP_Olambda__628,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_wt(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_628
tff(fact_8744_ATP_Olambda__629,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_od(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_629
tff(fact_8745_ATP_Olambda__630,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wp(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_630
tff(fact_8746_ATP_Olambda__631,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_wr(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_631
tff(fact_8747_ATP_Olambda__632,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : ( aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aca(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uu),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_632
tff(fact_8748_ATP_Olambda__633,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_bn(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_633
tff(fact_8749_ATP_Olambda__634,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : ( aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cm(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_634
tff(fact_8750_ATP_Olambda__635,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : ( aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_uy(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_635
tff(fact_8751_ATP_Olambda__636,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_sr(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_636
tff(fact_8752_ATP_Olambda__637,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_tp(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_637
tff(fact_8753_ATP_Olambda__638,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_yi(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_638
tff(fact_8754_ATP_Olambda__639,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_wc(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_639
tff(fact_8755_ATP_Olambda__640,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : ( aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_zq(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu),aa(B,A,Uua,Uub)) ) ) ).

% ATP.lambda_640
tff(fact_8756_ATP_Olambda__641,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : ( aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_oy(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),aa(B,A,Uua,Uub)) ) ) ).

% ATP.lambda_641
tff(fact_8757_ATP_Olambda__642,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_tf(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_642
tff(fact_8758_ATP_Olambda__643,axiom,
    ! [A: $tType] :
      ( topolo1287966508704411220up_add(A)
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : ( aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ach(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_643
tff(fact_8759_ATP_Olambda__644,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_mb(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_644
tff(fact_8760_ATP_Olambda__645,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bc(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_645
tff(fact_8761_ATP_Olambda__646,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( aa(A,$o,aa(set(nat),fun(A,$o),aTP_Lamp_oh(list(A),fun(set(nat),fun(A,$o)),Uu),Uua),Uub)
    <=> member(A,Uub,aa(list(A),set(A),set2(A),nths(A,Uu,Uua))) ) ).

% ATP.lambda_646
tff(fact_8762_ATP_Olambda__647,axiom,
    ! [Uu: real,Uua: real,Uub: product_unit] : ( aa(product_unit,real,aa(real,fun(product_unit,real),aTP_Lamp_adj(real,fun(real,fun(product_unit,real)),Uu),Uua),Uub) = powr_real(Uu,Uua) ) ).

% ATP.lambda_647
tff(fact_8763_ATP_Olambda__648,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: fun(A,$o),Uub: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aTP_Lamp_afe(fun(A,B),fun(fun(A,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,Uua,aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),Uu),Uub)) ) ).

% ATP.lambda_648
tff(fact_8764_ATP_Olambda__649,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_dx(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_649
tff(fact_8765_ATP_Olambda__650,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_dw(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_650
tff(fact_8766_ATP_Olambda__651,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat,Uub: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afq(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_651
tff(fact_8767_ATP_Olambda__652,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_vi(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_652
tff(fact_8768_ATP_Olambda__653,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_vd(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_653
tff(fact_8769_ATP_Olambda__654,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_uc(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_654
tff(fact_8770_ATP_Olambda__655,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_qv(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_655
tff(fact_8771_ATP_Olambda__656,axiom,
    ! [Uu: fun(nat,real),Uua: nat,Uub: nat] : ( aa(nat,real,aa(nat,fun(nat,real),aTP_Lamp_ace(fun(nat,real),fun(nat,fun(nat,real)),Uu),Uua),Uub) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_656
tff(fact_8772_ATP_Olambda__657,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_wo(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_657
tff(fact_8773_ATP_Olambda__658,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] : ( aa(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_pc(fun(nat,set(A)),fun(nat,fun(nat,set(A))),Uu),Uua),Uub) = aa(nat,set(A),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_658
tff(fact_8774_ATP_Olambda__659,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bm(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_659
tff(fact_8775_ATP_Olambda__660,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_vc(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_660
tff(fact_8776_ATP_Olambda__661,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_yr(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_661
tff(fact_8777_ATP_Olambda__662,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_ax(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_662
tff(fact_8778_ATP_Olambda__663,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_cp(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_663
tff(fact_8779_ATP_Olambda__664,axiom,
    ! [A: $tType,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_664
tff(fact_8780_ATP_Olambda__665,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,$o),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_wx(fun(A,$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ) ).

% ATP.lambda_665
tff(fact_8781_ATP_Olambda__666,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_ub(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_666
tff(fact_8782_ATP_Olambda__667,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_ur(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_667
tff(fact_8783_ATP_Olambda__668,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_qw(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_668
tff(fact_8784_ATP_Olambda__669,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_il(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),product_Pair(A,B,Uua),Uub)) ) ).

% ATP.lambda_669
tff(fact_8785_ATP_Olambda__670,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_ug(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_670
tff(fact_8786_ATP_Olambda__671,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_ud(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_671
tff(fact_8787_ATP_Olambda__672,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_cr(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_672
tff(fact_8788_ATP_Olambda__673,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_aal(fun(B,A),fun(fun(num,B),fun(num,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(num,B,Uua,Uub)) ) ).

% ATP.lambda_673
tff(fact_8789_ATP_Olambda__674,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_ka(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(nat,B,Uua,Uub)) ) ).

% ATP.lambda_674
tff(fact_8790_ATP_Olambda__675,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(A,$o),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ww(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_675
tff(fact_8791_ATP_Olambda__676,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : ( aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_wa(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_676
tff(fact_8792_ATP_Olambda__677,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_wv(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ) ).

% ATP.lambda_677
tff(fact_8793_ATP_Olambda__678,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,nat),Uua: fun(nat,A),Uub: nat] : ( aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_add(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ) ).

% ATP.lambda_678
tff(fact_8794_ATP_Olambda__679,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,nat),Uua: fun(nat,A),Uub: nat] : ( aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_adc(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ) ).

% ATP.lambda_679
tff(fact_8795_ATP_Olambda__680,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_ue(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_680
tff(fact_8796_ATP_Olambda__681,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_jd(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_681
tff(fact_8797_ATP_Olambda__682,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_ja(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_682
tff(fact_8798_ATP_Olambda__683,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_1(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : ( aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_mi(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_683
tff(fact_8799_ATP_Olambda__684,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_sp(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_so(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub)) ) ) ).

% ATP.lambda_684
tff(fact_8800_ATP_Olambda__685,axiom,
    ! [B: $tType] :
      ( real_V4867850818363320053vector(B)
     => ! [Uu: set(B),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_aew(set(B),fun(B,fun(B,$o)),Uu),Uua),Uub)
        <=> ( real_V7696804695334737415tation(B,real_V4986007116245087402_basis(B,Uu),Uua,Uub) != zero_zero(real) ) ) ) ).

% ATP.lambda_685
tff(fact_8801_ATP_Olambda__686,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_dr(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_686
tff(fact_8802_ATP_Olambda__687,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : ( aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_wh(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_687
tff(fact_8803_ATP_Olambda__688,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_mc(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_688
tff(fact_8804_ATP_Olambda__689,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_sl(fun(B,real),fun(fun(real,A),fun(B,real)),Uu),Uua),Uub) = aa(int,real,ring_1_of_int(real),aa(A,int,archim6421214686448440834_floor(A),aa(real,A,Uua,aa(B,real,Uu,Uub)))) ) ) ).

% ATP.lambda_689
tff(fact_8805_ATP_Olambda__690,axiom,
    ! [A: $tType,B: $tType,Uu: list(A),Uua: list(B),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_nb(list(A),fun(list(B),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ? [I3: nat] :
          ( ( Uub = aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Uu),I3)),aa(nat,B,nth(B,Uua),I3)) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(list(B),nat,size_size(list(B)),Uua))) ) ) ).

% ATP.lambda_690
tff(fact_8806_ATP_Olambda__691,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( aa(A,$o,aa(set(nat),fun(A,$o),aTP_Lamp_oe(list(A),fun(set(nat),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [I3: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uu),I3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Uu))
          & member(nat,I3,Uua) ) ) ).

% ATP.lambda_691
tff(fact_8807_ATP_Olambda__692,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: A,Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ael(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [K3: real] : member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),real_V8093663219630862766scaleR(A,K3,Uu)),real_Vector_span(A,Uua)) ) ) ).

% ATP.lambda_692
tff(fact_8808_ATP_Olambda__693,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_wl(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> ! [A5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),A5)
             => ! [B5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A5),B5)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or3652927894154168847AtMost(nat,A5,B5)))),aa(nat,real,Uua,A5)) ) ) ) ) ).

% ATP.lambda_693
tff(fact_8809_ATP_Olambda__694,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_zl(fun(A,A),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ? [N4: nat] : ( Uub = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N4),Uu),Uua) ) ) ).

% ATP.lambda_694
tff(fact_8810_ATP_Olambda__695,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aek(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [X4: A,Y5: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Y5) )
              & member(A,X4,real_Vector_span(A,Uu))
              & member(A,Y5,real_Vector_span(A,Uua)) ) ) ) ).

% ATP.lambda_695
tff(fact_8811_ATP_Olambda__696,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_agf(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [X4: A,Y5: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Y5) )
              & member(A,X4,Uu)
              & member(A,Y5,Uua) ) ) ) ).

% ATP.lambda_696
tff(fact_8812_ATP_Olambda__697,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat] : ( aa(nat,A,aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_afl(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),aTP_Lamp_afk(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uub),Uuc)),set_ord_atMost(nat,Uuc)) ) ).

% ATP.lambda_697
tff(fact_8813_ATP_Olambda__698,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat] : ( aa(nat,A,aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_afs(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),Uub,Uuc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uuc)) ) ).

% ATP.lambda_698
tff(fact_8814_ATP_Olambda__699,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat] : ( aa(nat,A,aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_afx(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),Uub,Uuc)),set_ord_lessThan(nat,Uuc)) ) ).

% ATP.lambda_699
tff(fact_8815_ATP_Olambda__700,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_gq(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
              ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)
              & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uuc) ),
              aa(A,A,aa(A,fun(A,A),times_times(A),real_V8093663219630862766scaleR(A,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))),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_700
tff(fact_8816_ATP_Olambda__701,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_gm(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
              ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)
              & ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uuc) ),
              aa(A,A,aa(A,fun(A,A),times_times(A),real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(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))),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_701
tff(fact_8817_ATP_Olambda__702,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_go(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),real_V8093663219630862766scaleR(A,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))),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_702
tff(fact_8818_ATP_Olambda__703,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_fv(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_703
tff(fact_8819_ATP_Olambda__704,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_fx(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_704
tff(fact_8820_ATP_Olambda__705,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_fy(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_705
tff(fact_8821_ATP_Olambda__706,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_fw(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_706
tff(fact_8822_ATP_Olambda__707,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
      ( aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_afn(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_707
tff(fact_8823_ATP_Olambda__708,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_jc(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_708
tff(fact_8824_ATP_Olambda__709,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_je(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_709
tff(fact_8825_ATP_Olambda__710,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: B,Uuc: A] :
          ( aa(A,B,aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_jf(A,fun(fun(A,B),fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),Uub) ) ) ).

% ATP.lambda_710
tff(fact_8826_ATP_Olambda__711,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: B,Uub: B,Uuc: A] :
      ( aa(A,B,aa(B,fun(A,B),aa(B,fun(B,fun(A,B)),aTP_Lamp_acn(set(A),fun(B,fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uuc,Uu),Uua,Uub) ) ).

% ATP.lambda_711
tff(fact_8827_ATP_Olambda__712,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_afv(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_712
tff(fact_8828_ATP_Olambda__713,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_lz(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_713
tff(fact_8829_ATP_Olambda__714,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_ly(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_714
tff(fact_8830_ATP_Olambda__715,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: list(B),Uuc: nat] : ( aa(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_kt(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uuc),aa(A,fun(A,A),times_times(A),Uua)),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))) ) ) ).

% ATP.lambda_715
tff(fact_8831_ATP_Olambda__716,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(real,A),Uub: fun(real,B),Uuc: real] : ( aa(real,C,aa(fun(real,B),fun(real,C),aa(fun(real,A),fun(fun(real,B),fun(real,C)),aTP_Lamp_aef(fun(A,fun(B,C)),fun(fun(real,A),fun(fun(real,B),fun(real,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,aa(real,A,Uua,Uuc)),aa(real,B,Uub,Uuc)) ) ) ).

% ATP.lambda_716
tff(fact_8832_ATP_Olambda__717,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: fun(D,B),Uuc: D] : ( aa(D,C,aa(fun(D,B),fun(D,C),aa(fun(D,A),fun(fun(D,B),fun(D,C)),aTP_Lamp_acz(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,B),fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,Uuc)),aa(D,B,Uub,Uuc)) ) ) ).

% ATP.lambda_717
tff(fact_8833_ATP_Olambda__718,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: fun(D,B),Uuc: D] : ( aa(D,C,aa(fun(D,B),fun(D,C),aa(fun(D,A),fun(fun(D,B),fun(D,C)),aTP_Lamp_acw(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,B),fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,Uuc)),aa(D,B,Uub,Uuc)) ) ) ).

% ATP.lambda_718
tff(fact_8834_ATP_Olambda__719,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: B,Uuc: D] : ( aa(D,C,aa(B,fun(D,C),aa(fun(D,A),fun(B,fun(D,C)),aTP_Lamp_acx(fun(A,fun(B,C)),fun(fun(D,A),fun(B,fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,Uuc)),Uub) ) ) ).

% ATP.lambda_719
tff(fact_8835_ATP_Olambda__720,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,B),Uub: A,Uuc: D] : ( aa(D,C,aa(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_acy(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,Uub),aa(D,B,Uua,Uuc)) ) ) ).

% ATP.lambda_720
tff(fact_8836_ATP_Olambda__721,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_em(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_el(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_721
tff(fact_8837_ATP_Olambda__722,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_gb(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_722
tff(fact_8838_ATP_Olambda__723,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B,Uuc: B] :
          ( aa(B,A,aa(B,fun(B,A),aa(fun(B,A),fun(B,fun(B,A)),aTP_Lamp_aev(set(B),fun(fun(B,A),fun(B,fun(B,A))),Uu),Uua),Uub),Uuc) = real_V8093663219630862766scaleR(A,real_V7696804695334737415tation(B,real_V4986007116245087402_basis(B,Uu),Uub,Uuc),
              $ite(member(B,Uuc,Uu),aa(B,A,Uua,Uuc),zero_zero(A))) ) ) ).

% ATP.lambda_723
tff(fact_8839_ATP_Olambda__724,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_fu(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,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_ft(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_724
tff(fact_8840_ATP_Olambda__725,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_fo(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_fn(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_725
tff(fact_8841_ATP_Olambda__726,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_qj(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),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_726
tff(fact_8842_ATP_Olambda__727,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_qh(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),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_727
tff(fact_8843_ATP_Olambda__728,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_qf(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),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_728
tff(fact_8844_ATP_Olambda__729,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_qg(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),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_729
tff(fact_8845_ATP_Olambda__730,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_eb(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_730
tff(fact_8846_ATP_Olambda__731,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_ky(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uub))) ) ).

% ATP.lambda_731
tff(fact_8847_ATP_Olambda__732,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_el(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_732
tff(fact_8848_ATP_Olambda__733,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_we(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = 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_733
tff(fact_8849_ATP_Olambda__734,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_acs(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
          & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Uub),Uuc),Uua) ) ) ) ).

% ATP.lambda_734
tff(fact_8850_ATP_Olambda__735,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_fr(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_735
tff(fact_8851_ATP_Olambda__736,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_fm(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_736
tff(fact_8852_ATP_Olambda__737,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_fq(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_737
tff(fact_8853_ATP_Olambda__738,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_gx(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(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))),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_738
tff(fact_8854_ATP_Olambda__739,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_ed(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_739
tff(fact_8855_ATP_Olambda__740,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_ec(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_740
tff(fact_8856_ATP_Olambda__741,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_afc(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_741
tff(fact_8857_ATP_Olambda__742,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_mj(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_742
tff(fact_8858_ATP_Olambda__743,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_ee(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_743
tff(fact_8859_ATP_Olambda__744,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_fg(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_744
tff(fact_8860_ATP_Olambda__745,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_le(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_745
tff(fact_8861_ATP_Olambda__746,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_kn(int,fun(int,fun(int,fun(int,$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_746
tff(fact_8862_ATP_Olambda__747,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_lc(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_747
tff(fact_8863_ATP_Olambda__748,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_lg(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)) ) ).

% ATP.lambda_748
tff(fact_8864_ATP_Olambda__749,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_li(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ) ).

% ATP.lambda_749
tff(fact_8865_ATP_Olambda__750,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_ic(A,fun(B,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_750
tff(fact_8866_ATP_Olambda__751,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_aam(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua) ) ) ).

% ATP.lambda_751
tff(fact_8867_ATP_Olambda__752,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_ir(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_752
tff(fact_8868_ATP_Olambda__753,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_ip(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_753
tff(fact_8869_ATP_Olambda__754,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_wm(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = real_V8093663219630862766scaleR(B,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_754
tff(fact_8870_ATP_Olambda__755,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_ca(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_755
tff(fact_8871_ATP_Olambda__756,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_ju(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_756
tff(fact_8872_ATP_Olambda__757,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_so(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),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_757
tff(fact_8873_ATP_Olambda__758,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_si(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)),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_758
tff(fact_8874_ATP_Olambda__759,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_gj(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_759
tff(fact_8875_ATP_Olambda__760,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_nh(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_760
tff(fact_8876_ATP_Olambda__761,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_ft(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_761
tff(fact_8877_ATP_Olambda__762,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_fk(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_762
tff(fact_8878_ATP_Olambda__763,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_fn(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_763
tff(fact_8879_ATP_Olambda__764,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,nat),Uua: fun(B,nat),Uub: A,Uuc: B] : ( aa(B,nat,aa(A,fun(B,nat),aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_abm(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,Uu,Uub)),aa(B,nat,Uua,Uuc)) ) ).

% ATP.lambda_764
tff(fact_8880_ATP_Olambda__765,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : ( aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_zo(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ) ).

% ATP.lambda_765
tff(fact_8881_ATP_Olambda__766,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : ( aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_ow(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ) ).

% ATP.lambda_766
tff(fact_8882_ATP_Olambda__767,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_rg(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_767
tff(fact_8883_ATP_Olambda__768,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_sk(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_768
tff(fact_8884_ATP_Olambda__769,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_sa(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_769
tff(fact_8885_ATP_Olambda__770,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_rc(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_770
tff(fact_8886_ATP_Olambda__771,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_rw(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,uminus_uminus(real),sin(real,aa(A,real,Uu,Uub)))) ) ) ).

% ATP.lambda_771
tff(fact_8887_ATP_Olambda__772,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_re(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_772
tff(fact_8888_ATP_Olambda__773,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_xl(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = 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_773
tff(fact_8889_ATP_Olambda__774,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_xh(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = 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_774
tff(fact_8890_ATP_Olambda__775,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_xe(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(A,C,Uua,Uuc))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uuc))),Uub)) ) ) ).

% ATP.lambda_775
tff(fact_8891_ATP_Olambda__776,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_xo(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),Uu),Uua),Uub),Uuc) = 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_xn(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_xn(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_xn(A,A)))))) ) ) ).

% ATP.lambda_776
tff(fact_8892_ATP_Olambda__777,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_xi(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = 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_777
tff(fact_8893_ATP_Olambda__778,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_xj(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = 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_778
tff(fact_8894_ATP_Olambda__779,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : ( aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_mk(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(A),nat,finite_card(A),collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_mj(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ) ).

% ATP.lambda_779
tff(fact_8895_ATP_Olambda__780,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_xm(fun(nat,A),fun(nat,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uub),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_bx(fun(nat,A),fun(A,fun(nat,A)),Uu),Uuc)),set_ord_atMost(nat,Uua)))) ) ) ).

% ATP.lambda_780
tff(fact_8896_ATP_Olambda__781,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_agi(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_781
tff(fact_8897_ATP_Olambda__782,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_jl(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_782
tff(fact_8898_ATP_Olambda__783,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_jj(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_783
tff(fact_8899_ATP_Olambda__784,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_afr(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_784
tff(fact_8900_ATP_Olambda__785,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_ay(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_785
tff(fact_8901_ATP_Olambda__786,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_ct(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_786
tff(fact_8902_ATP_Olambda__787,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_afp(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_787
tff(fact_8903_ATP_Olambda__788,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : ( aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_ox(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Sup_Sup(A),image(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_ow(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc),Uub)) ) ) ).

% ATP.lambda_788
tff(fact_8904_ATP_Olambda__789,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : ( aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_zp(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Inf_Inf(A),image(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_zo(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc),Uub)) ) ) ).

% ATP.lambda_789
tff(fact_8905_ATP_Olambda__790,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_kk(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(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_790
tff(fact_8906_ATP_Olambda__791,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_ki(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(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_791
tff(fact_8907_ATP_Olambda__792,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_rt(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_792
tff(fact_8908_ATP_Olambda__793,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat,Uud: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A))),aTP_Lamp_afu(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),aTP_Lamp_aft(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uub),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uuc)) ) ).

% ATP.lambda_793
tff(fact_8909_ATP_Olambda__794,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,A),Uuc: nat,Uud: nat] : ( aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,A),fun(nat,fun(nat,A))),aTP_Lamp_afw(fun(A,fun(A,A)),fun(A,fun(fun(nat,A),fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),Uub),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uud),Uuc),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uud),Uuc)),Uuc))) ) ).

% ATP.lambda_794
tff(fact_8910_ATP_Olambda__795,axiom,
    ! [A: $tType,Uu: A,Uua: nat,Uub: fun(nat,A),Uuc: fun(nat,A),Uud: nat] :
      ( aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aa(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),aTP_Lamp_afm(A,fun(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uud),Uua),
          aa(nat,A,Uub,Uud),
          $ite(Uud = Uua,Uu,aa(nat,A,Uuc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uud),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_795
tff(fact_8911_ATP_Olambda__796,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_gg(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aa(A,fun(A,fun(nat,fun(nat,A))),aTP_Lamp_gf(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_796
tff(fact_8912_ATP_Olambda__797,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_qi(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,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_qh(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),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_797
tff(fact_8913_ATP_Olambda__798,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_gk(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,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gj(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_798
tff(fact_8914_ATP_Olambda__799,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_ge(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_799
tff(fact_8915_ATP_Olambda__800,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_fz(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_800
tff(fact_8916_ATP_Olambda__801,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_ga(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_801
tff(fact_8917_ATP_Olambda__802,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_gf(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_802
tff(fact_8918_ATP_Olambda__803,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_ry(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_803
tff(fact_8919_ATP_Olambda__804,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(B,A),Uua: B,Uub: fun(B,A),Uuc: int,Uud: B] : ( aa(B,A,aa(int,fun(B,A),aa(fun(B,A),fun(int,fun(B,A)),aa(B,fun(fun(B,A),fun(int,fun(B,A))),aTP_Lamp_sg(fun(B,A),fun(B,fun(fun(B,A),fun(int,fun(B,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uub,Uud)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Uuc)),power_int(A,aa(B,A,Uu,Uua),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uuc),one_one(int))))) ) ) ).

% ATP.lambda_804
tff(fact_8920_ATP_Olambda__805,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_rr(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) = 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_805
tff(fact_8921_ATP_Olambda__806,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_ri(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),real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub),aa(A,B,Uud,Uue))),real_V8093663219630862766scaleR(B,aa(A,real,Uua,Uue),aa(A,B,Uuc,Uub))) ) ) ).

% ATP.lambda_806
tff(fact_8922_ATP_Olambda__807,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_rp(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_807
tff(fact_8923_ATP_Olambda__808,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_sc(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))))),divide_divide(B,aa(A,B,Uua,Uue),aa(A,B,Uuc,Uub))) ) ) ).

% ATP.lambda_808
tff(fact_8924_ATP_Olambda__809,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT] : ( aa(vEBT_VEBT,A,aa(list(vEBT_VEBT),fun(vEBT_VEBT,A),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A))),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),aTP_Lamp_agh(fun(B,A),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,A,Uu,aa(vEBT_VEBT,B,aa(list(vEBT_VEBT),fun(vEBT_VEBT,B),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B))),Uua,Uub),Uuc),Uud),Uue)) ) ).

% ATP.lambda_809
tff(fact_8925_ATP_Olambda__810,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: fun(D,A),Uuc: D,Uud: fun(D,B),Uue: fun(D,B),Uuf: D] : ( aa(D,C,aa(fun(D,B),fun(D,C),aa(fun(D,B),fun(fun(D,B),fun(D,C)),aa(D,fun(fun(D,B),fun(fun(D,B),fun(D,C))),aa(fun(D,A),fun(D,fun(fun(D,B),fun(fun(D,B),fun(D,C)))),aa(fun(D,A),fun(fun(D,A),fun(D,fun(fun(D,B),fun(fun(D,B),fun(D,C))))),aTP_Lamp_ada(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,A),fun(D,fun(fun(D,B),fun(fun(D,B),fun(D,C)))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,Uuc)),aa(D,B,Uue,Uuf))),aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uub,Uuf)),aa(D,B,Uud,Uuc))) ) ) ).

% ATP.lambda_810
tff(fact_8926_ATP_Olambda__811,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ng(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uue),Uuf)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuf),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu))
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uud))
           => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),I3)),X_13)
            <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Uub),I3) ) )
        & $ite(
            Uue = Uuf,
            ! [X4: vEBT_VEBT] :
              ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Uua))
             => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X7) ),
            ( vEBT_V5917875025757280293ildren(Uuc,Uua,Uuf)
            & ! [X4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu))
               => ( vEBT_V5917875025757280293ildren(Uuc,Uua,X4)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uue),X4)
                    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_811
tff(fact_8927_ATP_Olambda__812,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : ( aa(A,set(B),aTP_Lamp_abn(set(B),fun(A,set(B)),Uu),Uua) = Uu ) ).

% ATP.lambda_812
tff(fact_8928_ATP_Olambda__813,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : ( aa(list(A),set(A),aTP_Lamp_acr(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ) ).

% ATP.lambda_813
tff(fact_8929_ATP_Olambda__814,axiom,
    ! [Uu: rat,Uua: nat] : ( aa(nat,rat,aTP_Lamp_mt(rat,fun(nat,rat),Uu),Uua) = Uu ) ).

% ATP.lambda_814
tff(fact_8930_ATP_Olambda__815,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: B,Uua: A] : ( aa(A,B,aTP_Lamp_rj(B,fun(A,B),Uu),Uua) = Uu ) ) ).

% ATP.lambda_815
tff(fact_8931_ATP_Olambda__816,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_pt(A,fun(nat,A),Uu),Uua) = Uu ) ) ).

% ATP.lambda_816
tff(fact_8932_ATP_Olambda__817,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: real] : ( aa(real,A,aTP_Lamp_ady(A,fun(real,A),Uu),Uua) = Uu ) ) ).

% ATP.lambda_817
tff(fact_8933_ATP_Olambda__818,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_br(A,fun(nat,A),Uu),Uua) = Uu ) ) ).

% ATP.lambda_818
tff(fact_8934_ATP_Olambda__819,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : ( aa(A,A,aTP_Lamp_qp(A,fun(A,A),Uu),Uua) = Uu ) ) ).

% ATP.lambda_819
tff(fact_8935_ATP_Olambda__820,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A,Uua: B] : ( aa(B,A,aTP_Lamp_ci(A,fun(B,A),Uu),Uua) = Uu ) ) ).

% ATP.lambda_820
tff(fact_8936_ATP_Olambda__821,axiom,
    ! [B: $tType,A: $tType] :
      ( ( zero(A)
        & topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu: A,Uua: B] : ( aa(B,A,aTP_Lamp_ua(A,fun(B,A),Uu),Uua) = Uu ) ) ).

% ATP.lambda_821
tff(fact_8937_ATP_Olambda__822,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : ( aa(nat,A,aTP_Lamp_nl(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_822
tff(fact_8938_ATP_Olambda__823,axiom,
    ! [Uu: complex] : ( aa(complex,complex,aTP_Lamp_dp(complex,complex),Uu) = Uu ) ).

% ATP.lambda_823
tff(fact_8939_ATP_Olambda__824,axiom,
    ! [Uu: real] : ( aa(real,real,aTP_Lamp_adz(real,real),Uu) = Uu ) ).

% ATP.lambda_824
tff(fact_8940_ATP_Olambda__825,axiom,
    ! [Uu: nat] : ( aa(nat,nat,aTP_Lamp_az(nat,nat),Uu) = Uu ) ).

% ATP.lambda_825
tff(fact_8941_ATP_Olambda__826,axiom,
    ! [Uu: int] : ( aa(int,int,aTP_Lamp_bj(int,int),Uu) = Uu ) ).

% ATP.lambda_826
tff(fact_8942_ATP_Olambda__827,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A] : ( aa(A,A,aTP_Lamp_xn(A,A),Uu) = Uu ) ) ).

% ATP.lambda_827
tff(fact_8943_ATP_Olambda__828,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A] : ( aa(A,A,aTP_Lamp_qq(A,A),Uu) = Uu ) ) ).

% ATP.lambda_828
tff(fact_8944_ATP_Olambda__829,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A] : ( aa(A,A,aTP_Lamp_xq(A,A),Uu) = Uu ) ) ).

% ATP.lambda_829
tff(fact_8945_ATP_Olambda__830,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : ( aa(A,A,aTP_Lamp_xw(A,A),Uu) = Uu ) ) ).

% ATP.lambda_830
tff(fact_8946_ATP_Olambda__831,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: A] : ( aa(A,A,aTP_Lamp_mx(A,A),Uu) = Uu ) ) ).

% ATP.lambda_831
tff(fact_8947_ATP_Olambda__832,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : ( aa(A,A,aTP_Lamp_os(A,A),Uu) = Uu ) ) ).

% ATP.lambda_832
tff(fact_8948_ATP_Olambda__833,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : ( aa(A,A,aTP_Lamp_ag(A,A),Uu) = Uu ) ) ).

% ATP.lambda_833
tff(fact_8949_ATP_Olambda__834,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: A] : ( aa(A,A,aTP_Lamp_afj(A,A),Uu) = Uu ) ) ).

% ATP.lambda_834
tff(fact_8950_ATP_Olambda__835,axiom,
    ! [A: $tType,Uu: A] : ( aa(A,A,aTP_Lamp_afb(A,A),Uu) = Uu ) ).

% ATP.lambda_835
tff(fact_8951_ATP_Olambda__836,axiom,
    ! [A: $tType,B: $tType,Uu: A] : ( aa(A,set(B),aTP_Lamp_abt(A,set(B)),Uu) = top_top(set(B)) ) ).

% ATP.lambda_836
tff(fact_8952_ATP_Olambda__837,axiom,
    ! [Uu: nat] : ( aa(nat,rat,aTP_Lamp_aac(nat,rat),Uu) = zero_zero(rat) ) ).

% ATP.lambda_837
tff(fact_8953_ATP_Olambda__838,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat] : ( aa(nat,A,aTP_Lamp_bl(nat,A),Uu) = zero_zero(A) ) ) ).

% ATP.lambda_838
tff(fact_8954_ATP_Olambda__839,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: nat] : ( aa(nat,A,aTP_Lamp_bg(nat,A),Uu) = zero_zero(A) ) ) ).

% ATP.lambda_839
tff(fact_8955_ATP_Olambda__840,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : ( aa(B,A,aTP_Lamp_ce(B,A),Uu) = zero_zero(A) ) ) ).

% ATP.lambda_840
tff(fact_8956_ATP_Olambda__841,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: B] : ( aa(B,A,aTP_Lamp_nd(B,A),Uu) = zero_zero(A) ) ) ).

% ATP.lambda_841
tff(fact_8957_ATP_Olambda__842,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: A] : ( aa(A,B,aTP_Lamp_rk(A,B),Uu) = zero_zero(B) ) ) ).

% ATP.lambda_842
tff(fact_8958_ATP_Olambda__843,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [Uu: A] : ( aa(A,B,aTP_Lamp_aep(A,B),Uu) = zero_zero(B) ) ) ).

% ATP.lambda_843
tff(fact_8959_ATP_Olambda__844,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : ( aa(A,A,aTP_Lamp_ac(A,A),Uu) = zero_zero(A) ) ) ).

% ATP.lambda_844
tff(fact_8960_ATP_Olambda__845,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: A] : ( aa(A,B,aTP_Lamp_agb(A,B),Uu) = zero_zero(B) ) ) ).

% ATP.lambda_845
tff(fact_8961_ATP_Olambda__846,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : ( aa(A,B,aTP_Lamp_aex(A,B),Uu) = zero_zero(B) ) ) ).

% ATP.lambda_846
tff(fact_8962_ATP_Olambda__847,axiom,
    ! [Uu: nat] : ( aa(nat,rat,aTP_Lamp_aad(nat,rat),Uu) = one_one(rat) ) ).

% ATP.lambda_847
tff(fact_8963_ATP_Olambda__848,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B] : ( aa(B,A,aTP_Lamp_bh(B,A),Uu) = one_one(A) ) ) ).

% ATP.lambda_848
tff(fact_8964_ATP_Olambda__849,axiom,
    ! [A: $tType,Uu: A] : ( aa(A,real,aTP_Lamp_do(A,real),Uu) = one_one(real) ) ).

% ATP.lambda_849
tff(fact_8965_ATP_Olambda__850,axiom,
    ! [A: $tType,Uu: A] : ( aa(A,nat,aTP_Lamp_dm(A,nat),Uu) = one_one(nat) ) ).

% ATP.lambda_850
tff(fact_8966_ATP_Olambda__851,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_kb(nat,$o),Uu)
    <=> $false ) ).

% ATP.lambda_851
tff(fact_8967_ATP_Olambda__852,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_kc(nat,$o),Uu)
    <=> $true ) ).

% ATP.lambda_852
tff(fact_8968_ATP_Olambda__853,axiom,
    ! [A: $tType,Uu: A] : ( aa(A,fun(nat,nat),aTP_Lamp_adk(A,fun(nat,nat)),Uu) = suc ) ).

% ATP.lambda_853

% Type constructors (893)
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_1,axiom,
    bounded_lattice(extended_enat) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_2,axiom,
    ! [A13: $tType] : bounded_lattice(filter(A13)) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_3,axiom,
    bounded_lattice($o) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_4,axiom,
    ! [A13: $tType] : bounded_lattice(set(A13)) ).

tff(tcon_fun___Lattices_Obounded__lattice_5,axiom,
    ! [A13: $tType,A14: $tType] :
      ( bounded_lattice(A14)
     => bounded_lattice(fun(A13,A14)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A13: $tType,A14: $tType] :
      ( comple6319245703460814977attice(A14)
     => condit1219197933456340205attice(fun(A13,A14)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
    ! [A13: $tType,A14: $tType] :
      ( counta3822494911875563373attice(A14)
     => counta3822494911875563373attice(fun(A13,A14)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A13: $tType,A14: $tType] :
      ( comple592849572758109894attice(A14)
     => comple592849572758109894attice(fun(A13,A14)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
    ! [A13: $tType,A14: $tType] :
      ( comple489889107523837845lgebra(A14)
     => comple489889107523837845lgebra(fun(A13,A14)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A13: $tType,A14: $tType] :
      ( bounded_lattice(A14)
     => bounde4967611905675639751up_bot(fun(A13,A14)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A13: $tType,A14: $tType] :
      ( bounded_lattice(A14)
     => bounde4346867609351753570nf_top(fun(A13,A14)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A13: $tType,A14: $tType] :
      ( comple6319245703460814977attice(A14)
     => comple6319245703460814977attice(fun(A13,A14)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A13: $tType,A14: $tType] :
      ( boolea8198339166811842893lgebra(A14)
     => boolea8198339166811842893lgebra(fun(A13,A14)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__top,axiom,
    ! [A13: $tType,A14: $tType] :
      ( bounded_lattice(A14)
     => bounded_lattice_top(fun(A13,A14)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
    ! [A13: $tType,A14: $tType] :
      ( bounded_lattice(A14)
     => bounded_lattice_bot(fun(A13,A14)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A13: $tType,A14: $tType] :
      ( semilattice_sup(A14)
     => semilattice_sup(fun(A13,A14)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A13: $tType,A14: $tType] :
      ( semilattice_inf(A14)
     => semilattice_inf(fun(A13,A14)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A13: $tType,A14: $tType] :
      ( distrib_lattice(A14)
     => distrib_lattice(fun(A13,A14)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A13: $tType,A14: $tType] :
      ( order_top(A14)
     => order_top(fun(A13,A14)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A13: $tType,A14: $tType] :
      ( order_bot(A14)
     => order_bot(fun(A13,A14)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A13: $tType,A14: $tType] :
      ( preorder(A14)
     => preorder(fun(A13,A14)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A13: $tType,A14: $tType] :
      ( lattice(A14)
     => lattice(fun(A13,A14)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A13: $tType,A14: $tType] :
      ( order(A14)
     => order(fun(A13,A14)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A13: $tType,A14: $tType] :
      ( ord(A14)
     => ord(fun(A13,A14)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A13: $tType,A14: $tType] :
      ( uminus(A14)
     => uminus(fun(A13,A14)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A13: $tType,A14: $tType] :
      ( minus(A14)
     => minus(fun(A13,A14)) ) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_6,axiom,
    condit1219197933456340205attice(int) ).

tff(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
    bit_un5681908812861735899ations(int) ).

tff(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
    semiri1453513574482234551roduct(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
    euclid5411537665997757685th_nat(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
    euclid8789492081693882211th_nat(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
    euclid3128863361964157862miring(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
    euclid4440199948858584721cancel(int) ).

tff(tcon_Int_Oint___Rings_Onormalization__semidom__multiplicative,axiom,
    normal6328177297339901930cative(int) ).

tff(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
    unique1627219031080169319umeral(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
    euclid8851590272496341667cancel(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
    semiri6575147826004484403cancel(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
    strict9044650504122735259up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere580206878836729694up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere2412721322843649153imp_le(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
    bit_se359711467146920520ations(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
    linord2810124833399127020strict(int) ).

tff(tcon_Int_Oint___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___Limits_Otopological__comm__monoid__mult,axiom,
    topolo4987421752381908075d_mult(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_Osemidom__divide__unit__factor,axiom,
    semido2269285787275462019factor(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___GCD_Osemiring__gcd__mult__normalize,axiom,
    semiri6843258321239162965malize(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___Rings_Oordered__comm__semiring,axiom,
    ordere2520102378445227354miring(int) ).

tff(tcon_Int_Oint___Rings_Onormalization__semidom,axiom,
    normal8620421768224518004emidom(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
    linord6961819062388156250ring_1(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
    ordered_ab_group_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
    cancel_semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
    linordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
    ordered_semiring_0(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
    linordered_semidom(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__sup_7,axiom,
    semilattice_sup(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__inf_8,axiom,
    semilattice_inf(int) ).

tff(tcon_Int_Oint___Lattices_Odistrib__lattice_9,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_10,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_11,axiom,
    lattice(int) ).

tff(tcon_Int_Oint___Groups_Ogroup__add,axiom,
    group_add(int) ).

tff(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
    semiring_gcd(int) ).

tff(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
    semiring_Gcd(int) ).

tff(tcon_Int_Oint___Rings_Omult__zero,axiom,
    mult_zero(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
    comm_ring(int) ).

tff(tcon_Int_Oint___Orderings_Oorder_12,axiom,
    order(int) ).

tff(tcon_Int_Oint___Num_Oneg__numeral,axiom,
    neg_numeral(int) ).

tff(tcon_Int_Oint___Nat_Oring__char__0,axiom,
    ring_char_0(int) ).

tff(tcon_Int_Oint___Rings_Osemiring,axiom,
    semiring(int) ).

tff(tcon_Int_Oint___Rings_Osemidom,axiom,
    semidom(int) ).

tff(tcon_Int_Oint___Orderings_Oord_13,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_14,axiom,
    uminus(int) ).

tff(tcon_Int_Oint___Rings_Oring__1,axiom,
    ring_1(int) ).

tff(tcon_Int_Oint___Rings_Oabs__if,axiom,
    abs_if(int) ).

tff(tcon_Int_Oint___Groups_Ominus_15,axiom,
    minus(int) ).

tff(tcon_Int_Oint___GCD_Oring__gcd,axiom,
    ring_gcd(int) ).

tff(tcon_Int_Oint___Power_Opower,axiom,
    power(int) ).

tff(tcon_Int_Oint___Num_Onumeral,axiom,
    numeral(int) ).

tff(tcon_Int_Oint___Groups_Ozero,axiom,
    zero(int) ).

tff(tcon_Int_Oint___Groups_Oplus,axiom,
    plus(int) ).

tff(tcon_Int_Oint___Rings_Oring,axiom,
    ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom,axiom,
    idom(int) ).

tff(tcon_Int_Oint___Groups_Oone,axiom,
    one(int) ).

tff(tcon_Int_Oint___Rings_Odvd,axiom,
    dvd(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_16,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_17,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_18,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_19,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_20,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_21,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_22,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_23,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_24,axiom,
    normal6328177297339901930cative(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_25,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_26,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_27,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_28,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_29,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_30,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_31,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_32,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_33,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_34,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_35,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_36,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_37,axiom,
    topolo4987421752381908075d_mult(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_38,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_39,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_40,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_41,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_42,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_43,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_44,axiom,
    semido2269285787275462019factor(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_45,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_46,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_47,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_48,axiom,
    semiri6843258321239162965malize(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__mult_49,axiom,
    topolo1898628316856586783d_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_50,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_51,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_52,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_53,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_54,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_55,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_56,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_57,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Rings_Onormalization__semidom_58,axiom,
    normal8620421768224518004emidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_59,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_60,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_61,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_62,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_63,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_64,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_65,axiom,
    distrib_lattice(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_66,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_67,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_68,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_69,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_70,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_71,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_72,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_73,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_74,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_75,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_76,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_77,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_78,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_79,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_80,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_81,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_82,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_83,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_84,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_85,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_86,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_87,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_88,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_89,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_90,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_91,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_92,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_93,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_94,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_95,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_96,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_97,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_98,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_99,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom_100,axiom,
    semidom(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_101,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Groups_Ominus_102,axiom,
    minus(nat) ).

tff(tcon_Nat_Onat___Power_Opower_103,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_104,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_105,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_106,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_107,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_108,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_109,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_110,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_111,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_112,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_113,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_114,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_115,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_116,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_117,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_118,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_119,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_120,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_121,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_122,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_123,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_124,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_125,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_126,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_127,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_128,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_129,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_130,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_131,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_132,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_133,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_134,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_135,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_136,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_137,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_138,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_139,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_140,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_141,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_142,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_143,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_144,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_145,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_146,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_147,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_148,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_149,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_150,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_151,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_152,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_153,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_154,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_155,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_156,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_157,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_158,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_159,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_160,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_161,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_162,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_163,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_164,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_165,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_166,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_167,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_168,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_169,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_170,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_171,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_172,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_173,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_174,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_175,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_176,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_177,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_178,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_179,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_180,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_181,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_182,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_183,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_184,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_185,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_186,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_187,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_188,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom_189,axiom,
    semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_190,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_191,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_192,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_193,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Groups_Ominus_194,axiom,
    minus(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_195,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_196,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_197,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_198,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_199,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_200,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_201,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_202,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_203,axiom,
    ! [A13: $tType] : condit1219197933456340205attice(set(A13)) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_204,axiom,
    ! [A13: $tType] : counta3822494911875563373attice(set(A13)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_205,axiom,
    ! [A13: $tType] : comple592849572758109894attice(set(A13)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_206,axiom,
    ! [A13: $tType] : comple489889107523837845lgebra(set(A13)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_207,axiom,
    ! [A13: $tType] : bounde4967611905675639751up_bot(set(A13)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_208,axiom,
    ! [A13: $tType] : bounde4346867609351753570nf_top(set(A13)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_209,axiom,
    ! [A13: $tType] : comple6319245703460814977attice(set(A13)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_210,axiom,
    ! [A13: $tType] : boolea8198339166811842893lgebra(set(A13)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_211,axiom,
    ! [A13: $tType] : bounded_lattice_top(set(A13)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_212,axiom,
    ! [A13: $tType] : bounded_lattice_bot(set(A13)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_213,axiom,
    ! [A13: $tType] : semilattice_sup(set(A13)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_214,axiom,
    ! [A13: $tType] : semilattice_inf(set(A13)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_215,axiom,
    ! [A13: $tType] : distrib_lattice(set(A13)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_216,axiom,
    ! [A13: $tType] : order_top(set(A13)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_217,axiom,
    ! [A13: $tType] : order_bot(set(A13)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_218,axiom,
    ! [A13: $tType] : preorder(set(A13)) ).

tff(tcon_Set_Oset___Lattices_Olattice_219,axiom,
    ! [A13: $tType] : lattice(set(A13)) ).

tff(tcon_Set_Oset___Orderings_Oorder_220,axiom,
    ! [A13: $tType] : order(set(A13)) ).

tff(tcon_Set_Oset___Orderings_Oord_221,axiom,
    ! [A13: $tType] : ord(set(A13)) ).

tff(tcon_Set_Oset___Groups_Ouminus_222,axiom,
    ! [A13: $tType] : uminus(set(A13)) ).

tff(tcon_Set_Oset___Groups_Ominus_223,axiom,
    ! [A13: $tType] : minus(set(A13)) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_224,axiom,
    condit1219197933456340205attice($o) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_225,axiom,
    counta3822494911875563373attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_226,axiom,
    comple592849572758109894attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_227,axiom,
    comple489889107523837845lgebra($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_228,axiom,
    topolo4958980785337419405_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_229,axiom,
    topolo1944317154257567458pology($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_230,axiom,
    bounde4967611905675639751up_bot($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_231,axiom,
    bounde4346867609351753570nf_top($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_232,axiom,
    comple6319245703460814977attice($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_233,axiom,
    topolo2564578578187576103pology($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_234,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_235,axiom,
    bounded_lattice_top($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_236,axiom,
    bounded_lattice_bot($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_237,axiom,
    topological_t2_space($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_238,axiom,
    semilattice_sup($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_239,axiom,
    semilattice_inf($o) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_240,axiom,
    distrib_lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_241,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_242,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_243,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_244,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Lattices_Olattice_245,axiom,
    lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_246,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_247,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_248,axiom,
    uminus($o) ).

tff(tcon_HOL_Obool___Groups_Ominus_249,axiom,
    minus($o) ).

tff(tcon_List_Olist___Nat_Osize_250,axiom,
    ! [A13: $tType] : size(list(A13)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_251,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_252,axiom,
    condit1219197933456340205attice(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_253,axiom,
    semiri1453513574482234551roduct(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
    condit5016429287641298734tinuum(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_254,axiom,
    ordere1937475149494474687imp_le(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinear__continuum__topology,axiom,
    topolo8458572112393995274pology(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ofirst__countable__topology,axiom,
    topolo3112930676232923870pology(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__div__algebra,axiom,
    real_V8999393235501362500lgebra(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra__1,axiom,
    real_V2822296259951069270ebra_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_255,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_256,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_257,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_258,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_259,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_260,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_261,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_262,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_263,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_264,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_265,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_266,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_267,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_268,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_269,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_270,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_271,axiom,
    linord181362715937106298miring(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra__1,axiom,
    real_V2191834092415804123ebra_1(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space,axiom,
    real_V8037385150606011577_space(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__semigroup__mult_272,axiom,
    topolo4211221413907600880p_mult(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
    topolo8386298272705272623_space(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_273,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_274,axiom,
    semiri3467727345109120633visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Osgn__div__norm,axiom,
    real_V6567297691418259687v_norm(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_275,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_276,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_277,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_278,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_279,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_280,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_281,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_282,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_283,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_284,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_285,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Odist__norm,axiom,
    real_V6936659425649961206t_norm(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
    topolo1633459387980952147up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_286,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_287,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_288,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_289,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_290,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_291,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_292,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_293,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_294,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_295,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__inf_296,axiom,
    semilattice_inf(real) ).

tff(tcon_Real_Oreal___Lattices_Odistrib__lattice_297,axiom,
    distrib_lattice(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_298,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_299,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_300,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_301,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_302,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_303,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_304,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_305,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_306,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_307,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_308,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_309,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_310,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_311,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_312,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_313,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_314,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__abs__sgn_315,axiom,
    field_abs_sgn(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_316,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_317,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_318,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_319,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_320,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_321,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_322,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_323,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_324,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_325,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_326,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_327,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_328,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_329,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_330,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_331,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_332,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_333,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_334,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_335,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_336,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_337,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_338,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_339,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_340,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_341,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_342,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_343,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom_344,axiom,
    semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_345,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_346,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_347,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_348,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Groups_Ominus_349,axiom,
    minus(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_350,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_351,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_352,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_353,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_354,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_355,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_356,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_357,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_358,axiom,
    dvd(real) ).

tff(tcon_String_Ochar___Nat_Osize_359,axiom,
    size(char) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_360,axiom,
    ! [A13: $tType,A14: $tType] : size(sum_sum(A13,A14)) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_361,axiom,
    ! [A13: $tType] : condit1219197933456340205attice(filter(A13)) ).

tff(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_362,axiom,
    ! [A13: $tType] : counta3822494911875563373attice(filter(A13)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_363,axiom,
    ! [A13: $tType] : bounde4967611905675639751up_bot(filter(A13)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_364,axiom,
    ! [A13: $tType] : bounde4346867609351753570nf_top(filter(A13)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_365,axiom,
    ! [A13: $tType] : comple6319245703460814977attice(filter(A13)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_366,axiom,
    ! [A13: $tType] : bounded_lattice_top(filter(A13)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_367,axiom,
    ! [A13: $tType] : bounded_lattice_bot(filter(A13)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_368,axiom,
    ! [A13: $tType] : semilattice_sup(filter(A13)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_369,axiom,
    ! [A13: $tType] : semilattice_inf(filter(A13)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_370,axiom,
    ! [A13: $tType] : distrib_lattice(filter(A13)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_371,axiom,
    ! [A13: $tType] : order_top(filter(A13)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_372,axiom,
    ! [A13: $tType] : order_bot(filter(A13)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_373,axiom,
    ! [A13: $tType] : preorder(filter(A13)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_374,axiom,
    ! [A13: $tType] : lattice(filter(A13)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_375,axiom,
    ! [A13: $tType] : order(filter(A13)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_376,axiom,
    ! [A13: $tType] : ord(filter(A13)) ).

tff(tcon_Option_Ooption___Nat_Osize_377,axiom,
    ! [A13: $tType] : size(option(A13)) ).

tff(tcon_String_Oliteral___Groups_Osemigroup__add_378,axiom,
    semigroup_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Opreorder_379,axiom,
    preorder(literal) ).

tff(tcon_String_Oliteral___Orderings_Olinorder_380,axiom,
    linorder(literal) ).

tff(tcon_String_Oliteral___Groups_Omonoid__add_381,axiom,
    monoid_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Oorder_382,axiom,
    order(literal) ).

tff(tcon_String_Oliteral___Orderings_Oord_383,axiom,
    ord(literal) ).

tff(tcon_String_Oliteral___Groups_Ozero_384,axiom,
    zero(literal) ).

tff(tcon_String_Oliteral___Groups_Oplus_385,axiom,
    plus(literal) ).

tff(tcon_String_Oliteral___Nat_Osize_386,axiom,
    size(literal) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_387,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_388,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_389,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_390,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_391,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_392,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_393,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_394,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_395,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_396,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_397,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_398,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_399,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_400,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_401,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_402,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_403,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Osgn__div__norm_404,axiom,
    real_V6567297691418259687v_norm(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_405,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_406,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_407,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_408,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_409,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_410,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_411,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_412,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_413,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Odist__norm_414,axiom,
    real_V6936659425649961206t_norm(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__group__add_415,axiom,
    topolo1633459387980952147up_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_416,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_417,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_418,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_419,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_420,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_421,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_422,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_423,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_424,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_425,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_426,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_427,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_428,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_429,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_430,axiom,
    field_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_431,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_432,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_433,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_434,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_435,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_436,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_437,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_438,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_439,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_440,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_441,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_442,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_443,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_444,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_445,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_446,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_447,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_448,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_449,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_450,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom_451,axiom,
    semidom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_452,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_453,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ominus_454,axiom,
    minus(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_455,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_456,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_457,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_458,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_459,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_460,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_461,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_462,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_463,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_464,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_465,axiom,
    condit1219197933456340205attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_466,axiom,
    counta3822494911875563373attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_467,axiom,
    comple592849572758109894attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_468,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_469,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_470,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_471,axiom,
    bounde4967611905675639751up_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_472,axiom,
    bounde4346867609351753570nf_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder,axiom,
    comple5582772986160207858norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_473,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_474,axiom,
    comple6319245703460814977attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_475,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_476,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_477,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_478,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__top_479,axiom,
    bounded_lattice_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_480,axiom,
    bounded_lattice_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_481,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_482,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_483,axiom,
    semilattice_inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_484,axiom,
    distrib_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_485,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_486,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_487,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_488,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_489,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_490,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_491,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_492,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_493,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_494,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_495,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_496,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_497,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_498,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_499,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_500,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_501,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_502,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_503,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_504,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_505,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_506,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_507,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_508,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_509,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_510,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_511,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_512,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ominus_513,axiom,
    minus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_514,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_515,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_516,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_517,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_518,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_519,axiom,
    dvd(extended_enat) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_520,axiom,
    ! [A13: $tType,A14: $tType] :
      ( ( topolo4958980785337419405_space(A13)
        & topolo4958980785337419405_space(A14) )
     => topolo4958980785337419405_space(product_prod(A13,A14)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_521,axiom,
    ! [A13: $tType,A14: $tType] :
      ( ( topological_t2_space(A13)
        & topological_t2_space(A14) )
     => topological_t2_space(product_prod(A13,A14)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_522,axiom,
    ! [A13: $tType,A14: $tType] : size(product_prod(A13,A14)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_523,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_524,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_525,axiom,
    counta3822494911875563373attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_526,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_527,axiom,
    comple489889107523837845lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_528,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_529,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_530,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_531,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_532,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_533,axiom,
    bounded_lattice_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_534,axiom,
    bounded_lattice_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_535,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_536,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_537,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_538,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_539,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_540,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_541,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_542,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_543,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_544,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_545,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_546,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ominus_547,axiom,
    minus(product_unit) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_548,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_549,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_550,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_551,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_552,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_553,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_554,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_555,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_556,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_557,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_558,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_559,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_560,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_561,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_562,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_563,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_564,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_565,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_566,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_567,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_568,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_569,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_570,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_571,axiom,
    euclid5891614535332579305n_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_572,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_573,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_574,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_575,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_576,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_577,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_578,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_579,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_580,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_581,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_582,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_583,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_584,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_585,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_586,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_587,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_588,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_589,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_590,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_591,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_592,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_593,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_594,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_595,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_596,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_597,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_598,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_599,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_600,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_601,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_602,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_603,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_604,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_605,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_606,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_607,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_608,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_609,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_610,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_611,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_612,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_613,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_614,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_615,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_616,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_617,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_618,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_619,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_620,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_621,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_622,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_623,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_624,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_625,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_626,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_627,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_628,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_629,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_630,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_631,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_632,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_633,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_634,axiom,
    semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_635,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_636,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_637,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_638,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_639,axiom,
    minus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_640,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_641,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_642,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_643,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_644,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_645,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_646,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_647,axiom,
    dvd(code_integer) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_648,axiom,
    bit_un5681908812861735899ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_649,axiom,
    euclid5411537665997757685th_nat(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_650,axiom,
    ordere1937475149494474687imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_651,axiom,
    euclid3128863361964157862miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_652,axiom,
    euclid4440199948858584721cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_653,axiom,
    semiri6575147826004484403cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_654,axiom,
    strict9044650504122735259up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_655,axiom,
    ordere580206878836729694up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_656,axiom,
    ordere2412721322843649153imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_657,axiom,
    bit_se359711467146920520ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_658,axiom,
    linord2810124833399127020strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_659,axiom,
    strict7427464778891057005id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_660,axiom,
    ordere8940638589300402666id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_661,axiom,
    euclid3725896446679973847miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_662,axiom,
    linord4140545234300271783up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_663,axiom,
    semiri2026040879449505780visors(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_664,axiom,
    linord181362715937106298miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_665,axiom,
    linord8928482502909563296strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_666,axiom,
    semiri3467727345109120633visors(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_667,axiom,
    ordere6658533253407199908up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_668,axiom,
    ordere6911136660526730532id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_669,axiom,
    cancel2418104881723323429up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_670,axiom,
    cancel1802427076303600483id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_671,axiom,
    comm_s4317794764714335236cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_672,axiom,
    bit_semiring_bits(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_673,axiom,
    ordere2520102378445227354miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_674,axiom,
    cancel_semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_675,axiom,
    linordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_676,axiom,
    ordered_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_677,axiom,
    linordered_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_678,axiom,
    ab_semigroup_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_679,axiom,
    semiring_1_cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_680,axiom,
    algebraic_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_681,axiom,
    comm_monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_682,axiom,
    comm_monoid_diff(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_683,axiom,
    ab_semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_684,axiom,
    ordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_685,axiom,
    semiring_parity(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_686,axiom,
    comm_monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_687,axiom,
    semiring_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_688,axiom,
    comm_semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_689,axiom,
    comm_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_690,axiom,
    semigroup_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_691,axiom,
    semidom_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_692,axiom,
    semidom_divide(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_693,axiom,
    semiring_numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_694,axiom,
    semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_695,axiom,
    zero_less_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_696,axiom,
    comm_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_697,axiom,
    semiring_char_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_698,axiom,
    zero_neq_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Opreorder_699,axiom,
    preorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Olinorder_700,axiom,
    linorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_701,axiom,
    monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_702,axiom,
    monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_703,axiom,
    semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_704,axiom,
    semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Omult__zero_705,axiom,
    mult_zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oorder_706,axiom,
    order(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring_707,axiom,
    semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom_708,axiom,
    semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oord_709,axiom,
    ord(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ominus_710,axiom,
    minus(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Power_Opower_711,axiom,
    power(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Onumeral_712,axiom,
    numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ozero_713,axiom,
    zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oplus_714,axiom,
    plus(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oone_715,axiom,
    one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Odvd_716,axiom,
    dvd(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osize_717,axiom,
    size(code_natural) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_718,axiom,
    size(vEBT_VEBT) ).

% Helper facts (3)
tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X10: A,Y: A] :
      ( ( X10 != Y )
      | aa(A,$o,aa(A,fun(A,$o),fequal(A),X10),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X10: A,Y: A] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),fequal(A),X10),Y)
      | ( X10 = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
      = ( ? [X7: A] : aa(A,$o,P,X7) ) ) ).

% Free types (2)
tff(tfree_0,hypothesis,
    real_V4867850818363320053vector(a) ).

tff(tfree_1,hypothesis,
    semiring_1(a) ).

% Conjectures (1)
tff(conj_0,conjecture,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),m) ).

%------------------------------------------------------------------------------